Lecture III Higgs Bosons in Supersymmetric Models Outline The significance of the TeV-scale—Part 2 • The MSSM Higgs sector at tree-level • Saving the MSSM Higgs sector—the impact of radiative corrections • LHC searches for the heavy Higgs states of the MSSM • Alignment without decoupling in the MSSM? • Beyond the MSSM Higgs sector • The significance of the TeV scale—Part 2 Despite the tremendous success of the Standard Model (SM) of particle physics, we know that there are some missing pieces. Neutrinos are not massless—perhaps suggestive of a new high energy • (see-saw) scale. Dark matter is not accounted for. • There is no explanation for the baryon asymmetry of the universe. • There is no explanation for the inflationary period of the very early universe. • The gravitational interaction is omitted. • Consequently, the SM is an low-energy effective theory. New high energy scales must exist where more fundamental physics resides. Indeed, quantum gravitational effects are likely to be relevant only at the Planck scale, M =(c~/G )1/2 1019 GeV , PL N ≃ 2 which arises as follows. Consider the gravitational potential energy, GN M /r, of a particle of mass M evaluated at its Compton wavelength, r = ~/(Mc). If this energy is of order the rest mass, Mc2 (i.e., when M M ), then ∼ PL pair production is possible and quantum gravitational effects can no longer be ignored. Denote scale at which new physics enters by Λ. Predictions made by the SM depend on a number of parameters that must be taken as input to the theory. These parameters are sensitive to ultraviolet physics, and since the physics at very high energies is not known, one cannot predict their values. However, one can determine the sensitivity of these parameters to the ultra- violet scale (which one can take as the Planck scale or some other high energy scale at which new physics beyond the Standard Model enters). In the 1930s, it was already appreciated that a critical difference exists between bosons and fermions. Fermion masses are logarithmically sensitive to ultraviolet physics. Ultimately, this is due to the chiral symmetry of massless fermions, which implies that δm m ln(Λ2/m2 ) . f ∼ f f No such symmetry exists for bosons (in the absence of supersymmetry), and consequently we expect quadratic sensitivity of the boson squared-mass to ultraviolet physics, δm2 Λ2 . B ∼ 2 2 2 1 2 2 In the SM, mh = λv and mW = 4g v imply that 2 mh 4λ 2 = 2 , mW g which one would expect to be roughly of (1). A 125 GeV Higgs boson O satisfies this expectation. However, the Higgs boson is a consequence of a spontaneously broken scalar potential, 2 1 2 V (Φ) = µ (Φ†Φ) + λ(Φ†Φ) , − 2 2 1 2 where we identify µ = 2λv in terms of the vacuum expectation value v of the Higgs field. The parameter µ2 is quadratically sensitive to Λ. Hence, to obtain v = 246 GeV in a theory where v Λ requires a significant fine-tuning ≪ of the ultraviolet parameters of the fundamental theory. Indeed, the one-loop contribution to the squared-mass parameter µ2 would be expected to be of order (g2/16π2)Λ2. Setting this quantity to be of order of v2 (to avoid an unnatural fine-tuning of the tree-level parameter and the loop contribution) yields Λ 4πv/g (1 TeV) ≃ ∼ O A natural theory of electroweak symmetry breaking (EWSB) would seem to require new physics at the TeV scale to govern the EWSB dynamics. The Principle of Naturalness In 1939, Weisskopf announces in the abstract to this paper that “the self-energy of charged particles obeying Bose statistics is found to be quadratically divergent”…. …. and concludes that in theories of elementary bosons, new phenomena must enter at an energy scale of order m/e (e is the relevant coupling)—the first application of naturalness. Principle of naturalness in modern times How can we understand the magnitude of the EWSB scale? In the absence of new physics beyond the SM, its natural value would be the Planck scale (or perhaps the GUT scale or seesaw scale that controls neutrino masses). The alternatives are: Naturalness is restored by a symmetry principle—supersymmetry—which • ties the bosons to the more well-behaved fermions. The Higgs boson is an approximate Goldstone boson—the only other known • mechanism for keeping an elementary scalar light. The Higgs boson is a composite scalar, with an inverse length of order the • TeV-scale. The naturalness principle does not apply. The Higgs boson is very • unlikely, but unnatural choices for the EWSB parameters arise from other considerations (landscape?). Avoiding quadratic sensitivity to Λ with elementary scalars A lesson from