Pramana – J. Phys. (2017) 89:53 © Indian Academy of Sciences DOI 10.1007/s12043-017-1448-2 Hierarchy problem and BSM physics GAUTAM BHATTACHARYYA Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700 064, India E-mail: [email protected] Published online 5 October 2017 Abstract. The ‘hierarchy problem’ plagues the Standard Model of particle physics. The source of this problem is our inability to answer the following question: Why is the Higgs mass so much below the GUT or Planck scale? A brief description about how ‘supersymmetry’ and ‘composite Higgs’ address this problem is given here. Keywords. Higgs boson; supersymmetry; composite Higgs. PACS Nos 14.80.Bn; 11.30.Pb; 12.60.Rc 1. Introduction are talking about large finite corrections arising from -scale states (fermions and bosons) which couple to In quantum field theory, the mass of a spin-zero ele- the Higgs. Even though turns out to be very large, mentary particle is not ‘protected’ from receiving large one can exploit the sign difference between fermion quantum corrections. If the Higgs-like object, discov- and boson loops, and tune the magnitude of the cou- ered by the CMS and ATLAS Collaborations of LHC plings, to keep the net contribution to the Higgs mass at CERN, indeed turns out to be the Higgs boson in 100 GeV range. But even if couplings are adjusted of the Standard Model (SM), then why is its mass in a given order in perturbation theory, the adjustment around 125 GeV, and not close to the maximum pos- is offset in the next order, and so tuning has to be sible energy scale of the theory, e.g. grand unification done order by order in perturbation theory. This is what scale (1016 GeV) or the Planck scale (1019 GeV)? This is called the hierarchy problem. Importantly, as Higgs question constitutes the so-called ‘hierarchy problem’ mass is not protected, the electroweak vacuum expec- [1–4]. Does it mean that the SM is just a good effective tation value (vev) v is also not protected (mh ∼ λv, theory at weak scale, which needs to be supplemented where λ is the quartic coupling), and hence the entire by a more fundamental beyond the Standard Model SM is plagued by the hierarchy problem. To sum up, (BSM) theory at higher scale? Which kinds of BSM we are facing a situation where we have to accept a theory elegantly address the hierarchy problem? In this miraculous cancellation of one part in 1028 in getting 2 = O ( )2 talk, I shall describe what the hierarchy problem is all mh 100 GeV – which is really a big hier- about, and very briefly describe how ‘supersymmetry’ archy. In QED, electron mass is protected by chiral [5] and ‘composite Higgs’ [6], as two broad scenar- symmetry. Similarly, the photon always remains mass- ios, address this problem. I shall avoid details, and only less thanks to gauge symmetry. Can we associate the highlight the essential points relevant in the present con- Higgs boson with a similar symmetry that can protect text. its mass? In the SM, there is no such symmetry. Can this be a guiding principle behind looking for BSM physics? 2. Hierarchy problem 3. Supersymmetry One-loop quantum corrections to the Higgs boson mass 2( )=−( 2 2)/ π 2 from fermion loops go like mh f y f 2 16 Supersymmetry relates matter particles with force par- 2( ) = and from the bosonic (e.g. scalar) loops as mh s ticles, i.e. it relates fermions with bosons, in the sense +(λ2 2)/ π 2 S 16 ,where is the highest scale of the the- that for every known fermion there is a bosonic partner ory. These are not merely calculation of divergences, as and vice versa. No superparticle has been found so far, they can be removed through regularizations. Here we which means that supersymmetry is very badly broken 53 Page 2 of 3 Pramana – J. Phys. (2017) 89:53 (in masses, not in couplings). But even then, the quadrat- The first equality in the above equation is actually a ically growing large contributions to the Higgs mass, near-equality because we have assumed tan β>3, leading to the big hierarchy, get cancelled between loops a choice that substantially reduces the dependence with virtual particles carrying opposite spins (particles on m . Note that m2 has flipped sign in the sec- Hd Hu vs. sparticles). This happens because the couplings in ond near-equality. It receives large corrections from the fermion and boson loops are related. Minimal super- the stop mass at one loop, but the loop suppres- symmetric model has two Higgs doublets and therefore sion factor is quite offset by a large log (of ratio five physical scalar degrees of freedom – three neutral of high scale to weak scale, where the high scale (two CP-even and one