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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 is
obtained

(Q) =
 (39)

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

m_{H} = 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 finite as
Q forces (v) to
zero. This implies a noninteracting 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
the requirement can be loosened to something like
thus setting a maximal value on (v) giving an upper limit on
the Higgs mass from (2.81) and (2.82).
Setting
at the Planck scale
m_{Pl} 10^{19} GeV and requiring that the pertubative approach is valid for
all energies gives the low limit
m_{H} 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 is in the
pertubative regime for all values of below
.
Figure:
The maximal possible Higgs mass with a given scale
for new physics. The dashed line marks
m_{H}^{max} = .

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
m_{H} O() 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
m_{H} 1000 GeV as seen in
fig. 2.12 having the value

(m_{H}^{max})^{2} =
 (42)

where the top quark coupling is ignored. The line
m_{H}^{max} = is also plotted.
However, using (2.82) to define m_{H} is wrong since the
tree level value will have large radiative corrections for m_{H} v . Using (2.81) to define the Higgs mass instead from
(m_{H}) results in no upper limit on the Higgs mass but the
result is not reliable since pushing m_{H} to the maximal value will
exactly evaluate in the nonpertubative region to get the
value m_{H} . 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.

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
m_{H} 710 GeV thus giving a upper limit similar to the limit from
perturbation theory. The theoretical uncertainty on the upper mass is
evaluated to 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: The width of the
Up: A theoretical limit on
Previous: A theoretical limit on
Ulrik Egede
1/8/1998