Gustavo Burdman 1 New Solutions to the Hierarchy Problem∗ Gustavo Burdman Instituto de F´ısica, Universidade de S˜ao Paulo Cidade Universitaria, S˜ao Paulo SP 05508-900, Brazil (Received on 2 of June, 2006) After summarizing the status of the Standard Model, we focus on the Hierarchy Problem and why we believe this strongly suggests the need for new physics at the TeV scale. We then concentrate on theories with extra dimensions and their possible manifestations at this scale. Keywords: beyond the standard model, electroweak bymmetry breaking, extra dimensions I. INTRODUCTION short range inside a superconductor, by getting an effective photon mass. The Standard Model (SM) is a very successful description The first question is: at what scale should this phase transi- of particle physics up to the weak scale. It agrees with all the tion occur ? In order to answer this question within the SM, experimental data we have our disposal today. However, it we can look –for instance– at the scattering of massive gauge has several shortcomings. We do not know the origin of the bosons. These amplitudes (without including the Higgs) grow fermion masses, which in the SM arise from the Yukawa in- like teractions with the Higgs doublet. These are extremely varied, s 2 (1) with a Yukawa coupling for the top of order one, and for the MW electron about 106 times smaller. This constitutes the fermion mass hierarchy problem. Its solution might lie at high energy violate unitarity unless something happens before 1 TeV. In scales so that experimentsat the TeV scale might not necessar- the contextof the SM, this means that we need mh < O(1) TeV ily explore it. Another question unanswered by the SM, refers to restore unitarity. to the so called gauge hierarchy problem: why is the weak The SM introduces the scalar doublet which has a La- scale so much smaller than the Planck scale. We believe that grangian: the answer to this problem is at the TeV scale and therefore † µ L = (DµΦ) (D Φ) V(Φ) (2) accessible to the Large Hadron Collider (LHC). We will first − briefly summarize the status of the SM in Section II. In Sec- tion III we will expose the gauge hierarchy problem in more with the covariant derivative of the scalar doublet is given by detail. In Section IV we will presents a review of solutions to ig the gauge hierarchy problem involving compact extra spatial D Φ = ∂ + igtaW a + ′ B Φ (3) µ µ µ 2 µ dimensions. The vacuum expectation value (VEV) ΦT = (0 v/√2) h i breaks the SM gauge group down to U(1)EM, leaving the pho- II. THE STANDARD MODEL ton massless and giving the weak gauge bosons masses: gv g2 + g′2 v arXiv:hep-ph/0703194v1 19 Mar 2007 The SM gauge group, SU(2) U(1) , must be sponta- M = ; M = (4) L Y W 2 Z 2 neously broken to electromagnetism.× This is achieved through p the Higgs mechanism. The spontaneous breaking results in The minimization of the Higgs potential results in mh = a set of 3 Nambu-Goldstone Bosons (NGB), which are not √2λv, with λ the Higgs boson self-coupling. physical states since in the unitary gauge they are gauged + + away and appear as the longitudinal components of the W ± W W 0 W+ W+ and the Z . Within the SM, a complex scalar doublet, corre- W+ W+ sponding to 4 degrees of freedom, results in one of them be- γ ing left out, while the other are the NGBs. But in general, we γ can think of this as analogous to superconductivity: the Higgs − W − W mechanism results in a superconducting phase, where the or- W − W − − der parameter of the phase is a scalar field, which may or may W W − not be an elementary particle. As a result, the weak interac- tions become short range, just as electromagnetism becomes FIG. 1: Contributions to WW scattering amplitude. It violates uni- tarity at about 1 TeV The SM has been extremely successful when compared ∗Presented at the First Latin American Workshop on High Energy Phe- with experiment. The discovery of neutral currents in the 70’s, nomenology, Porto Alegre, December 3-5 2005 and the discovery of the W and Z gauge bosons, where just the 2 Brazilian Journal of Physics, vol. 37, 2A, June, 2007 prelude to the spectacularly precise tests started in the 90’s at LEP and SLAC. A large number of electroweak observables 6 are predicted with the input of only three parameters. The fun- Theory uncertainty ∆α(5) = damental