Electroweak Breaking in Supersymmetric Models

arXiv:hep-ph/9204201v1 1 Apr 1992 CERN−TH.6412/92 Electroweak Breaking in Supersymmetric Models* Luis E. Ibáñez** CERN, 1211 Geneva 23, Switzerland and Graham G. Ross Dep. Theoretical Physics, Oxford University, England ABSTRACT We discuss the mechanism for electroweak symmetry breaking in supersymmetric versions of the standard model. After briefly reviewing the possible sources of supersymmetry breaking, we show how the required pattern of symmetry breaking can automatically result from the structure of quantum corrections in the theory. We demonstrate that this radiative breaking mechanism works well for a heavy top quark and can be combined in unified versions of the theory with excellent predictions for the running couplings of the model. * To appear in “Perspectives in Higgs Physics”, G. Kane editor. ** Adress after 1st October 1992: Departamento de Física Teórica C-XI, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain. CERN−TH.6412/92 −1− 1 Introduction The principle motivation which suggests the standard model should be extended to make it supersymmetric through the addition of new light supersymmetric states, accessible to particle accelerators, is the need to solve the hierarchy problem [1], [2], [3]. This is the problem of explaining why the W and Z-boson masses are so light in comparison with the unification or Planck scale despite the fact that radiative corrections associated with elementary Higgs scalar fields typically drive the electroweak breaking scale close to the largest mass scale in the theory. In this article we will describe how a low-energy supersymmetric extension of the standard model [4], [5] solves the hierarchy problem by limiting the magnitude of the radiative corrections. We will also discuss how the breaking of the electroweak symmetry appears as a consequence of the quantum corrections of the theory and show how this may easily be included in a unification scheme. In the standard model the hierarchy problem is due to the existence of elementary Higgs scalar fields. These fields have pointlike couplings associated with their gauge, gravitational and other interactions which lead to large radiative corrections to their mass and the expectation that they should have a mass comparable to the largest mass scale in the theory. As a result the electroweak breaking scale which is related to the Higgs mass is also expected to be unacceptably large. Two solutions to the hierarchy problem have been suggested. It could be that the Higgs (and other) fields may not be elementary and the pointlike interaction only applies up to the scale at which the composite structure appears. This has the effect of cutting off the radiative corrections at this composite scale and provided this is small enough the hierarchy problem is avoided. Examples of composite solutions to the hierarchy problem will be found elsewhere in this volume. The second possible solution is that the Higgs scalars are indeed elementary with pointlike interactions up to the Planck scale but that a symmetry protects the Higgs scalars from large radiative corrections to their masses. It has been shown that the only possible such symmetry is supersymmetry, hence the interest in low energy supersymmetric versions of the standard model [6], [7], [8]. Although it appeared that there was considerable freedom in building supersymmetric theories corresponding to the possibility of several supersymmetry generators, it was soon clear that the chiral nature of the standard −2− model probably only allows a supersymmetric extension with only a single (N=1) supersymmetry. ∗ Thus we turn now to a consideration of how to build such an N=1 extension of the standard model. 1.1 The supermultiplet content of the supersymmetric standard model In order to create a supersymmetric version of the standard model it is necessary to assign its states to N=1 supermultiplets. These consist of pairs of states differing in helicity, λ, by 1/2 . The gauge bosons of the standard model are assigned to gauge supermultiplets which contain a λ = 1, 1/2 helicity pair and thus to every gauge boson there is assigned a fermion partner called a gaugino. Since the supersymmetry generator commutes with the Yang Mills generators the gauge quantum numbers of the gaugino are the same as its gauge boson partner. The resulting set of gauge supermultiplets needed is given in Table 1. VECTOR MULTIPLETS CHIRAL MULTIPLETS J =1 J = 1/2 J = 1/2 J =0 g g̃ QL , ULc , DLc Q̃L , ŨLc , D̃Lc W ±, W 0 W̃ ± , W̃ 0 LL , ELc L̃L , ẼLc B B̃ H̃1 , H̃2 H1 , H2 Table 1 The matter fields of the standard model, the quarks and leptons, must also be assigned to supermultiplets. Because, in the standard model, the gauge quantum ∗ Matter must belong to vectorlike representations in N > 1 theories corresponding to the existence of mirror fermions. The difficulty in making these (so-far unobserved) states heavy has essentially killed attempts to build viable models with N > 1. −3− numbers of the gauge bosons and gauginos are different from the matter fields it is not possible to identify any of the latter with any of the gauginos. Thus we must introduce further supermultiplets and since we cannot allow additional gauge bosons without enlarging the gauge group we are forced to assign the matter states to chiral supermultiplets which contain a λ = 1/2, 0 helicity pair. Thus each quark and each lepton is assigned to a (left-handed) chiral supermultiplet with a scalar partner, a squark and a slepton respectively. These are also given in Table 1 and again the gauge quantum numbers of the scalar states are identical to those of their fermion partners. Finally it is necessary to assign the Higgs scalars of the standard model to a supermultiplet. Although there are colour singlet, electroweak doublet slepton states that appear to have the correct quantum numbers to be identified with the Higgs doublet, this proves to be unsuitable. The reason is that the constraints of supersymmetry on the Yukawa couplings of the theory prevent one from coupling the matter fields in the chiral supermultiplets of Table 1 to the conjugate scalar fields of the chiral supermultiplets, only coupling to non- conjugate fields is allowed. As a result it is necessary to have two (left-handed) chiral superfields containing two Higgs doublets, H1,2 with opposite hypercharge Y=-1,1 to allow for the couplings needed to give both up and down quarks a mass. Thus even if we identify H1 with a slepton supermultiplet it is necessary to introduce an additional chiral supermultiplet to accommodate H2 . In this case its fermion partner will introduce an anomaly coupled to hypercharge. To avoid this it is necessary to introduce two additional supermultiplets with hypercharge Y=1,-1 and then it is usual to identify the two Higgs doublets with the scalar components of these supermultiplets as is shown in Table 1 . While this is not unique the alternative identification of H1 with one of the slepton doublets introduces lepton number violation and is avoided in the minimal supersymmetric version of the standard model (the MSSM). 1.2 The couplings of the MSSM To complete the definition of the MSSM it is necessary to specify the couplings of the theory. The vector boson gauge couplings to the new states are all specified since the gauge quantum numbers of the new supersymmetric part- −4− ners of the standard model states have all been specified to be the same as their standard model partners. Operation by the supersymmetry generator induces new couplings involving the gaugino partner of the gauge boson with a strength given in terms of the original gauge coupling. The resulting Feynman rules are well known [4],[5] and we will not reproduce them here. In addition to the gauge couplings the MSSM must have the Yukawa couplings necessary to give mass to the quarks and leptons. These are associated with new scalar couplings related by the operation of the supersymmetry generator. The totality of these terms is most conveniently derived from the superpotential, P. In order to generate the required Yukawa couplings P must contain the terms P = hijk Li H1j Ek + h′ ijk Qi H1j Dk + h′′ ijk Qi H2j Uk (1) where L and E (Q and U, D ) are the (left-handed) lepton doublet and antilepton singlet (quark doublet and antiquark singlet) chiral superfields respectively and H1,2 are (left-handed) Higgs superfields. The supersymmetric couplings correspond to the F terms of the superpotential P . These give both Yukawa couplings and pure scalar couplings. For example, the Yukawa couplings following from the first term of eq.