Constraints from the Renormalisation Group

The bare self-coupling ${\lambda }^0$ of the Higgs Boson requires renormalisation. We obtain the renormalisation group equations (RGE).

For a theory involving a scalar field with quartic self-interactions, the RGE self-coupling $\lambda$ reads

${\mu }^2\frac{d}{d{\mu }^2}\lambda (\mu )={\beta }_{\lambda }(\lambda (\mu ))={\beta }_{\lambda }^{(0)}{\lambda }^2+\mathcal{O}({\lambda }^3),\text{ with }{\beta }_{\lambda }^{(0)}=\frac{3{\lambda }^2}{32{\pi }^2}$.

This defines the $\beta$-function of the running self-coupling, which describes the running of the coupling with respect to the renormalisation scale. The $\beta$-function does not explicitly depend on the renormalisation scale; it depends on it implicitly through the coupling itself. To order ${\lambda }^2$, one obtains

$\ln \left[\frac{{\mu }^2}{{\mu }_0^2}\right]={\int }_{\lambda ({\mu }_0)}^{\lambda (\mu )}\frac{d\lambda }{{\beta }_{\lambda }^{(0)}{\lambda }^2}=\frac{1}{{\beta }_{\lambda }^{(0)}}{\int }_{\lambda ({\mu }_0)}^{\lambda (\mu )}\frac{d\lambda }{{\lambda }^2}=\frac{1}{{\beta }_{\lambda }^{(0)}}\left(\frac{1}{\lambda (\mu )}-\frac{1}{\lambda ({\mu }_0)}\right)$.

In terms of the initial condition $\lambda ({\mu }_0)$,

$\lambda (\mu )=\frac{\lambda ({\mu }_0)}{1-{\beta }_{\lambda }^{(0)}\lambda ({\mu }_0^2)\ln \left(\frac{{\mu }^2}{{\mu }_0^2}\right)}$.

(diagram)

RGE in the Standard Model

In SM, the $\beta$-function does not solely depend on $\lambda$; ${\beta }_{\lambda }={\beta }_{\lambda }(\lambda ,y_t,g,g^{\prime })$ depends on all the other couplings of the theory:

• top Yukawa coupling $y_t$
• $SU(2{)}_L$ gauge coupling $g$
• $U(1{)}_Y$ gauge coupling $g’$

The RGE for the self-coupling $\lambda$ in the SM to order ${\lambda }^2,y_t^4,g^3,g^{\prime 4}$ reads

${\mu }^2\frac{d}{d{\mu }^2}\lambda (\mu )={\beta }_{\lambda }(\lambda (\mu ),y_t(\mu ),g(\mu ),g^{\prime }(\mu ))$

$=\frac{1}{16{\pi }^2}\left[12{\lambda }^2-3y_t^4+6\lambda y_t^2-\frac{3\lambda }{2}(3g^2-g^{\prime 2})+\frac{3}{16}(2g^4+(g^2+g^{\prime 2}{)}^2)\right]+\mathcal{O}({\lambda }^3)$.

Triviality Bound

At large values of $\lambda$,

${\mu }^2\frac{d}{d{\mu }^2}\lambda (\mu )={\beta }_{\lambda }(\lambda )=\frac{3{\lambda }^2}{4{\pi }^2}$.

Then, using the solution of the RGE, we find that the Landau pole of the Higgs self-coupling is located at

${\mu }^{\ast }=v\text{ exp }\left(\frac{2{\pi }^2}{3\lambda (v)}\right)=v\text{ exp }\left(\frac{4{\pi }^2{v}^2}{3m_h^2}\right)$.

Alternatively, we can claim that our theory should remain perturbative up to a given scale ${\mu }^{\ast }$. Then, the Higgs mass should satisfy the bound $m_h\lesssim \sqrt{\frac{4{\pi }^2{v}^2}{3\ln ({\mu }^{\ast }/v)}}$.

This bound is commonly known as the Triviality Bound, since the only way to avoid a Landau pole altogether and to ensure that our theory remains perturbative at all scales is to have $\lambda (v)=0$, i.e., a trivial theory without interactions.

Stability Bounds

The running of $\lambda$ poses another issue. We have assumed that $\lambda >0$ since the potential $V(\Phi )$ is assumed to be bounded from below. This is known as the stability condition. For $\lambda \ll 1$, the SM RGE might lead to $\lambda$ turning negative as the scale $\mu$ changes. Then, we can approximate the $\beta$-function as

${\mu }^2\frac{d}{d{\mu }^2}\lambda (\mu )=-\frac{1}{16{\pi }^2}\left(3y_t^4-\frac{3}{16}(2g^4+(g^2+g^{\prime 2}{)}^2)\right)$.

The solution to the RGE reads

$\lambda (\mu )=\lambda (v)-\frac{1}{16{\pi }^2}\left(3y_t^4-\frac{3}{16}(2g^4+(g^2+g^{\prime 2}{)}^2)\right)\ln \left(\frac{{\mu }^2}{v^2}\right)$.

The stability condition $\lambda (v)>0$ then implies

$m_h^2>\frac{v^2}{8{\pi }^2}\left(3y_t^4-\frac{3}{16}(2g^4+(g^2+g^{\prime 2}{)}^2)\right)\ln \left(\frac{{\mu }^2}{v^2}\right)$.

This gives us a lower bound on the Higgs mass of $m_h\gtrsim 70\text{ GeV}$ for scales $\mu <1\text{ TeV}$ and $m_h\gtrsim 130\text{ GeV}$ for scales $\mu <10\text{ TeV}$.