I. THE HIGGS POTENTIAL AND THE LIGHT HIGGS BOSON In the previous chapter, it was demonstrated that a negative mass squared in the Higgs potential is generated radiatively for a large range of boundary conditions. We are now in position to write and minimize the Higgs potential and examine the mass eigenvalues and eigenstates and their characteristics. A. Minimization of the Higgs potential In principle, it is far from clear that the Higgs bosons, rather than some sfermion, receive VEVs. Aside from the sneutrino, whose VEV only breaks lepton number, leading to the generation of neutrino masses, all other sfermions cannot have non-vanishing expectation values, or else QED and possibly QCD would be spontaneously broken. Furthermore, there could be some direction in this many-scalar-field space, in which the complete scalar potential (involving Higgs and sfermion fields) is not bounded from below. These considerations lead to constraints on the parameter space, for example A f /m f < 3 6. For the purpose of this chapter, we simply assume that these constraints are satisfied, and we focus on the Higgs potential. The Higgs part of the MSSM scalar potential is given by V (H 1, H 2 ) = (m 2 H 1 + µ 2 ) H 1 2 + (m 2 H 2 + µ 2 ) H 2 2 b(h 1 H 2 + h.c.) + g2 + g 2 ( H 2 2 H 1 2 ) 2 + V. (1) 8 The terms proportional to µ 2 are supersymmetric; they come from the F-terms. The quartic terms are given by the SU(2) L and U(1) Y D-terms. The terms proportional to m 2 H 1, m 2 H 2, b are the soft mass parameters (normalized down to the weak scale). The one loop correction, V = V one loop (which, in fact, is a threshold correction to the one-loop improved tree-level potential) can be absorbed to a good approximation in redefinitions of the tree level parameters. The only term in the scalar potential that depends on the phases of the fields is the b-term. Therefore, a redefinition of the phase of H 1 or H 2 can absorb any phase in b, so we can take b to be real and positive. Then it is clear that a minimum of the potential V requires that H1 0H0 2 is also real and positive, so that H0 1 and H0 2 must have opposite 1
phases. We can therefore use a U(1) Y gauge transformation to make them both real and positive without loss of generality, since H 1 and H 2 have opposite hypercharges (±1/2). It follows that CP cannot be spontaneously broken by the Higgs scalar potential, since the VEVs and b can be simultaneously chosen real, as a convention. This means that the Higgs scalar mass eigenstates can be assigned well-defined eigenvalues of CP, at least at tree level. CP violating phases in other couplings can induced loop-suppressed CP violation in the Higgs sector, but do not change the fact that b, H1 0 and H0 2 can always be chosen real and positive. In order for the MSSM scalar potential to be viable, we must first make sure that it is bounded from below for arbitrarily large values of the scalar fields, so that V will really have a minimum. (Scalar potentials in supersymmetric theories are automatically non-negative and so clearly bounded from below. But now, that we have introduced supersymmetry breaking, we must be careful.) The scalar quartic interactions in V stabilize the vacuum for almost all arbitrarily large values of H 0 1 and H 0 2. However, for the special directions in field space H 0 1 = H 0 2, the quartic contributions to V are identically zero. Such directions in field space are called D-flat directions, because along them the part of the scalar potential coming from the D-terms vanishes. In order for the potential to be bounded from below, we need the quadratic part of the scalar potential to be positive along the D-flat directions. This requirement amounts to m 2 H 1 + m 2 H 2 + 2 µ 2 2b. (2) A broken SU(2) U(1) requires (m 2 H 1 + µ 2 )(m 2 H 2 + µ 2 ) b 2. (3) If this inequality is not satisfied, then H1 0 = H0 2 = 0 will be a stable minimum of the potential (or there will be no stable minimum at all). Given m 2 H 1 > 0, Eq. (3) is automatically satisfied for m 2 H 2 + µ 2 < 0. This is the situation discussed in the previous chapter. Having established the conditions necessary for H2 0 and H0 1 to get non-zero VEVs, we can now require that they are compatible with the observed phenomenology of electroweak symmetry breaking. Let us write v 2 = Hu, 0 v 1 = Hd. 0 (4) 2
