Pramana – J. Phys. (2016) 87: 37 c Indian Academy of Sciences DOI 10.1007/s12043-016-1246-2 Why supersymmetry? Physics beyond the standard model ROMESHKKAUL The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113, India E-mail: [email protected] Published online 23 August 2016 Abstract. The Naturalness Principle as a requirement that the heavy mass scales decouple from the physics of light mass scales is reviewed. In quantum field theories containing elementary scalar fields, such as the Standard Model of electroweak interactions containing the Higgs particle, mass of the scalar field is not a natural parameter as it receives large radiative corrections. How supersymmetry solves this Naturalness Problem is outlined. There are also other contexts where the presence of elementary scalar fields generically spoils the high–low mass scales decoupling in the quantum theory. As an example of this, the non-decoupling of possible Planck scale violation of Lorentz invariance due to quantum gravity effects from the physics at low scales in theories with elementary scalar fields such as the Higgs field is described. Here again supersymmetry provides a mechanism for ensuring that the decoupling of heavy–light mass scales is maintained. Keywords. Naturalness problem; hierarchy problem; decoupling; supersymmetry; Standard Model; Higgs particle; violation of Lorentz invariance. PACS Nos 11.30.Pb; 12.15.−y; 12.60.−i; 12.10.−g; 04.60.−m 1. Introduction this scale are not acceptable at all. Another name for this requirement is ‘Naturalness Principle’. Till recently, all elementary particles that were known A quantum field theory containing only spin-half to exist in Nature were only spin-half fermions and fermions and gauge fields exhibits precisely this decou- spin-one gauge particles. With the discovery of Higgs pling. Such theories are called natural theories and the particle at the LHC in 2012, we now have the first ele- masses and the gauge couplings are natural parameters. mentary spin-zero particle. An elementary scalar field, Examples of such theories are: Quantum electrody- such as the Higgs field, introduces a completely new fea- namics (QED) and quantum chromodynamics (QCD). ture in quantum field theories containing such a field. The notion of Naturalness emerged in the late 1970s This new feature is a generic non-decoupling of the from the work of Wilson, Gildener and Weinberg and heavy mass scales from the physics of low mass scales. ’t Hooft [1,2]. A concise formulation is provided by ’t Hooft’s Doctrine of Naturalness [2]: A parameter A quantum field theoretic description for physical α(μ) at any energy scale μ in the description of physical processes with a characteristic smaller mass scale m L reality can be small, if and only if, there is an enhanced should not depend sensitively on the physics of larger symmetry in the limit α(μ) → 0. This implies a rule of mass scales m . This decoupling requirement is a rea- H thumb: quantum corrections to the parameter α (masses sonable expectation so that whatever low mass scale and couplings) should be proportional to a positive quantum theory we have, can describe the physics at n power of that parameter itself: (α)quantum ∼ α , n ≥ 1. that scale reliably. The only possible allowed depen- This would be ensured by the associated approxi- dence of the physics at low mass scale mL on the higher mate symmetry. Further, this property implies that the mass scale mH is in the form of its inverse powers enhanced symmetry holds even at the quantum level as and at the worst, a milder dependence through loga- α → 0. rithms of the high scale, but, as shall be discussed in Let us look at QED in some detail. The theory detail in the following, those with positive powers of describes the interaction of fermions λ of charge qf 1 37 Page 2 of 9 Pramana – J. Phys. (2016) 87: 37 such as an electron with electromagnetic radiation 2 In fact, such corrections to mL appear with quadratic through a Lagrangian density: divergences: L =−1 μν k2+m2 QED F Fμν m2 ∼−y2 d4k H 4 L (k 2−m2)2 +λ¯ iγ μ ∂ −ieq A − m λ. (1) H μ f μ e ∼− 2 2 2 2 y mH ln(mH/μ ), (5) Various parameters here, the electromagnetic coupling where we have used dimensional regularization and e and electron mass me, can be naturally small. Limit m → minimal subtraction in the last step. Notice that this e 0 leads to an enhanced symmetry, the chiral 2 2 symmetry: separate conservation of the number of left- correction is proportional to mH and not to mL.There- and right-handed electrons. It is for this reason that fore, there is no decoupling of the heavy mass scale one-loop correction to the electron mass in this theory from the light mass scale theory. Even in the limit → is given by logarithmically divergent diagrams and is mL 0, this correction does not go away. proportional to the electron mass itself. 