history The electron self-energy in classical E&M goes like e2/a (a 0), i.e., it → is linearly divergent. In quantum theory, fluctuations of the electromagnetic fields (in the “single electron theory”) generate a quadratic divergence. If these divergences are not canceled, one would expect QED to break down at an energy of order me/e far below the Planck scale. The linear and quadratic divergences will cancel exactly if one makes a bold hypothesis: the existence of the positron (with a mass equal to that of the electron but of opposite charge). Weisskopf was the first to demonstrate this cancellation in 1934...well, actually he initially got it wrong, but thanks to Furry, the correct result was presented in an erratum. A remarkable result: e e e e e a b The linear and quadratic divergences of a quantum theory of elementary fermions are precisely canceled if one doubles the particle spectrum—for every fermion, introduce an anti-fermion partner of the same mass and opposite charge. In the process, we have introduced a new CPT-symmetry that associates a fermion with its anti-particle and guarantees the equality of their masses. TeV-scale supersymmetry to the rescue Take the SM and double the particle spectrum. Introduce supersymmetry (SUSY), which dictates that for every boson, there is a fermion superpartner of equal mass and vice versa. Supersymmetry relates the self-energy of the boson to the self-energy of its fermionic partner. Since the latter is only logarithmically sensitive to Λ, we conclude that the quadratic sensitivity of the boson squared-mass to ultraviolet physics must exactly cancel. Naturalness is restored! Since no superpartners exist that are degenerate in mass with the corresponding SM particle, SUSY must be a broken symmetry. Conclusion: The effective scale of SUSY-breaking cannot be much larger than of order 1 TeV, if SUSY is responsible for the EWSB scale. The minimal supersymmetric extension of the SM The minimal supersymmetric extension of the Standard Model (MSSM) consists of the fields of the two-Higgs-doublet extension of the Standard Model and the corresponding superpartners. A particle and its superpartner together form a supermultiplet. The corresponding field content of the supermultiplets of the MSSM and their gauge quantum numbers are shown in the following table. The enlarged Higgs sector of the MSSM constitutes the minimal structure needed to guarantee the cancellation of anomalies from the introduction of the higgsino superpartners. Moreover, without a second Higgs doublet, one cannot generate mass for both “up”-type and “down”-type quarks (and charged leptons) in a way consistent with a holomorphic superpotential. To account for supersymmetry (SUSY) breaking, one adds the most general set of soft-SUSY-breaking terms consistent with the SM gauge symmetry. Field content of the MSSM Super- Super- Bosonic Fermionic multiplets field fields partners SU(3) SU(2) U(1) gluon/gluino V8 g g 8 1 0 0 0 gauge/ V W ± , W W ± , W 1 3 0 b e gaugino Vb′ B f Bf 1 1 0 slepton/ L (ν , e−) (ν, e−) 1 2 1 b L L e L − lepton Ec e˜+ ec 1 1 2 b e Re L squark/ Qb (uL, dL) (u, d)L 3 2 1/3 c c quark U u∗ u 3¯ 1 4/3 b e Re L − c c D d∗ d 3¯ 1 2/3 b eR L 0 0 Higgs/ Hd (H e, H−) (H , H−) 1 2 1 b d d d d − + 0 + 0 higgsino Hbu (Hu , Hu) (Heu , He u) 1 2 1 b e e The fields of the MSSM and their SU(3) SU(2) U(1) quantum numbers are listed. The electric charge is × × 1 given in terms of the third component of the weak isospin T3 and U(1) hypercharge Y by Q = T3 + 2Y . For simplicity, only one generation of quarks and leptons is exhibited. For each lepton, quark, and Higgs super- multiplet, there is a corresponding anti-particle multiplet of charge-conjugated fermions and their associated scalar partners. The L and R subscripts of the squark and slepton fields indicate the helicity of the corresponding fermionic superpartners. More on anomaly cancellation One-loop VVA and AAA triangle diagrams with three external gauge bosons, with fermions running around the loop, can contribute to a gauge anomaly. (Here V refers to a γµ vertex and A refers to a γµγ5 vertex.) A theory that possesses gauge anomalies is inconsistent as a quantum theory. To cancel the gauge anomalies, we must satisfy certain group theoretical constraints. W iW jB triangle Tr(T 2Y ) = 0 , ⇐⇒ 3 BBB triangle Tr(Y 3) = 0 . ⇐⇒ Example: contributions of the fermions to Tr(Y 3) Tr(Y 3) = 3 1 + 1 64 + 8 1 1+8=0 . SM 27 27 − 27 27 − − + 0 Suppose we only add the higgsinos (Hu , Hu). The resulting anomaly factor 3 3 is Tr(Y ) = Tr(Y )SM + 2, leading to a gauge anomaly.