These VEVs are related to the known masses of the Z 0 boson and the electroweak gauge couplings: v1 2 + v2 2 = v 2 = 2m 2 Z/(g 2 + g 2 ) (174 GeV) 2. (5) The ratio of the VEVs is traditionally written as tanβ v 2 /v 1. (6) The value of tanβ is not fixed by present experiments, but it depends on the Lagrangian parameters of the MSSM in a calculable way. Since v 1 = v sin β and v 2 = v cos β were taken to be real and positive by convention, we have 0 < β < π/2, a requirement that will be sharpened below. The minimization conditions read m 2 H 2 + µ 2 b cot β (m 2 Z /2) cos 2β = 0, m 2 H 1 + µ 2 b tan β + (m 2 Z/2) cos 2β = 0. (7) These equations allow us ro eliminate two of the Lagrangian parameters b and µ in flavor of tan β, but do not determine the phase of µ: µ 2 = m2 H 1 m 2 H 2 tan 2 β tan 2 β 1 m2 Z 2, (8) b = 1 2 sin 2β ( m 2 H 1 + m 2 H 2 + 2 µ 2). (9) Conversely, taking µ 2, b, m 2 H 1 amd m 2 H 2 as input parameters, one can obtain β and m 2 Z as output parameters: sin 2β = 2b m 2 H 1 + m 2 H 2 + 2 µ 2, m 2 Z = m2 H 2 m 2 H 1 cos 2β m 2 H 1 m 2 H 2 2 µ 2. (10) By writing Eq. (8) we subscribe to the convenient notion that µ is determined by the precisely known value m Z = 91.19 GeV. This is mere convenience. Renormalization cannot mix the supersymmetric µ parameter (which is protected by non-renormalization theorems which apply to the superpotential, dµ/d lnq µ) and the SSB parameters. Hence, the independent µ can be treated as a purely low-energy parameter. Nevertheless, it highlights the µ-problem: Why is a supersymmetric mass parameter exactly of the order of the SSB 3
parameters (rather than M U, for example). We touched upon this point in the context of supersymmetric GUTs and the doublet-triplet splitting, but it is a much more general puzzle whose solution must encode some information on the ultraviolet theory that explains this relation. The above form of Eq. (8) also highlights the fine-tuning issue whose rough measure is the ratio µ/m Z. Typically, m 2 H 2 is a relatively large parameter, controlled by the stop renormalization, which itself is controlled by QCD and gluino loops. One often finds that a phenomenologically acceptable value of µ is µ(m Z ) M 3 (m Z ) and that m Z is then determined by a cancellation between two O(TeV ) parameters, e.g. m 2 Z = 2(m 2 H 2 + µ 2 ) in the large tanβ limit. Clearly, this is a product of our decision to fix m Z rather than extract it. All it tells us is that m Z is a special rather than arbitrary (generic) value. The fine tuning problem is instead in the relation µ M 3 which is difficult to understand. B. The Higgs spectrum and its symmetries Without taking into account electroweak symmetry breaking, the Higgs mass matrix is MH 2 = m2 H 1 + µ 2 b. (11) b m 2 H 2 + µ 2 The CP-even and charged Higgs receive also contributions from the D-terms or, equivalently, from the quartic terms. But for the pseudoscalar (that is, CP-odd) Higgs fields, one can use (11) directly. Using the minimization equations, the pseudoscalar mass-squared matrix reads MPS 2 = b tanβ 1. (12) 1 cot β The determinant vanishes due to the massless Goldstone boson. It has a positive masssquared eigenvalue, m 2 A = Tr(M 2 PS) = 2b sin 2β = m2 1 + m 2 2, (13) where m 2 i m2 H i +µ 2 for i = 1, 2. The angle β is, in this context, the rotation angle between the interaction and mass eigenstates: 2Im(H 0 u ) = cosβa 0 + sin βg 0, 2Im(H 0 d ) = sin βa 0 cosβg 0, (14) 4