(1) are LY ukawa = hijk (LiH1j Ek + L̃i H̃1j Ek + Li H̃1j Ẽk ) (2) where we denote by a supertwiddle the supersymmetric partners to the quarks, leptons and Higgs bosons, namely the squarks, sleptons and Higgsinos. The first term is the usual term in the standard model needed to give charged leptons a mass. The new couplings associated with the supersymmetric states related to the first term by the operation of the supersymmetry generator are given by the second and third terms. The scalar couplings associated with eq.(1) are Lscalar = X i,j | ∂ 2P 2 | ∂φi φj (3) where φi are chiral superfields and after differentiation of the superpotential only the scalar components of the remaining chiral superfields are kept. The full set −5− of Feynman rules resulting from the superpotential of eq.(1)may be found e.g. in ref.[4] . There is one further coupling needed to complete the couplings of the minimal supersymmetric version of the standard model. In order to generate a mass for the Higgsinos associated with the Higgs doublets H1,2 it is necessary to add a term to the superpotential given by P ′ = µH1 H2 (4) In addition to giving a mass µ to the Higgsinos, this term plays an important role in determining the Higgs scalar potential and the pattern of electroweak symmetry breaking. As we will discuss in more detail in section 3 the scalar term following from eq.(4) aligns the vacuum expectation values (vevs) of the two Higgs fields so that the photon is left massless, obviously a crucial ingredient for a viable theory. We note that this term is the only one involving a coupling with dimensions of mass. If the theory is to avoid the hierarchy problem µ must be small, of order the electroweak breaking scale, for the Higgs scalars also get a contribution µ2 to their mass squared. Thus any complete explanation of the electroweak breaking scale must explain the origin of µ. There is a simple modification of the theory that avoids the introduction of a mass at this stage. In this variation of the MSSM eq.(4) is replaced by P ′ = λH1 H2 N + λ′ N 3 (5) where N is a gauge singlet chiral superfield. As we will see this term generates a vev, < N >, for the scalar component of N which then gives the superpotential term of eq.(4) with µ = λ < N >. The advantage of starting with eq.(5) is that < N > is automatically driven to be of the same magnitude as the Higgs vevs, avoiding the need to introduce a new mass scale by hand. −6− 1.3 R-parity and discrete symmetries Although we will be primarily concerned with the minimal supersymmetric standard model defined above it is not the only way a supersymmetric version of the standard model with minimal particle content can be constructed. For completeness we add here a discussion of possible variants of the supersymmetric version of the standard model. The ambiguity arises because the couplings of eq.(1) and eq.(4) or eq.(5) are not the only ones allowed by SU (3)⊗SU (2)⊗U (1) for we may add to the superpotential the terms [9] [λijk Li Lj Ek + λ′ijk Li Qj Dk + λ′′ijk Ui Dj Dk + µi ′ Li H2 ]F (6) These terms violate baryon or lepton number and, if all are present in the Lagrangian, they generate via graphs such as in fig. 1 an unacceptably large amplitude for proton decay suppressed only by the inverse supersymmetry-breaking Fig. 1 Graph giving proton decay in R-parity broken models mass scale squared. For this reason the MSSM requires an additional discrete symmetry called matter parity to forbid them. Under this symmetry the quark and lepton superfields appearing in the superpotential change sign while the Higgs superfields are left invariant. Thus the four terms of eq.(6) change sign under this symmetry and are forbidden while the terms of eq.(1) are invariant −7− and allowed. Using this symmetry the allowed couplings of the MSSM are only those of eq.(1). Note that supersymmetric states only occur in pairs in e.g., eq.(2) , giving rise to a conserved “ R-parity” under which the standard model states are R even and the new supersymmetric states are R odd. (R-parity is broken if any of the terms of eq.(6) are present). An immediate consequence of R-parity is that the new supersymmetric states may only be produced in pairs and the lightest supersymmetric state (the LSP) is stable. However it is not necessary to forbid all the terms of eq.(6) to stabilise the nucleon [10], [11] . The graph of Fig.1 is the dominant one responsible for nucleon decay and it may be seen from it that (redefining fields so that µ′ i = 0), provided at least the second or third baryon- or lepton- number violating vertices of eq.(6) is absent, the graph vanishes giving an acceptable model in which the remaining operators will violate matter- and R- parity. For example the first term gives rise to the Yukawa couplings λijk (Li Lj ẽk + Li L̃j Ek + L̃i Lj Ek ) (7) involving a coupling to single sparticle states. The effect of these terms is to significantly change the phenomenology associated with the new supersymmetric states for the LSP may decay changing ET missing signals to visible energy [12], [13], [14] so it is important to ask whether such models are reasonable. This amounts to a discussion whether a discrete symmetry can lead to a non-zero subset of the terms of eq.(6). Indeed this is the case and a general classification of discrete symmetries forbidding or allowing different couplings has been recently worked out [15]. The main difference between the discrete symmetry leading to the MSSM and those leading to R-parity breaking schemes is that in the latter the quarks and leptons transform differently. Is this to be expected? The origin of discrete symmetries must lie in the underlying (Grand) unified theory. In string theories it is known that discrete symmetries arise on compactification under which quarks and leptons do transform differently. Similarly many Unified theories do have −8− gauge symmetries under which quarks and leptons transform differently and, on spontaneous breakdown, may lead to such discrete symmetries. A survey of ZN (flavour-independent) symmetries (for low N ) [15] shows it is easy to obtain any of the allowed possibilities which forbid fast proton decay, namely M atter parity λ̃ = λ̃′ = λ˜′′ = 0; Lepton “parity” λ̃′ = λ˜′′ = 0; ∆B = ∆L = 0 ∆B 6= 0, ∆L = 0 Baryon “parity” λ̃ = 0; ∆B = 0, ∆L 6= 0 Overall there are many possibilities (including flavour quantum numbers there are 45 distinct new operators in eq.(6) ! ). Although constrained by their virtual corrections to standard model processes [12] and by baryogenesis [16] their coefficients may be large enough to give rise to completely new SUSY phenomenology. It has been suggested [17] that the possible discrete symmetries should be further constrained by the requirement that they be stable against large gravitational corrections [18],[15] and this is only possible if the symmetries come from an underlying gauge symmetry. If this is the case, the discrete symmetries have to obey certain “discrete anomaly” cancellation conditions [18]. One then finds that the condition for anomaly cancellation in the underlying gauge symmetry can only be satisfied by a restricted set of discrete symmetries if one demands only the minimal light spectrum of states. In this case there are only two preferred discrete symmetries [15], ZN , for N < 9, namely the usual Z2 matter parity leading to the MSSM and a Z3 baryon “parity” under which g(Q, U, D, L, E) = (1, α2, α, α2, α2 ) 2π ψi → gi ψi ; α = exp(i ) 3 (8) The baryon parity has the advantage that it additionally forbids the dangerous dimension 5 QQQL operators which may give fast proton decay. At the very least it should be considered on a par with the MSSM as a minimal supersymmetric extension of the standard model. One can also find a Z3 anomaly-free lepton parity but it requires that the underlying theory posseses a large Zm9 , m ∈ Z+ symmetry which looks rather unlikely [15]. −9− In what follows we will be mainly concerned with the MSSM as a minimal supersymmetric extension of the standard model. In fact the mechanism of electroweak breaking does not really depend on which symmetry one is imposing to the Yukawa couplings and practically all the results obtained below do also apply to the R-parity violating models. 