CP-odd) and one pair of charged can be the GUT scale). The mass parameter μ is scalars. Without going into the details, I shall mention a supersymmetry preserving coefficient of Hu Hd two generic but crucial features that capture the tension term in the superpotential. It implies a cancellation in supersymmetric model building: between μ and the supersymmetry breaking param- eter (namely, the stop mass) to eventually generate (1) In supersymmetry, the quartic couplings are related MZ = 91 GeV.As the stop mass has been pushed to to gauge couplings. This is why at tree level the TeV range by the requirement mh = 125 GeV, we Higgs mass is at most the Z boson mass. How- μ face a cancellation between mt˜ and to the level ever, significantly large loop corrections are arising of one part in 103. This is what we call the ‘lit- mainly from the large top mass. At one loop, the tle hierarchy’ problem. Although the big hierarchy approximate estimate of the lighter CP-even Higgs has been solved by the very structure of supersym- mass is given by metry, this little hierarchy continues to haunt us. √ This is basically the tension! It arises not so much m2 M2 cos2 2β+3m4 ln(m2/m2)/( 2π 2v2). h Z t t˜ t from the non-observation of the superparticles in (1) colliders, but from the requirement of reproduc- ing 125 GeV mass of the Higgs. Supersymmetric Here tan β = v2/v1, the ratio of the two vevs. If tan β>3 or so, the tree-level contribution max- model building is all about addressing this prob- lem. imizes but it is limited by MZ 91 GeV. So, to reach up to 125 GeV, one needs a large supersym- metry breaking soft mass, more specifically a really ∼ 4. Composite Higgs heavystop(mt˜ 10 TeV). Even though we have not explicitly displayed it, there is a prominent role of trilinear scalar coupling At in determining the Suppose we ask the following question: Can the Higgs Higgs mass. Large value of At (∼ TeV) can gener- be a pseudo-Goldstone boson of some sort, e.g. like ate sizable mixing between the left- and right-type a pion? Here the big hierarchy problem is not there stop squarks, and this mixing can bring down the because the Higgs would be a composite object, and desired value of the average stop mass necessary beyond the compositeness scale (which is not large) to reproduce mh = 125 GeV. Higher loop compu- the Higgs would dissolve. This is precisely why we tation of the Higgs mass further brings down the were never bothered about the Planck scale correction required stop mass (e.g. the conclusion from lead- to the pion mass! Indeed, we could employ the same ing 3-loop computation is that a 3 GeV stop can mechanism of chiral symmetry breaking responsible for reproduce mh = 125 GeV even without the neces- pion mass generation in the case of electroweak break- sity of large left–right scalar mixing. Nevertheless, ing. The problem would be that the W-mass would be the overall message is that we need rather heavy gfπ /2 = 29 MeV – a phenomenological disaster! stop to reproduce 125 GeV for the Higgs mass. A way out is to employ not the standard QCD group (2) One elegant feature of supersymmetry is that one of but a QCD-like group, which was called ‘technicolour’, the (soft) mass-squared parameters (more precisely, whose slow (logarithmic) running creates TeV scale. m2 ) starting from a positive value at high scale Then fπ → f ∼ 1 TeV, which helps to reproduce Hu can become negative near the weak scale under the the correct W-mass. But technicolour models faced influence of large top Yukawa coupling. This neg- many troubles, e.g. unacceptably large flavour changing ative mass-square triggers electroweak breaking. neutral currents. But the real killer was the oblique elec- From the electroweak symmetry breaking condi- troweak S-parameter, namely, the coefficient of W3 − B tions, it follows that quantum mixing. It turns out that S ∼ v2/f 2,where v = 246 GeV is the electroweak vev. In technicolour, 0.5M2 −|μ|2 − M2 −|μ2|+O(1) m2. Z Hu t˜ the strongly interacting sector ‘directly’ participates in (2) electroweak breaking. This is why v ∼ f , and hence, Pramana – J. Phys. (2017) 89:53 Page 3 of 3 53 S ∼ 1. But LEP electroweak precision observables put The elementary fermions may couple linearly with the a limit, S < 0.01. This was a big blow for technicolour. strong sector composite operators – a framework known But we can still employ the technicolour idea by ask- as ‘partial fermion compositeness’. Thus, the Higgs can ing the following question: Can v and f be treated as sense the elementary fermions through composite inter- independent parameters? Suppose we do not allow the mediate states.