Editing Hierarchy problem (section)

==== Supersymmetry ====
Some physicists believe that one may solve the hierarchy problem via [[supersymmetry]]. Supersymmetry can explain how a tiny Higgs mass can be protected from quantum corrections. Supersymmetry removes the power-law divergences of the radiative corrections to the Higgs mass and solves the hierarchy problem as long as the supersymmetric particles are light enough to satisfy the [[Riccardo Barbieri|Barbieri]]–[[Gian Francesco Giudice|Giudice]] criterion.<ref>{{cite journal |last1=Barbieri |first1=R. |last2=Giudice |first2=G. F. |year=1988 |title=Upper Bounds on Supersymmetric Particle Masses |url=https://cds.cern.ch/record/180560 |journal=Nuclear Physics B |volume=306 |issue=1 |page=63 |bibcode=1988NuPhB.306...63B |doi=10.1016/0550-3213(88)90171-X}}</ref> This still leaves open the [[mu problem]], however. The tenets of supersymmetry are being tested at the [[Large Hadron Collider|LHC]], although no evidence has been found so far for supersymmetry.

Each particle that couples to the Higgs field has an associated [[Yukawa coupling]] <math display="inline">\lambda_f</math>. The coupling with the Higgs field for fermions gives an interaction term <math display="inline">\mathcal{L}_{\mathrm{Yukawa}}=-\lambda_f\bar{\psi}H\psi</math>, with <math display="inline">\psi</math> being the [[Dirac field]] and <math display="inline">H</math> the [[Higgs field]]. Also, the mass of a fermion is proportional to its Yukawa coupling, meaning that the Higgs boson will couple most to the most massive particle. This means that the most significant corrections to the Higgs mass will originate from the heaviest particles, most prominently the top quark. By applying the [[Feynman diagram#Feynman rules|Feynman rules]], one gets the quantum corrections to the Higgs mass squared from a fermion to be:

<math display="block">\Delta m_{\rm H}^{2} = - \frac{\left|\lambda_{f} \right|^2}{8\pi^2} [\Lambda_{\mathrm{UV}}^2+ \dots].</math>

The <math display="inline">\Lambda_{\mathrm{UV}}</math> is called the ultraviolet cutoff and is the scale up to which the Standard Model is valid. If we take this scale to be the Planck scale, then we have the quadratically diverging Lagrangian. However, suppose there existed two complex scalars (taken to be spin 0) such that:

<math display="block">\lambda_S= \left|\lambda_f\right|^2</math>

(the couplings to the Higgs are exactly the same).

Then by the Feynman rules, the correction (from both scalars) is:

<math display="block">\Delta m_{\rm H}^{2} = 2 \times \frac{\lambda_{S}}{16\pi^2} [\Lambda_{\mathrm{UV}}^2+ \dots].</math>

(Note that the contribution here is positive. This is because of the spin-statistics theorem, which means that fermions will have a negative contribution and bosons a positive contribution. This fact is exploited.)

This gives a total contribution to the Higgs mass to be zero if we include both the fermionic and bosonic particles. [[Supersymmetry]] is an extension of this that creates 'superpartners' for all Standard Model particles.<ref>{{cite book |last=Martin |first=Stephen P. |title=Perspectives on Supersymmetry |year=1998 |isbn=978-981-02-3553-6 |series=Advanced Series on Directions in High Energy Physics |volume=18 |pages=1–98 |chapter=A Supersymmetry Primer |publisher=World Scientific |doi=10.1142/9789812839657_0001 |arxiv=hep-ph/9709356 |s2cid=118973381}}</ref>