2 Supersymmetry breaking and the MSSM sparticle masses. Supersymmetry must be broken in a realistic theory for the suppersymmetric partners have not been observed so far. However the scale of supersymmetry breaking and the associated masses of the supersymmetric states cannot be far above the electroweak breaking scale otherwise the hierarchy problem will reappear and radiative corrections will drive up the electroweak breaking scale. Considerable effort has been devoted to understanding the origin of supersymmetry breaking. If no mass scale is to be added by hand to the theory just to trigger the breaking then the scale of supersymmetry breaking must be related to the underlying scales in the theory, the Planck scale or the unification scale. In this case it is necessary to understand why the supersymmetry breaking scale is so small. It is known that if supersymmetry is broken through quantum corrections it must be through non- perturbative effects. This leads to the hope that the large difference between the Planck scale and the supersymmetry breaking scale may be understood due to the appearance of a small factor, exp(−a/g 2 ), associated with such non- perturbative effects. (Here a is a constant , g is a coupling and the expansion of the exponential around g=0 vanishes, characteristic of non-perturbative processes.) In our opinion this is the only promising explanation that has been advanced to explain the large mass hierarchy and so we concentrate on the implications for the spectrum of supersymmetric states that results from models implementing this idea. Nonperturbative effects occur when an interaction become large and in supersymmetric gauge theories there is a very natural source of such effects leading to supersymmetry breaking. This follows because an asymptotically free (nonabelian) gauge interaction, initially small, increases in strength as the energy scale decreases and at some scale will become large and in the non-perturbative domain. In analogy with QCD it may be expected that the gauginos will be − 10 − strongly bound by this interaction to form a gaugino condensate, < λλ > which breaks (local [19] ) supersymmetry generating a gravitino mass m3/2 ∝ <λλ> MP2 [20], [21], [22], [23], [24], [25]. Since the scale determining the gaugino condensate is given by the scale, Λ, at which the gaugino binding becomes non- perturbative we have m3/2 ∝ Λ3 ( 2b 3g2 ) 0 = M e P MP2 (9) where b0 (which is assumed to be negative) is the coefficient of the one loop β-function and the last equality follows from using the running of the gauge coupling from its value, g, at the Planck scale. This has the form anticipated in our discussion of non-perturbative effects and, for suitable values of g and b0 , offers an explanation for the large magnitude of the mass hierarchy. In order to build a realistic theory using this mechanism it is usual to associate the gaugino condensate with an “hidden” sector which couples to the visible sector only via gravitational interactions. This assumption is convenient in order to obtain universal supersymmetry-breaking effects [26], [27]. It is also the natural situation in 4-D strings [21], [22], [23],[24],[25]. in which the gauge group is always larger than the one of the standard model or its GUT extensions. Very often there are in these theories “hidden sectors” of particles which couple to hidden sector (confining) gauge interactions which couple only gravitationally to the “observable” world. Hidden sector breaking has the advantage of reducing the effect of supersymmetry breaking in the visible sector because the gravitational strength couplings generate visible sector masses of order the gravitino mass and not of order the much larger gaugino condensate scale. Apart from triggering supersymmetry breaking the hidden sector plays no role in low energy phenomenology for the states of this sector are confined with mass of order the gaugino condensate scale (∼ 1013 Gev). The most important implication for the MSSM of such hidden sector breaking is that supersymmetry breaking is communicated to the visible sector only via gravitational interactions. The precise effect on the visible sector depends on the details of these gravitational interactions but the independence of these interactions from the gauge and Yukawa interactions of the MSSM leads to a useful − 11 − parameterisation of the supersymmetry breaking effects in terms of flavour symmetric breaking terms [26],[27], [28]. This amounts to including a common mass m1/2 for the gauginos and another common mass, mo , for the scalars. The gauge bosons and fermions do not acquire mass at this stage due to residual unbroken gauge and chiral symmetries. It is also found in specific supergravity models that there are additional [29],[26], [27] supersymmetry breaking terms given by (A0 P3 + B0 P2 ) where A0 and B0 are masses of order m3/2 and P3 and P2 are the trilinear and quadratic terms of the superpotential with the supermultiplets replaced by their scalar components. The expectation for A0 and B0 depends on the metric [28]. The flavour independence of the supersymmetry breaking terms is broken by radiative corrections [30] involving the gauge and Yukawa couplings of the standard model. These corrections may be calculated explicitly and are most conveniently included via the renormalisation group equations [31],[26], [32], [33] for the masses (cf Appendix) and the A and B parameters using for initial values at MX the common gaugino and scalar masses m1/2 and mo and the common A0 and B0 parameters. Note that even if some of these initial values are zero the related terms may be generated at low scales via radiative corrections. ∗ The final form for the effective potential from the SUSY breaking soft terms for one family and keeping only the up quark Yukawa coupling is Vef f = [hA(Q̃Ũ H2 ) + h.c.] + [Bµ(H1 H2 ) + h.c.] µ21 | H1 |2 +µ22 | H2 |2 +m2L̃ | L̃ |2 +m2Ẽ | Ẽ |2 (10) + m2Q̃ | Q̃ |2 +m2Ũ | Ũ |2 +m2D̃ | D̃ |2 At the unification (Planck) mass one assumes m2Q̃ = m2Ũ = m2D̃ = m2L̃ = m2Ẽ = m20 µ21 = µ22 = m20 + µ20 (11) ∗ Recently [34] it has been observed that in string theories the universality of supersymmetry breaking mass terms may be broken if the fields have different modular weights. Although possible the necessity to avoid large flavour changing neutral currents strongly constrains the amount of such non- universality and suggests that, in a viable model, flavour blind scalar masses at the unification scale is a good approximation. On the other hand, non-universal gaugino masses may also be pressent in certain string models [34] . − 12 − where µ0 is the boundary value for µ. One also has universal gaugino masses M1 = M2 = M3 = m1/2 . (12) Using this parameterisation for the soft supersymmetry breaking mass terms at the unification scale it is now straightforward to determine the spectrum of the light states of the MSSM, for the couplings and soft supersymmetry breaking terms at low energy scales may be determined in terms of their values at the unification scale using the coupled renormalisation group equations to determine the radiative corrections. We turn now to a discussion of these equations and their implications for the generation of a further stage of spontaneous symmetry breaking this time in the gauge sector. 3 Electroweak breaking We discussed in the previous section the possible origin of the required breaking of supersymmetry. The resulting Lagrangian includes supersymmetryconserving pieces plus extra terms breaking supersymmetry explicitly (but softly [35]). The next problem to address is the breaking of the weak interactions symmetry SU (2)×U (1). In principle, this breaking is completely independent of that of supersymmetry but the fact that the masses of sparticles have to be of order the weak scale makes one suspect that both symmetry-breaking processes should be somehow related. Indeed, as we will now describe, once one has broken supersymmetry the breaking of SU (2) × U (1) appears as an automatic consequence of quantum corrections. To study the process of electroweak symmetry breaking let us consider the piece of the scalar potential involving just the Higgs doublets. From the gauge interactions and the interactions following from the superpotential of eq(4) one has g2 2 g1 2 τa τa (H1∗ H1 + H2∗ H2 )2 + (|H1 |2 − |H1 |2 )2 2 2 2 8 + µ1 2 |H1 |2 + µ2 2 |H2 |2 − µ3 2 (H1 H2 + h.c. ) V (H1 , H2 ) = (13) − 13 − where τ a , a = 1, 2, 3 are the SU (2)L Pauli matrices and µ1 2 ≡ m20 + µ20 ; µ2 2 ≡ m20 + µ20 ; µ3 2 ≡ B0 µ0 . (14) This is the SUSY version of the ‘mexican hat’ Higgs potential of the standard model. However, this potential as it stands looks problematic. Indeed, in order to get a non-trivial minimum we need to have a negative mass 2 eigenvalue in the Higgs mass matrix, i.e., we need µ21 µ22 −µ43 < 0. However, since µ21 = µ22 > 0, this may only happen if µ41 = µ42 < µ43 in which case the scalar potential is unbounded below in the direction < H1 >=< H2 >→ ∞. 