where G 0 is the would-be Goldstone boson eaten by the Z 0. The charged Higgs mass is given by m 2 H ± = m2 A + m 2 W. (15) Again, β is here the rotation angle between interaction and mass bases: H + u = cosβh+ + sin βg +, H d = sin βa cosβg, (16) where G ± are the would-be Goldstone bosons eaten by the W ±. The CP-even Higgs treelevel mass-squared matrix is MH 2 = 0 m2 A It gives the following eigenvalues: s2 β s β c β s β c β c 2 β + m 2 Z c2 β s β c β s β c β s 2 β. (17) m 2(tree) h 0,H = 1 ] [m 2A 0 2 + m2z (m 2 A + m2 Z )2 4m 2 A m2 Z cos2 2β. (18) Note that, at tree level, there is a sum rule for the neutral Higgs mass eigenvalues: m 2 H 0 + m2 h 0 = m2 A + m2 Z. (19) There are two particularly interesting limits to Eq. (19). In the limit tan β 1, one has µ, and the SU(2) U(1) breaking is driven by the b term. In practice, one avoids the divergent limit by taking tanβ > 1.1, as is also required by the perturbativity of the top-yukawa coupling and by the experimental lower bound on the Higgs mass (discussed in the next section). For tanβ, one has b 0, so that the symmetry breaking is driven by m 2 H 2 < 0. The tan β 1 case corresponds to an approximate SU(2) L+R custodial symmetry of the vacuum: Turning off hypercharge and flavor mixing, and taking y t = y b = y, one can rewrite the t and b Yukawa terms in an SU(2) L SU(2) R invariant form, y t L b L a ǫ ab H0 1 H 2 + H1 H2 0 bc bc L t c L c, (20) where in the SM H 2 = iσ 2 H 1. For v 1 = v 2 (as in the SM or in the tan β 1 limit), the symmetry is spontaneously broken, SU(2) L SU(2) R SU(2) L+R. However, y t y b, and 5
the different hypercharges of t c and b c explicitly break the left-right symmetry, and therefore the residual custodial symmetry. In the MSSM, on the other hand, H 1 is distinct from H 2, and if v 1 v 2, SU(2) L SU(2) R U(1) T3L +T 3R. The SU(2) L+R symmetry is preserved if β = π/4 (v 1 = v 2 ) and maximally broken for β = π/2 (v 1 v 2 ). The symmetry is broken at the loop level, so one expects in any case tanβ above unity. As a result of the symmetry, MH 2 = 1 1 0 µ2, (21) 1 1 and it has a massless eigenvalue. This is of course a well known result of the tree level formula when taking β π/4. The mass is then determined by the loop corrections, which are well known (to two loops): m 2 h 0 2 h 0 h 2 tm 2 t (see next section). The heavier CP-even Higgs boson mass eigenvalue equals approximately 2 µ. (The loop corrections are less relevant here as typically m 2 H 0 2 H 0.) The custodial symmetry (or the large µ parameter) dictates in this case a degeneracy m A m H 0 m H + 2 µ. (The tree level corrections to that relation are O[(m Z /m A ) 2 ].) In other words, at a scale Λ 2 µ, the heavy Higgs doublet H is decoupled, and the effective field theory below Λ has only one, SM-like, Higgs doublet, which contains a light physical state. This is a special case of the MSSM, in which all other Higgs bosons (and possibly sparticles) decouple. The decoupling limit typically holds for m A > 300 GeV, and is realized more generally. The Higgs sector in the large tanβ limit exhibits an approximate O 4 O 4 symmetry. For b 0, there is no mixing between H 1 and H 2 and the Higgs sector respects the O 4 O 4 symmetry (up to gauge coupling corrections), i.e. invariance under independent rotations of each doublet. The symmetry is broken to O 3 O 3 for v 1 v 2 0 and the six Goldstone bosons are the SM Goldstone bosons, A 0 and H ±. The symmetry is explicitly broken for g 2 0 (so that m H + = m W ), and is not exact even when neglecting gauge couplings (i.e. b 0). Thus, A 0 and H ± are the massive pseudo-goldstone bosons, m 2 H + m 2 W m 2 A = C b 2. However, C = 2/ sin 2β and it can be large, which is a manifestation of the fact that O 4 O 4 O 4 O 3 for v 1 = 0. (The limit b 0 corresponds also to a U(1) PQ symmetry under which the combination H 1 H 2 is charged.) In the case β π/2, one has m (tree) h 0 (assuming m A m Z ). When adding up loop corrections, m h 0 2m Z 130 GeV. m Z 6
