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The triviality bound

 An upper limit on the Higgs mass can be given from what is called the triviality bound. The simplified Goldstone model (section 2.1.1) having a single scalar is used to illustrate the process. Using the renormalisation group equations as in [20] a running value of the coupling $\lambda$ is obtained

 
$\displaystyle\lambda$(Q) = $\displaystyle{\frac{\lambda(v)}{1-\frac{3}{4\pi^3}\log\frac{Q^2}{v^2}\lambda(v)}}$ (39)

where v is the vacuum expectation value (2.27) and the Higgs mass (at tree level) given as

 
mH = v$\displaystyle\sqrt{2\lambda(v)}$. (40)

It can be seen that if no new physics enters equation (2.81) has to be valid for all values of Q ; but keeping $\lambda$ finite as Q $\rightarrow$ $\infty$ forces $\lambda$(v) to zero. This implies a non-interacting theory called a trivial theory. If instead imagining that the standard model is embedded in a more complete theory having new physics at an energy scale $\Lambda_{NP}^{}$ the requirement can be loosened to something like

 
$\displaystyle\lambda$($\displaystyle\Lambda_{NP}^{}$) $\displaystyle\lesssim$ 1 (41)

thus setting a maximal value on $\lambda$(v) giving an upper limit on the Higgs mass from (2.81) and (2.82). Setting $\Lambda_{NP}^{}$ at the Planck scale mPl $\simeq$ 1019 GeV and requiring that the pertubative approach is valid for all energies gives the low limit mH $\lesssim$ 140 GeV. Defining a lower energy of the cutoff will give a larger allowed range for the Higgs mass. The implementation of lattice theory also makes it possible to loosen the requirement that $\lambda$ is in the pertubative regime for all values of $\lambda$ below $\Lambda_{NP}^{}$ .


  
Figure: The maximal possible Higgs mass with a given scale $\Lambda_{NP}^{}$ for new physics. The dashed line marks mHmax = $\Lambda_{NP}^{}$ .
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Using the full electroweak Lagrangian (2.28) does not change the conclusion from the simplified discussion above, but it is necessary to use it to get better numerical values. To imagine the largest possible Higgs mass the scale of new physics has to be pressed far down. It seems natural that mH $\lesssim$ $\cal$O($\Lambda_{NP}^{}$) as a Higgs with mass above the scale of new physics can not be defined as a standard model Higgs any longer. Just using this requirement sets mH $\lesssim$ 1000 GeV as seen in fig. 2.12 having the value

 
(mHmax)2 = $\displaystyle{\frac{8 \pi^2 v^2}{3 \log\frac{\Lambda_{NP}}{v^2}}}$ (42)

where the top quark coupling is ignored. The line mHmax = $\Lambda_{NP}^{}$ is also plotted.

However, using (2.82) to define mH is wrong since the tree level value will have large radiative corrections for mH $\gg$ v . Using (2.81) to define the Higgs mass instead from $\lambda$(mH) results in no upper limit on the Higgs mass but the result is not reliable since pushing mH to the maximal value will exactly evaluate $\lambda$ in the non-pertubative region to get the value mH . In [28] this problem has been treated with care taking into account the high top quark mass and up to two loop effects. In fig. 2.13, taken from the article, a band is given for the maximal Higgs mass with a given scale of new physics. The width of the band shows the uncertainty in the value of the Higgs mass where the pertubative approach breaks down.

  
Figure 2.13: The Higgs mass where the pertubative approach breaks down including two loop effects. The upper band reflects the uncertainty in the calculation. The lower band is a lower limit on the Higgs mass from stability arguments of the theory and will not be commented on further.
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The work presented in [29] uses lattice gauge theory on the scalar sector of the Standard Model to set an absolute bound on the Higgs mass and not just a limit from a breakdown of perturbation theory. From numerical analysis it is concluded that mH $\lesssim$ 710 GeV thus giving a upper limit similar to the limit from perturbation theory. The theoretical uncertainty on the upper mass is evaluated to $\pm$ 60 GeV. The bounding condition on the new physics scale was that the standard model can describe physics accurately below the double Higgs mass.


next up previous contents
Next: The width of the Up: A theoretical limit on Previous: A theoretical limit on
Ulrik Egede
1/8/1998