3.1 One loop radiatively corrected potential The puzzle is resolved [30] by noting that the boundary conditions eq.(14) apply only at the unification or Planck scale. At any scale below one has to consider the quantum corrections to the scalar potential which can be substantial. Consider for example the one-loop corrections to the masses of the Higgs fields. There are graphs as in fig.2 Fig. 2 − 14 − which involve the Yukawa couplings of the Higgs field H2 to the u-type quarks and squarks. Of course, these corrections will be negligible except for the ones involving the top quark which has a relatively large Yukawa coupling (for simplicity we ignore here the possibility of a large bottom Yukawa coupling). While supersymmetry is a good symmetry the first graph in fig. 1 leads to a (negative) quadratically divergent contribution which is exactly cancelled by the second graph. Once susy is broken the sparticles get masses and the right-hand diagram is suppressed compared to the left-hand one leaving an overall uncancelled negative contribution [30] δµ22 ≃ − 3 2 2 2 h m log(MX /m2Q̃ ) . 16π 2 t Q̃ (15) If ht is large enough (i.e., if the top quark is heavy enough) this negative contribution may overwhelm the original positive contribution and trigger electroweak symmetry breaking. Similar diagrams exist for the other Higgs field H1 but those are expected to give small contributions since they will be proportional to the bottom Yukawa coupling. One may worry that one can find similar graphs involving squarks such as t̃, b̃ that could drive their mass 2 negative leading to minima with broken charge and colour. Indeed these graphs exist but coloured scalars also get large (positive) contributions to their mass 2 from loops involving gluinos which are proportional to the large strong coupling constant, preventing SU (3)⊗U (1)em breaking. Thus one sees that the structure of the minimal supersymmetric standard model is such that quantum corrections select the desired pattern of SU (3) ⊗ SU (2) ⊗ U (1) symmetry breaking in a natural and elegant way. With µ22 6= µ21 the potential in eq.(13)is perfectly well behaved and one can see it is minimized for [36] ν 2 ≡ ν12 + ν22 = 2(µ21 − µ22 − (µ21 + µ22 )cos2β) (g22 + g12 )cos2β (16) 0 > and sin2β ≡ 2µ2 /(µ2 + µ2 ). The existence of a nonwhere ν1,2 =< H1,2 3 1 2 vanishing µ23 forces the two vevs to be aligned in such a way that electric charge − 15 − 2 /g 2 . This condition remains unbroken. The W mass is related to ν via ν 2 = 2MW may be equivalently written [37], [38] µ21 + 21 MZ2 ν22 = ν12 µ22 + 21 MZ2 (17) where µ21,2 should be evaluated at the weak scale. Thus in a model with the correct SU (2) × U (1) breaking the parameters are constrained in such a way that µ21,2 (MW ) and µ23 (MW ) satisfy the above conditions. As we discussed in section(2) the free parameters in the minimal model are just m0 , m1/2 , A0 , µ203 ≡ B0 µ0 , µ0 (18) plus the Yukawa couplings, of which ht is likely to be the only one playing an important role in the running of the soft terms. In order to see how all these parameters are constrained we need to use the renormalization group equations which relate the values of couplings and masses at the unification scale with their values at the weak scale. The renormalisation group equations generalise the analysis of radiative corrections given in eq.(15) allowing for the summation of all powers of the logarithmic corrections. For completeness the renormalization group equations are provided in the appendix. Semi-analytic solutions to those equations may be found in ref.[37],[38]. Before describing the renormalization group running let us make a few comments on the general analysis presented here. First, one should not keep the evolution determined from a renormalization group equation below the threshold ∗ of any particle involved in the given equation . In other words, care has to be taken with the thresholds of the different sparticles. Usually it is enough to use a step function for each threshold. These threshold effects are discussed e.g. in [39]. Secondly, for a sufficiently large A parameter, there can appear other minima in the scalar potential favourably compeating with that in eq.(16) . They involve vacuum expectation values for sleptons and/or squarks and they break ∗ As may be seen from eq.(15) this is necessary to reproduce the true result obtained from the evaluation of the Feynman graphs on shell. − 16 − charge conservation. To avoid that it is enough to constraint | A | to sufficiently small values (typically | A |≤ 3 [40],[32],[37], [41].) Thirdly, similar results to those presented below are obtained in the case with an additional singlet N as discussed in section 1, eq.(5). For small λ′ the analysis of the electroweak symmetry breaking [42], [43] is rather similar although the existence of a term | H1 H2 |2 in the scalar potential allows for symmetry breaking only for mt ≥ 70 GeV (for a numerical analysis of this case see [43], [44] .) Finally, in the discussion below we have considered the most probable case in which ht is assumed to be much bigger than hb . Numerical studies including a non-negligible hb were presented in [37], [45]. We will not describe all these fine details here but merely give a general discussion of the most prominent physical implications and direct the reader to those references for further details. 3.2 Renormalisation Group analysis The most extensively studied renormalization group equations are those concerning the three gauge coupling constants. The extrapolation of the measured values of αe.m. and α3 to high energies shows that for the MSSM the couplings meet for a value of the Glashow-Weinberg angle sin2 θW ≃ 0.23 [46], [47] in extremely good agreement with recent LEP data [48], [49]. We discuss this point in some detail in the next section. The renormalization group equations for the Yukawa couplings [36] (see the appendix) are also of considerable interest. They can be integrated analitically in the case in which one only keeps the top-quark Yukawa coupling. For the third generation one finds [37] E1 (t) 1 + 6Yt (0)F1 (t) E2 (t) h2b (t) = h2b (0) (1 + 6Yt (0)F1 (t))1/6 h2t (t) = h2t (0) (19) h2τ (t) = h2τ (0) E3 (t) , where t ≡ 2 log(MX /Q) and E1,2,3 and F1 are known functions given in the appendix and Q is the scale at which the couplings are evaluated. The Ei functions give just the usual gauge anomalous dimension enhancement whereas the effect − 17 − of the top Yukawa coupling in the running gives the extra factor. Let us first discuss the case of the top quark. Notice that for small ht (0) one recovers the well-known gauge anomalous dimension result. However, for Yt (0) → ∞ one gets h2t (t) = (4π)2E1 (t) 6F1 (t) (20) independently of the original value of Yt (0), i.e., there is an infrared fixed point. At the weak scale (t ≃ 67) one obtains E1 ≃ 13 and F1 ≃ 290 which gives an upper bound for the top-quark mass mt = ht ν2 ≤ ht ν ≤ 190 GeV . One also observes in eq. (19) that the bottom-quark mass decreases as mt increases [36],[37]whereas mτ does not depend directly on mt (see below). Fig. 3-a − 18 − Let us now consider the running of the mass parameters which are the ones of direct relevance to the SU (2) × U (1)-breaking process. In particular, consider the running of the squarks, sleptons and Higgs masses (see the appendix). Fig. 3-b The renormalization group equations describing the mass2 evolution have a gauge contribution proportional to gaugino masses and a second contribution proportional to the top-Yukawa coupling2 . The gauge piece makes the mass2 increase as the energy decreases. In particular, squarks get more and more massive as we go to low energies since their equation is proportional to α3 . The piece in the equations proportional to the top Yukawa coupling has the opposite effect and decreases the mass2 as the scale decreases. This effect is normally not big enough to overwhelm the large positive contribution to the mass2 of squarks involving the QCD coupling but may be sufficiently large to overwhelm the positive contribution of weakly interacting scalars which only involve the electroweak couplings. The only such scalar in which this can happen is H2 since it is the only one (unlike sleptons) which couples directly to the top Yukawa. This is nothing but the renormalization group improved version [26],[37],[32],[33] of the − 19 − mechanism in eq.