Naturalness

Let $\Lambda$ denote the energy scale associated with the onset of Beyond the Standard Model (BSM) effects, i.e., the emergence of new heavy particles or new interactions. Let us consider a scenario where there is a huge difference between $\Lambda$ and $v$. The Lagrangian takes the general form

$\mathcal{L}=\text{const }+{\Lambda }^2{\mathcal{L}}_2+\Lambda {\mathcal{L}}_3+{\mathcal{L}}_4+\dots$.

Here, ${\mathcal{L}}_d$ are field operators of mass dimension $d$. The Lagrangian consists of at least the following terms:

• ${\mathcal{L}}_2$ comprises the Higgs mass term ${\mu }^2{\Phi }^{†}\Phi$
• ${\mathcal{L}}_3$ comprises the Dirac fermion mass terms like $y_d{d}_R{\Phi }^{†}{Q}_L$, which appear in the SM only after EW SSB
• ${\mathcal{L}}_4$ comprises all dimension-four gauge and Higgs interactions.

These terms describe the renormalisable part. The baseline power for the BSM scale $\Lambda$ is fixed by the mass dimension of the operator, since mass dimension of $\mathcal{L}$ is chosen to be four by construction. The Dirac fermion mass terms emerge only due to the breaking of the electroweak symmetry, associated with the scale $v$. Thus, the baseline power for ${\mathcal{L}}_3$ actually reads $\mathcal{L}\supset v{\mathcal{L}}_3$.

The so-called Hierarchy problem now arises from the fact that there is no symmetry to change the baseline power multiplying the scalar mass term ${\mathcal{L}}_2$ to something other than ${\Lambda }^2$. Since this term appears in the Higgs potential, it should also fix $v$, and hence the masses of the Dirac fermions, and the W and Z bosons.

One of the raison d’être of particle physics is the unnatural size of the Higgs vev (in the absence of BSM), found up to scales of 2 TeV.