2. Naturalness of electroweak theory 2 (me)1 loop ∼ e me ln . (2) Standard Model (SM) of particle physics has an ele- → In the limit me 0, this correction disappears reflect- mentary scalar particle, the Higgs particle. Discussion ing the fact that chiral symmetry is preserved by such above then implies that its mass is not protected by any perturbative quantum corrections in this limit. Also, symmetry against large radiative corrections. e → 0 results in enhanced symmetry: there is no Tree-level masses for the Higgs particle, gauge par- ± interaction and hence particle number of each type is ticles W ,√Z0 and the fermions in SM are given by: conserved. Again, quantum corrections to the coupling = = = mHiggs √λv, mW gv/2, mZ gv/(2cosθW ), e are logarithmically divergent and are, in the lowest m = Y v/ 2, where v is the vacuum expectation 3 f f order, proportional to e : value of the scalar field, λ is the quartic scalar coupling, 3 g is the gauge coupling, Yf is the Yukawa coupling of (e)1 loop ∼ e ln . (3) the fermions f to the scalar field and θW is the weak It is for this reason that atomic physics described by the mixing angle. → interactions of electrons and photons is not disturbed Note that the limit v 0 does enhance classical by the fact there are other heavier charged fermions symmetry: (i) all particles being massless in this limit, such as the muon, tau-lepton, top quarks and others in we have scale invariance of the classical theory; (ii) the weak gauge bosons are massless resulting in restored Nature: mμ ∼ 200me, mτ ∼ 3500me, ..., mtop ∼ 3.4 SU(2) gauge symmetry and additionally (iii) there is × 1011m . Effects of heavier fermions are decoupled e chiral symmetry due to zero masses of the fermions. from the physics of electrons and photons; showing up Yet, v is not a natural parameter. Consequently, masses at best through logarithmic dependence on them. of the Higgs particle, weak gauge bosons and fermions As against this, theories with elementary scalar are not natural. This is due to the Coleman–Weinberg fields have completely different behaviour. Elementary mechanism of radiative breaking of symmetry: quan- scalar fields spoil heavy–light decoupling: quantum tum fluctuations generate a non-zero quantum vacuum field theories with scalar fields are not natural. As an expectation value for the scalar field even when clas- example, consider a Yukawa theory of an elementary sically v is zero, breaking all these symmetries. There scalar field φ of low mass m coupled to a heavy L is no enhancement of symmetry at the quantum level fermion λ of mass m : H in the limit where classical vacuum expectation value → 1 μ 1 2 2 ¯ μ v 0. L = ∂ φ∂μφ − m φ + λ iγ ∂μ − mH λ 2 2 L In the SM, one-loop radiative corrections to the ¯ + yλλφ; mH mL. (4) Higgs mass come from the diagrams of the type where fermions f , gauge fields (W±,Z)and Higgs field H go In this theory, the light scalar mass mL is not a natu- around the loop (see figure 1). The diagrams contribute ral parameter. Smallness of mL cannot be protected by correction to the Higgs mass as any approximate symmetry against perturbative quan- 2 = 2 tum corrections involving heavy fermions in the loops. mHiggs α Pramana – J. Phys. (2016) 87: 37 Page 3 of 9 37 W+_, Z H f ++ H H + H H HH f W+_, Z f H H H H + + H H + H H H H Figure 1. Diagrams showing fermions f , Higgs field H and gauge fields (W±,Z) in the loop. with Naturalness breakdown scale of the SM: As we have 1 seen above, one-loop correction to Higgs boson mass α = (Aλ + Bg2 − CY2), 2 f due to quantum fluctuations of a size characterized 16π 2 = 2 by the scale may be written as: mHiggs α . where A, B, C are numerical constants respectively Square of the vacuum expectation value of the scalar associated with the diagrams with scalar fields, gauge field, and also the masses of vector bosons W ± and fields and the fermions in the loops. Use dimensional Z0 and fermions would obtain similar corrections. For −1 regularization (with minimal subtraction) to write this coupling α ∼ (100) and mHiggs ∼ 100 GeV, if we correction as require that these radiative corrections to this mass do 2 ∼ 2 2 not exceed its value, mHiggs mHiggs,wehave 1 mHiggs m2 ∼ Aλm2 ln Higgs 2 Higgs 2 2 2 16π μ mHiggs (100 GeV) 2 = ∼ = (1000 GeV)2 m2 N α (100)−1 2 2 gauge +Bg m ln 2 gauge μ2 ≡ (1TeV) .