(15) . We thus see that the quantum structure of the MSSM leads automatically to the desired pattern of symmetry breaking in a natural way. The qualitative behaviour of the running of scalars is shown in fig. 3. A quantitative analysis of the process was given in [37], [32],[33], [50],[38]. Apart of obtaining the desired pattern of symmetry breaking one is interested in finding out the spectrum of sparticles in this scheme. Since there are only a few free parameters one has strong predictive power. In the case of the squarks and sleptons integration of the renormalization group equations (neglecting Yukawa couplings) leads to the following result [37] m2Ũ L m2D̃ L m2Ũ R m2D̃ R m2Ẽ L m2ν̃L m2Ẽ R 4 3 1 −1 2 = m20 + 2m21/2 ( α̃3 f3 + α̃2 f2 + α̃1 f1 ) + cos(2β)MZ2 ( + sin2 θW ) 3 4 36 2 3 1 4 3 1 1 = m20 + 2m21/2 ( α̃3 f3 + α̃2 f2 + α̃1 f1 ) + cos(2β)MZ2 ( − sin2 θW ) 3 4 36 2 3 4 4 2 2 2 2 2 = m0 + 2m1/2 ( α̃3 f3 + α̃1 f1 ) − cos(2β)MZ ( sin θW ) 3 9 3 4 1 2 2 2 1 = m0 + 2m1/2 ( α̃3 f3 + α̃1 ) + cos(2β)MZ ( sin2 θW ) 3 9 3 3 1 1 = m20 + 2m21/2 ( α̃2 f2 + α̃1 f1 ) + cos(2β)MZ2 ( − sin2 θW ) 4 4 2 1 1 3 = m20 + 2m21/2 ( α̃2 f2 + α̃1 f1 ) − cos(2β) MZ2 4 4 2 2 2 2 2 = m0 + 2m1/2 (α̃1 f1 ) + cos(2β)MZ sin θW (21) where θW is the weak angle, MZ is the Z 0 mass and m0 , m1/2 and tgβ = ν2 /ν1 are related to the free parameters in eq.(18). In this equation α̃i ≡ αi (MX ) (2 + bi α̃i t) ; fi ≡ t (4π) (1 + bi α̃i t)2 (22) where bi = (−3, 1, 11) are the one-loop coefficients of the β-function of the SU (3) ⊗ SU (2) ⊗ U (1) interactions. The above equations assume universal soft masses m0 for all the scalars in the theory at the unification scale as well as universal gaugino masses m1/2 . The rightmost term in eqs.(21) does not in fact come from the integration of the r.g.e.’s but from the contribution of the D2 -term in the scalar potential of sfermions once SU (2) × U (1) is broken. As noted above the squarks will be heavier than the sleptons since α3 ≫ α2 . − 20 − When computing physical masses the scale Q should be chosen equal to the mass. Fot masses in the TeV scale this corresponds to t ≃ 67. With this value the coefficients of the m21/2 -terms in eq.(21)are approximately 7.6 for ŨL , D̃L , 7.1 for ŨR , D̃R , 0.53 for ẼL , ν̃L and 0.15 for ẼR . The formulae for the stop and sbottom are modified by the top Yukawa coupling effects which can be evaluated numerically. Furthermore there are two additional contributions to the scalar masses: one is just equal to the mass of the corresponding fermionic partner mf and the other mixes left and right-handed sfermions and is proportional to mf A. These two contributions are only relevant for the heaviest quarks and may be included in a numerical analysis as discussed in section 4. Let us also briefly discuss the masses of physical Higgs particles. The MSSM contains two doublets of Higgses H1 = (H1+ , H20 ), H2 = (H2− , H20 ) with eight degrees of freedom. Three are swallowed by the W s and the Z 0 as they become massive. There remain one charged Higgs scalar H ± , two neutral scalars h, H and one neutral pseudoscalar P . Using the scalar potential (13) one finds for their masses [36] 2 m2H ± = MW + µ21 + µ22 m2P = µ21 + µ22 1 m2H,h = (m2P + MZ2 ± ((m2P + MZ2 )2 − 4m2P MZ2 cos2 2β)1/2 ) 2 (23) Using the above expressions one obtains the following constraints on Higgs masses [36], [51] 0 ≤ mh ≤| cos2β | MZ mh ≤ mP ≤ mH mH ≥ MZ (24) mH ± ≥ MW so that one observes that the scalar h is always necessarily lighter than the Z 0 . Although this is an important constraint, one must emphasize that for large mt these tree level Higgs formulae get important loop corrections which may modify the above inequalities [52]. − 21 − Let us now briefly discuss the spectrum of the fermionic sparticles. For the case of the gluino it is very easy to relate its mass to the SUSY-breaking parameters. It is simply given by Mg̃ = ( α3 ) M. αGU T (25) The mass spectrum for the W̃ , B̃ and Higgsinos is more complicated because once SU (2) × U (1) is broken they mix amongst themselves. Apart from this mixing, there are the usual Majorana mass for the gauginos and the direct Higgsino mass µ coming from the superpotential. In this way, we have a mass matrix for the charged winos-Higgsinos: ( W̃ − H̃ − ) M2 g2 ν2 g2 ν1 µ ! W̃ + H̃ + ! (26) where M2 is the direct wino mass and ν1,2 =< H1,2 >. This matrix has two ± eigenstates χ̃± 1 and χ̃2 (the ‘charginos’) with mass eigenvalues 1 M2,1 2 = (M2 2 + µ2 + 2MW 2 2q 1 ± (M2 2 − µ2 )2 + 4MW 4 cos2 2β + 4MW 2 (M2 2 + µ2 + 2M2 µsin2β)). 2 (27) The lightest of them, χ̃± 1 , is very often lighter than MW and has a good chance of being the lightest charged SUSY particle. The mass matrix for the ‘neutralinos’ is more complicated. In a basis spanned by (W̃ 0 , B̃ 0 , H̃10, H̃20 ) one finds [4] : Mχ0  M2   0 =   −g√2 ν1  2 g√ 2 ν2 2 0 M1 g√ 1 ν1 2 −g √1 ν2 2 −g √2 ν1 2 g√ 1 ν1 2 0 µ g√ 2 ν2 2 −g √1 ν2 2     µ   (28) 0 where M2 and M1 are related through the renormalization group equations by M1 = (3α1/5α2 )M2 . − 22 − This matrix has four eigenvalues and four corresponding eigenstates χ̃0i (i = 1 − 4), the neutralinos. The lightest of them (say χ̃01 ) has good chances of being the lightest supersymmetric particle (LSP). It will be a linear combination of the original fields: χ̃01 = U11 W̃ 0 + U12 B̃ + U13 H̃10 + U14 H̃20 (29) where the unitary matrix Uij relates the χ̃0i fields to the original ones. The entries of that matrix will depend only on tgβ,M2 and µ and, depending on those parameters, χ̃01 will be more ‘Higgsino-like’ or ‘photino-like’ etc. For example, for U13 = U14 = 0 the LSP would be the photino γ̃ = sinθW B̃ + cosθ W̃ 0 . In general one has to diagonalize the matrix (28) case by case in order to identify the eigenstates and mixing angles of the neutralinos. Notice that the couplings of neutralinos to the other particles will thus be affected by mixing angles. Examples of SUSY spectra consistent with appropriate electroweak symmetry breaking are provided in table 2. Finally we comment on the renormalisation group improvement of Grand unified (GUT) predictions [53] for quark and lepton masses. For example, in SU(5) mb and mτ are equal at the GUT scale. The gauge corrections to that relationship in the supersymmetric case [47] are numerically similar to the nonSUSY case. However, for a heavy top quark using eq.(19) one gets the result [37] mb (t) α3 (t) 8/9 α1 (t) 10/99 = ( ) ( ) (1 + 6Yt (0)F1 (t))−1/12 . mτ (t) α3 (0) α1 (0) (30) Of course, below the supersymmetric and top thresholds one has to use the equivalent non-supersymmetric (top-less) equations. This equation shows that for a sufficiently heavy top-quark the mb /mτ ratio is substantially decreased. This effect is numerically displayed in fig. 4. Notice that if one had an arbitrarily good precision of α3 (and a reliable definition of the b-quark current-mass) one should be able to extract a prediction for the mass of the top [37]. Unfortunately this is not the case. More recent analysis of the effect of a large t-quark mass on the mb /mτ ratio may be found in [54]. − 23 − . Fig. 4 4 Numerical analysis Using the parameterisation for the soft supersymmetry breaking mass terms at the unification scale given in section 2 together with the renormalisation group equations of section 3 and the appendix, it is straightforward to determine the supersymmetry breaking terms at low energy scales. As discussed in the last section this may lead automatically to a stage of electroweak breaking allowing for the complete determination of the mass spectrum of the states of the MSSM, including squarks, sleptons and the W and Z masses, in terms of the couplings of the theory and the parameters determining the supersymmetry breaking terms[49]. In this section we will generalize the discussion of section 3 to include all the terms in the RGEs by numerical integration and show that the resulting pattern is in excellent agreement with the precision measurements of the parameters of the standard model. We will also show that the MSSM leads to predictions for − 24 − the running gauge couplings in excellent agreement with a minimal unification picture in which these couplings are related at a high unification scale. While this latter result is not a necessary condition for the viability of the MSSM (unification could be non-minimal with several stages at high scale) its success does provide some circumstantial evidence for the need of low-energy supersymmetry. Perhaps more significant will be the very strong constraint on the supersymmetry breaking scale that results merely from the condition that there is a large scale of unification. As we will discuss, this is insensitive to the details of the unification and offers the most sensitive test of the idea there should be a low energy supersymmetry to solve the hierarchy problem, for it provides a strong upper bound to the masses of the new supersymmetric states. Although unification is not an absolute necessity for supersymmetry in our opinion it is the main reason for preferring the supersymmetric solution to the hierarchy problem over the composite solution, and for this reason we consider it important to consider the MSSM within the framework of an underlying unified theory [8]. 