parameters of the theory, g, g and v, can be traded had ′ 5 ± by quantities that are more accurately known, typically M , α 0.02758 0.00035 Z 0.02749±0.00012 and GF . In Figure (2) we see a sample of the observables and 4 incl. low Q2 data the agreement with the SM fit is very good. The precision of 2 3 ∆χ 2 Measurement Fit |Omeas−Ofit|/σmeas 0 1 2 3 ∆α(5) ± had(mZ) 0.02758 0.00035 0.02767 1 [ ] ± mZ GeV 91.1875 0.0021 91.1874 Γ [GeV] 2.4952 ± 0.0023 2.4959 Excluded Z 0 σ0 [ ] ± had nb 41.540 0.037 41.478 30100 300 R 20.767 ± 0.025 20.742 l m [GeV] 0,l ± H Afb 0.01714 0.00095 0.01643 ± Al(Pτ) 0.1465 0.0032 0.1480 ± Rb 0.21629 0.00066 0.21579 ± FIG. 3: The Higgs mass constraint in the SM. Rc 0.1721 0.0030 0.1723 0,b ± Afb 0.0992 0.0016 0.1038 0,c ± Afb 0.0707 0.0035 0.0742 ± Ab 0.923 0.020 0.935 baryon asymmetry in the universe, or of dark matter and dark A 0.670 ± 0.027 0.668 c energy? Why is the cosmologicalconstantso small ? So there A (SLD) 0.1513 ± 0.0021 0.1480 l is no shortage of shortcomings. However, we do not know at sin2θlept(Q ) 0.2324 ± 0.0012 0.2314 eff fb what energy scales these questions are answered. It could be [ ] ± mW GeV 80.410 0.032 80.377 Γ [ ] ± that the physics associated with them lies at a scale unreach- W GeV 2.123 0.067 2.092 [ ] ± able experimentally, at least for the moment. mt GeV 172.7 2.9 173.3 0 1 2 3 III. THE HIERARCHY PROBLEM FIG. 2: Comparison of electroweak observables with the SM fit (Summer 2005) However, the question that requires physics beyond the SM at the TeV, is the so called gauge hierarchy problem. the measurementshas reach the sub-percentlevel, and in order One way to state it is simply by asking: why is the weak to make predictions matching this one needs to go to one loop scale ( 100 GeV) so much smaller than the Planck scale calculations. This means that the measurements are sensitive ( 1019≃ GeV)? The weak scale is given by the VEV of to heavy physics running around loops. The most important the∼ Higgs, v 246 GeV, the only dimensionfull parameter example is the Higgs itself. Since it has not been observed, in the SM. However,≃ it is not naturally stable under radia- been sensitive to its loop effects is very interesting. The sensi- tive corrections. If we consider the radiative corrections to tivity of the electroweak observables to the Higgs mass is only the Higgs mass, coming from its couplings to gauge bosons, logarithmic. Yet, the precision is enough to turn this sensitiv- Yukawa couplings to fermions and its self-couplings, result in ity into a bound on the Higgs mass. In Figure (3) we show a quadratic sensitivity to the ultraviolet cutoff. Thus, if the χ2 the latest fit of the Higgs mass in the SM. The has a mini- SM were valid up to the Planck scale, then mh and therefore mum at around 100 GeV, although the direct search bound is the minimum of the Higgs, potential, v, would be driven to the mh > 114 GeV. The indirect limit implies that mh 246 GeV Planck scale by the radiative corrections. To avoid this, one ≤ at 95% C.L. Thus, we conclude that the SM is in very good has to adjust the Higgs bare mass in the SM Lagrangianto one agreement with experiment and that the Higgs must be light. part in 1017. This is quite unnatural, and is what we call the Then why do we want to have physics beyond the SM ? gauge hierarchy problem. The SM leaves many questions unanswered, and its answers In order for physics beyond the SM to regulate the Higgs to others are not very satisfactory. For instance, in the SM mass, and restore naturalness, its energy scale must be around fermion masses arise as a consequence of Yukawa couplings the TeV. Several alternative theories are potential candidates to the Higgs. But since there is a huge hierarchy of masses to do this job. Most of them, imply that new physics will be 6 (e.g. me/mt 10− ), then there should be a huge hierarchy of discoveredat the LHC. One of the most popularcandidate the- Yukawa couplings.≃ This is the fermion mass hierarchy prob- ories is weak scale supersymmetry. The superpartners of the lem. The SM gauge group being a product group, plus the SM particles, having different statistics, contribute to the ra- fact that the couplings get quite close to each other at high diative corrections to the Higgs mass with the opposite sign. energies, suggest grand unification.