C. The light Higgs boson Let us examine the lightness of the Higgs boson from a different perspective, and also the one-loop corrections to its mass. Including one-loop corrections, the general upper bound is [ ( m 2 h 0 m2 Z cos2 2β + 3αm4 t m 2 t ln 1 m 2 t ) ] 2 + 4πs 2 c 2 m 2 Z m 4 θt, (22) t where θt = ( m 2 t 1 m 2 t 2 ) sin 2 2θ t 2m 2 t ln m2 t 1 m 2 t 2 + ( m 2 t 1 m 2 t 2 ) 2 ( sin 2 ) 2 [ 2θ t 4m 2 t 2 m2 t 1 + m 2 t 2 m 2 t 1 m 2 t 2 ln m2 t 1 m 2 t 2 Here, m 2 t i are the eigenvalues of the stop mass-squared matrix, θ t is the left-right stop mixing angle, and we neglected other loop contributions. The tree-level mass-squared, m 2(tree) h, and 0 the loop correction, 2 h 0, are bounded by the first and second term on the RHS of Eq. (22), respectively. In the absence of mixing, θ t = 0. For tanβ 1, one has m 2(tree) h 0 hence m 2 h 0 2 h 0. ]. (23) 0, and Clearly, and as we observed before, the tree-level mass vanishes as tanβ 1 (cos 2β 0). In this limit, the D-term (expectation value) vanishes, as well as the tree-level potential which is now quadratic in the fields. It corresponds to a flat direction of the potential, and the massless real-scalar h 0 is its ground state. Now that we have identified the flat direction, it is clear that the upper bound must be proportional to cos 2β, so that h 0, which parameterizes this direction, is massless once the flat direction is realized. The proportionality to m Z is only the manifestation that the quartic couplings are the gauge couplings. Hence, the lightness of the Higgs boson is a model independent statement. The flat direction is always lifted by quantum corrections, the most important of which is given in Eq. (22). These corrections may be viewed as effective quartic couplings that have to be introduced to the effective theory once the stops, t i, are integrated out of the theory at a few hundred GeV or higher scale. These couplings are proportional to the large Yukawa couplings (for example, from integrating out loops induced by (y t th i ) 2 quartic F-terms in the scalar potential). Note that even though one finds in many cases O(100%) corrections to the light Higgs mass (and hence m h 0 m Z m h 0 < 2m Z ) this does not signal the breakdown of perturbation theory. It is only that the tree level mass (approximately) vanishes. Indeed, two-loop corrections are much smaller (and shift m h 0 by typically only a few GeV) and are often negative. 7
The above upper bound is modified if and only if the Higgs potential contains terms (aside from the loop corrections) that lift the flat direction. For example, this is the case in the NMSSM, or if the gauge structure is extended by an abelian factor, G SM G SM U(1). However, as long as one requires that all couplings remain perturbative in the ultraviolet, then the additional corrections to the Higgs mass are still modest, leading to m h0 < 150 200 GeV (including loop corrections). The existence of a model-independent light Higgs boson is therefore a prediction of the framework. It is encouraging to note that it seems to be consistent with current data. The electroweak precision measurements strongly indicate that the SM-like Higgs is light, m h 0 < 200 GeV, where the best fitted values are near 100 GeV. Searches at the LEP experiments bound the SM-like Higgs mass from below, m h 0 114 GeV. [1] H. E. Haber, R. Hempfling and A. H. Hoang, Z. Phys. C 75, 539 (1997) [arxiv:hep-ph/9609331]. [2] G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich and G. Weiglein, Eur. Phys. J. C 28, 133 (2003) [arxiv:hep-ph/0212020]. 8