4.1 Unification of gauge couplings The possibility for unification of gauge couplings was noted by Georgi, Quinn and Weinberg [55] who showed in the standard model that radiative corrections drive the strong, electromagnetic and weak couplings together as high energy scales. This may be seen from the renormalisation group equations for the SU (3) ⊗ SU (2) ⊗ U (1) gauge couplings, given in the Appendix. Those equations determine the gauge couplings at a scale Q2 in terms of the initial values 2 summing the leading and next to leading logarithmic terms in at a scale MX 2 ). The initial values are determined by the Grand Unified theory or log(Q2/MX 2 ) = α , where α = C g 2 /4π compactified (string) theory at MX . In SU(5) αi (MX i i i G and C2,3 = 1, C1 = 3/5. This also applies to SO(10) and E(6) GUTs. In four dimensional compactified string theories Ci are related to the Kac-Moody level − 25 − of the underlying conformal field theory [56] with the usual choice of level one leading to the same values given above. Fig. 5 Using these equations it is straightforward to check whether the low-energy couplings evolve in energy to meet at the unification value αG . In Fig.5 the evolution of the couplings (including two-loop effects) is plotted for the non- Fig. 6 − 26 − -supersymmetric case, ∗ showing that the predictions are inconsistent; the cou- plings fail to meet [48] by more than seven standard deviations (the figures are taken from Amaldi et al in ref.[48] ) . The situation is quite different when the effect of supersymmetric states are added when calculating the β functions [46], [47] . Including these gives the evolution of the couplings is shown in Fig. 6 and it may be seen that the couplings do meet in a point. This happens if the mass of the new supersymmetric states (assumed degenerate here) are low MSU SY ∼ 102.5±1 GeV. On the basis of these results it is tempting to argue that there is evidence both for new forms of (relatively) light supersymmetric matter and an underlying unified theory. The conclusion is so dramatic it is important to discuss the reliability of the prediction. At the two loop order of precision non-logarithmic corrections due to states with mass of O(MX ) are also important. These “threshold” corrections can be 2 analytically computed [57] and included in the boundary value at the scale MX 2 −1 α−1 i (MX ) = αX + 1 1 H 2 MX 1 MX 2 2 [(tiF ) + 4(tH ) + (tH )] iF ) ln( iS ) ln( 6π 2 MF 2 MS (31) These terms give the threshold corrections due to massive states; MX , MF and MS are the masses of the massive gauge, fermion and scalar fields respectively H H and tH iV , tiF and tiS are the matrices which represent the generators of the gauge group on the superheavy vector, fermion and scalar fields respectively. In general we expect a spectrum of heavy fermion and scalar fields in which case there will be several fermion and scalar terms contributing to eq.(31). Since some of these fields acquire mass through the Yukawa couplings in the theory their mass may be substantially different from MX . These effects were first studied [58] in the context of non-supersymmetric SU(5). Although individual terms are small, it was found that the totality could be large in non-minimal versions of the model in which there are scalars transforming as some high representation of SU (5) giving rise to large numbers of ∗ It is convenient to plot inverse couplings for, cf eq.(37), these are approximately linear in log µ. − 27 − heavy scalar fields contributing to eq.(2). Recently the same observation has been made by Barbieri and Hall [59] in the context of supersymmetric SU(5). It is clear the possible presence of such massive states at the unification threshold introduces an inherent uncertainty in the analysis, as does the assumption of a single scale of breaking at the unification scale directly to the MSSM with the standard model gauge group. Due to these uncertainties it is not possible to make any definitive statements about the need for unification and a stage of low energy supersymmetry. Nonetheless it still of interest to study the minimal unification possibility, for its success may indicate simplicity in the unification possibly due to the absence of large representations of heavy states as may happen in some compactified string schemes, or due to the degeneracy of such states. In any case the minimal analysis, in which the predictions are well defined, provides a benchmark to judge unification predictions and to test the detailed structure of the low energy supersymmetric theory. Apart from the possible threshold effects due to massive GUT states at MX there may also be corrections in compactified superstring models of unification due to the tower of Kaluza Klein and string states with mass quantised in units of the compactification and string scales [60], [61], [62], [63], [64], [34]. In such theories there is only one fundamental mass scale, the Planck scale, and the scale, MX , at which the gauge couplings are related is determined in terms of the Planck scale by these threshold corrections . In a class of orbifold four dimensional string theories the result of including these corrections yields boundary values given by [61],[62] 2 α−1 i (MX ) 2 Mstring bi 1 X ′m b ln(2ReTm | η(Tm ) |4 ) = ki α X + log − 2 4π 4π m i MX (32) where η(T ) is the Dedekind function which admits a large T expansion η(T ) ≃ e−πT /12 (1−e−2πT +...). The Tm are the three complex scalars (untwisted moduli) whose vevs give the mass scale of compactification. Here Mstring is given by [60] Mstring ≃ 0.7 × g × 1018 GeV m (33) and the constants b′ i depend on the particular orbifold model. We see from this − 28 − that string theories are potentially more predictive than Grand Unified theories for MX is not a free parameter; we will discuss this point shortly. The minimal unification assumption has no significant threshold corrections due to states at MX . However there are threshold corrections at the supersymmetry breaking scale that should be included in the analysis for, as we discussed in section 3, the supersymmetric spectrum is not expected to be the degenerate one assumed in obtaining Fig. 6. By integrating the renormalisation group equations it is easy to calculate the full spectrum as discussed above and to include it in the analysis of gauge coupling running. Fig.7 The solid lines show the contours in the m0 , m1/2 plane corresponding to the value of αs (MZ ) needed for unification of the gauge couplings. The dotted lines show the mt contours needed to give the correct electroweak breaking. − 29 − Once one has a mass spectrum the renormalisation group equations for the gauge couplings may be integrated up in energy using the experimentally determined values at MZ as boundary values. The beta functions change at the appropriate mass scales as the threshold for the supersymmetric states is passed and this can be done in detail using the results of the renormalisation group evaluation of the mass spectrum by including a particular supersymmetric contribution to the β functions only at scales above its mass. The result of this analysis [49] is shown in Fig. 7 for | µ0 /m0 |= 0.2, 0.4, 1.0 and 5.0 (figs. 7-(a),(b),(c),(d) ). From the figure it may be seen that the present measurement of the strong coupling constant in the range 0.108-0.118 [65] , is consistent with the unification prediction for a range of the supersymmetry breaking masses m1/2 and mo between 2 and 90 Tev (the results shown correspond to the case A0 = B0 = 0). The sensitivity to the value taken for µ0 is such that if it is reduced the mass scale for the remaining supersymmetric states is increased. Broadly the result of including the non-degenerate spectrum is to increase the effective SUSY scale (the average scale of the SUSY breaking masses) by a factor of 3-10 compared to the analysis of Amaldi et al. [48] the range corresponding to whether the higgsino is light or not [49] . It is perhaps appropriate to comment about the meaning of this fit, for the three gauge couplings are described in terms of three effective parameters MX , αX and the effective supersymmetry mass scale, meaning there will always be a fit and apparently no test of unification! However this is not quite fair for the resulting values of the parameters must be reasonable if the scheme is to make sense. Thus MX should be less than the Planck scale but large enough to inhibit proton decay in Grand Unified theories. Also αX must be positive and within the perturbative domain (although it may be sensible to contemplate nonperturbative unification). Finally the supersymmetry mass scale must be large enough to explain why no supersymmetric states have been found and small enough to avoid the hierarchy problem (as we will see the latter gives a very strong constraint). In the analysis presented above the values of the parameters satisfy these conditions consistent with the hypothesis of minimal unification. − 30 − 4.2 The electroweak breaking scale We now turn to the calculation of spontaneous breaking of electroweak symmetry by radiative effects. As discussed in section 3 in the MSSM the Higgs masses are also determined in terms of the supersymmetry breaking terms at the unification scale . For a viable theory it is necessary that the electroweak symmetry be spontaneously broken and the way that may come about is for the radiative corrections to drive the Higgs mass squared negative thus triggering spontaneous symmetry breaking. As noted above the gauge interactions increase the masses (squared) and only the (top) Yukawa interactions can drive the mass squared negative. The effect of the radiative corrections involving this coupling is to reduce the stop and the Higgs masses squared but due to the large positive QCD radiative corrections which affect only the stop it is the Higgs scalar mass squared that is driven negative as desired. Since the effective potential of the Higgs scalar is completely determined in a supersymmetric theory by its gauge and Yukawa couplings its resultant vacuum expectation value is fixed, corresponding to a prediction for MW in terms of the parameters of the model. The most significant radiative correction in this respect comes from the top Yukawa ∗ coupling so the value of MW is largely determined by the value of mt . The Higgs potential which determines the scale of electroweak breaking is given by eq.(13). Since, cf eq. (15) , µ2 depends on ht , we see from eq.(17) that tuning h0 ≡ ht (MX ) we may always obtain the correct value of MZ . Thus to each point of the solution plane of Fig.7 it is possible to assign a definite h0 (or equivalently m0 ) which gives the correct electroweak breaking scale. Using this the results of the analysis of electroweak breaking may be conveniently summarised by drawing contour plots of constant mt (needed to give the correct MZ ) on the m0 , m1/2 plane. These are shown [49] by the dotted lines in Fig.7 for the case mt = 100, 160 GeV from which it may be seen that part of the previously allowed region is excluded by the requirement of acceptable radiative electroweak breaking coupled with the LEP bounds on the top mass. (The solution has a potential ∗ In determining the structure of electroweak breaking we choose to start with the case A0 = B0 = 0. We do this for simplicity and also because it is also true in some string derived models. Despite being originally zero A and B are driven non-zero at scales below MX . − 31 − bounded from below and , as discussed in section 3, colour and charge remain unbroken). The origin of the variation with respect to m0 shown in fig. 7 may be easily understood because larger m0 requires larger h0 to drive µ22 negative at the correct scale. The same applies to increasing the value of µ. It may be seen from Fig. 7 that a heavy top quark leads to a very satisfactory explanation of both the existence of electroweak breaking and its magnitude for a wide range of supersymmetry breaking parameters. This provides a realisation of the structure discussed in section 3, namely that in the MSSM electroweak breaking is naturally triggered by radiative corrections. 4.3 The fine tuning problem It is remarkable that the intricacies of this radiatively generated mass spectrum are such that an acceptable scale of electroweak breaking results for values for mt within the range allowed by the precision LEP measurements. However there is a concealed fine tuning problem associated with this solution. The problem is that the value of the Higgs mass and hence of MW is sensitively dependent on the top Yukawa coupling. For example with the choice of parameters corresponding to Fig. 7, if MW is constrained within 0.4% (experimental uncertainty) then h0 must be tuned to within about two parts per million, or the top mass must be constrained to about 0.3MeV! The reason for this extreme sensitivity may be seen from the form of the term in the renormalisation group responsible for the Higgs mass µ22 , (see eq.(43) in the Appendix). As we have discussed the last term proportional to the top Yukawa coupling, h, drives the mass squared negative at some point Q0 far from MX . Expanding around this point and, for illustration, keeping only the dominant term involving the stop mass, below Q0 the Higgs mass squared is given by (here we also neglect the running of µ) µ22 (Q2 ) = 3 2 2 h (mt̃ + m2t̃c )log(Q2/Q20 ) . 16π 2 (34) It may be seen that the negative Higgs mass squared which sets the scale for the vacuum expectation value of H2 and hence MW is proportional to the stop mass squared and to the top Yukawa coupling squared. For larger squark masses − 32 − MW will be very sensitive to the value of the top Yukawa coupling, giving rise to the fine-tuning problem noted above. In ref. [66] “reasonable” scales for the supersymmetry thresholds were estimated by demanding that the sensitivity of the electroweak breaking scale to any of the parameters of the standard model should be less than some value, c. In the case we are interested in c is defined through 2 δMW δh2 = c 2 h2 MW (35) where the scale is the electroweak breaking scale. No fine tuning would correspond to c ∼ 1 but, more conservatively, a value of c = 10 was chosen as a measure of a reasonable theory i.e.h must be tuned to finer than 1/10 the W mass uncertainty. Using this measure of the fine tuning problem we may determine the implications for the supersymmetry breaking scale. Ignoring the gauge couplings in the renormalization group equation for µ22 and also the running of the squark and Yukawa coupling one may solve eqs. (34)and (35)to find [49] c∼ 2 /Q2 ) ln(MX . ln(Q20/Q2 ) (36) 2 /Q2 )/ln(Q2 /Q2 )) in the sensitivity of the W The enhancement of (ln(MX 0 mass to the top Yukawa coupling comes from the fact that Q0 is driven by a large logarithm involving the unification scale MX . Although eq.(36)is only a rough approximation it does serve to explain the origin of the extreme sensitivity of the Higgs mass to the top Yukawa coupling which may be found exactly from the full renormalisation group equations. This is shown [49] in Fig.8 using instead of the approximate form of eq.(36), the value of c determined from the full renormalisation group equations. − 33 − . Fig. 8 The solid lines give the contours of constant c in the m0 , m1/2 plane for | µ0 /m0 |= 1, A0 = B0 = 0. Also shown are the mb contours (dashed lines) following from the unification prediction mb (MX ) = mτ (MX ). The dotted lines are as in fig. 7. − 34 − It is found that the condition c ≤ 10 restricts us to the region | µ0 /m0 |∼ 1, αs = 0.118 in the m0 , m1/2 plane. The full SUSY spectra for each of the “extremes ” labelled by Z and X in Fig.8 are listed in Table 2 including the (small) contributions to the masses coming from the spontaneous breaking of electroweak symmetry eq.(21). The neutralino and chargino masses (labeled by their dominant content) are obtained by diagonalising the full mass matrices. The spectra are very strongly constrained and lie not far from the present experimental limits with all SUSY masses less than 1TeV and sleptons much less, of order 200GeV. Note that the fine tuning constraint is largely independent of the details of the unification at MX and may be expected to give similar limits on the SUSY spectrum in any unification scheme having a very large MX . We have seen that within the minimal unification scheme the MSSM with radiative corrections included is very predictive and that the predictions are in excellent agreement with experiment provided the supersymmetry breaking scale is low, with the new supersymmetric states within reach of the LHC and SSC colliders. As such the theory should be tested by these machines. In fact one can make further predictions for one can also determine the mass of the Higgs bosons from the results of this analysis. This requires minimisation of the full Higgs potential and determination of the masses at the minimum. For the parameter choice of Fig.8 at the extremes Z and X the resulting Higgs spectrum is given [49] in Table 2. However we should emphasise that these masses are sensitive to the value of tanβ, cf eq.(23) , and hence to the initial values A0 , B0 . As has recently been noted [52] there are also large radiative corrections to the quartic terms in the effective potential for large values of the top mass which increase the light scalar mass, but these have been evaluated and are easily included. The minimal unification of the gauge interactions also determines the unification scale which we find to be MX ≈ 1016 Gev, sufficient in Grand Unified supersymmetric theories to inhibit nucleon decay (through dimension 6 operators) below experimental limits. From the most optimistic point of view this may be seen as evidence (albeit circumstantial) for unification. Is there a realistic (string) model realising this simple scheme? In Grand Unification the minimal − 35 − Parameters m1/2 140 230 m0 190 120 µ0 190 -120 mt 160 100 tanβ 21 5 Gauginos γ̃ 57 83 Z̃; W̃ 99 , 99 120 , 112 g̃ 354 559 Sleptons L̃ 220 206 ẼR 195 146 Squarks ũL , c̃L ; d˜L, s̃L 365 ; 373 511 ; 517 ũR , c̃R 359 495 d˜R , s̃R , b̃R 358 491 t̃L ; b̃L 325 , 335 491 , 497 t̃R 273 452 Higgs, Higgsinos h, H 91 , 264 84 , 221 H ±; P 276 , 264 232 , 218 H̃ 0 205 , 225 139 , 226 H̃ ± 229 227 Table 2 − 36 − unification assumptions which lead to the successful predictions discussed above can be realised if the MSSM is embedded in a GUT such as SU(5) which breaks directly to SU (3) ⊗ SU (2) ⊗ U (1) at the unification scale MX and all the resulting massive states are nearly degenerate. There is no constraint in Grand Unification which prohibits such a pattern of unification and it provides a nice example of the minimal unification discussed here. In string theories, however, there are much stricter constraints for a given four dimensional string theory has definite multiplet structure. In principle one can construct a string model with a unification group like SU (5) or SO(10) although this requires the presence of higher Kac-Moody algebras [67] which is technically complicated [68] . This type of scenario would lead to results similar to any grand unification scheme. On the other hand the simplest realisation of minimal string unification is for the gauge group after compactification to be just SU (3) ⊗ SU (2) ⊗ U (1) with the minimal particle content i.e. there is no need for any additional heavy states. While no example of such a string theory has yet been constructed it seems likely that something quite close to it can be found. However the prediction for MX is difficult to satisfy for a recent survey [64],[34] of all abelian orbifold models has revealed that large classes of models do not admit this possibility ( although it is not excluded). The alternative, [69],[34] is to give up the minimal unification assumption and to add states beyond the minimal set. Although this is certainly a possibility one loses the simplicity of the minimal model. To summarize the main conclusions of this section, we have examined above the possibility of generating electroweak breaking via radiative corrections. The prediction is sensitive to the top quark mass and it is in remarkable agreement with experiment for a top quark mass in the range needed for consistency with the precision LEP experiments. However if the SUSY masses are near their upper value allowed by the coupling unification the top quark mass must be fine tuned to a very high degree i.e. the natural scale of W mass lies close to the supersymmetric mass. This fine tuning may be reduced only if the SUSY mass scale is low and asking that it be less than one part in ten (C < 10) forces m0 , m1/2 <∼ 102 GeV . In this case consistency with the prediction for the strong coupling constant is only possible for αs at its upper allowed value ∼ 0.118 as determined by jet analyses and LEP [65] . − 37 − 5 Outlook We have discussed in the present article the mechanism of electroweak symmetry breaking in the supersymmetric standard model. It is quite remarkable that the very structure of the MSSM is such that quantum loop corrections induce in a natural way the SU (2)L ⊗ U (1) symmetry breaking. We think it is fair to say that the electroweak symmetry breaking in the MSSM is much more satisfactory than that in the standard model, not only because of the solution that supersymmetry provides to the hierarchy problem but also for the natural way in which this mechanism takes place. Whereas in the minimal standard model one puts by hand a tachyonic mass for the Higgs field in order to generate the desired minimum, in the supersymmetric model this is just a consequence of the particle content of the model and the radiative corrections. The excellent agreement of the renormalization group running of the coupling constants with the measured values of sin2 θW is a further support in favour of the presence of low energy supersymmetry. Although some points remain unclear (e.g., the origin of supersymmetry breaking) the above mentioned successes of the MSSM do not seem to depend heavily on details but on the general structure of the low energy model. A very important property of low energy supersymmetry is that it should be possible to test it in forthcoming accelerators. We described above how indeed a relatively low mass spectrum is expected if we want to avoid fine-tuning. Thus the sparticles should be accesible to experimental production at LHC and SSC. If this is the case very exciting physics is waiting to be discovered using these machines. − 38 − Appendix We collect here the renormalization group equations for the couplings and parameters discussed in the main text. We assume here the minimal particle content of the SSM and neglect all Yukawa couplings except for that of the top quark. The more general case with non-vanishing tau and bottom Yukawa couplings may be found e.g. in ref. [37],[38]. One has for the gauge couplings dgi2 bi 4 = − g dt (4π)2 i (37) bi 2 dMi = − g Mi dt (4π)2 i (38) and for the gaugino masses 2 /Q2 ) and b = −3, b = +1 and b = +11 for the minimal where t ≡ log(MX 3 2 1 particle content of the SSM. The supersymmetric mass of the Higgsinos evolve according to the equation dµ2 = µ2 (3α̃2 + α̃1 − 3Yt ) dt (39) where one defines α̃i ≡ αi h2t ; Yt ≡ 4π (4π)2 (40) where ht is the Yukawa coupling of the top quark. The Yukawa coupling of the third generation particles have renormalization group equations 16 13 dYt = Yt ( α̃3 + 3α̃2 + α̃1 − 6Yt ) dt 3 9 dYb 16 7 = Yb ( α̃3 + 3α̃2 + α̃1 − Yt ) dt 3 9 dYτ = Yτ (3α̃2 + 3α̃1 ) . dt (41) The equivalent equations for the Yukawas of the first two generations are obtained by deleting the Yt factor inside the brackets except for the Yukawas for the u and − 39 − c quarks in which the factor six is replaced by a three. The running of the masses of squarks and sleptons are given by dm2Q 16 1 α̃3 M32 + 3α̃2 M22 + α̃1 M12 ) − Yt (m2Q + m2U + µ22 + A2U − µ2 ) 3 9 16 16 = ( α̃3 M32 + α̃1 M12 ) − 2Yt (m2Q + m2U + µ22 + A2t − µ2 ) 3 9 16 4 = ( α̃3 M32 + α̃1 M12 ) 3 9 = ( dt dm2U dt dm2D dt dm2L = (3α̃2 M22 α̃1 M12 ) dt dm2E = (4α̃1 ) . dt (42) For the scalars of the first two generations the same equations apply with Yt = 0. Concerning the Higgs doublets mass parameters one finds dµ21 = (3α̃2M22 + α̃1 M12 ) + (3α̃2 + α̃1 )µ2 − 3Yt µ2 dt dµ22 = (3α̃2M22 + α̃1 M12 ) + (3α̃2 + α̃1 )µ2 − 3Yt (m2Q + m2U + µ22 + A2t ) dt dµ23 3 1 3 = ( α̃2 + α̃1 − Yt )µ23 + 3µAt Yt − µ(3α̃2 M2 + α̃1 M1 ) . dt 2 2 2 (43) Finally, the trilinear soft terms associated to the third generation Yukawa couplings are 16 dAt = ( α̃3 M3 + 3α̃2 M2 + dt 3 dAb 16 = ( α̃3 M3 + 3α̃2 M2 + dt 3 dAτ = (3α̃2 M2 + 3α̃1 M1 ) . dt 13 M1 ) − 6Yt At 9 7 α̃1 M1 ) − Yt At 9 (44) For the soft terms associated to the first two generations the same equations apply setting Yt = 0 except for the soft terms Au and Ac in which one has to replace the factor six by one in the first of eq. 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