Dangerously irrelevant operator: Difference between revisions

Revision as of 17:53, 2 April 2010

Revision as of 06:40, 22 August 2010

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==Example==

Let us suppose there is a field <math>\phi</math> with a [[potential]] depending upon two parameters, <math>''a</math>'' and <math>''b</math>''.

:<math>V\left(\phi\right)=-a \phi^\alpha + b\phi^\beta</math>

Let us also suppose that <math>''a</math>'' is positive and nonzero and <math>\beta</math> > <math>\alpha</math>. If <math>''b</math>'' is zero, there is no stable equilibrium. If the [[scaling dimension]] of <math>\phi</math> is <math>''c</math>'', then the scaling dimension of <math>''b</math>'' is <math>d-\beta c</math> where <math>''d</math>'' is the number of dimensions. It is clear that if the scaling dimension of <math>''b</math>'' is negative, <math>''b</math>'' is an irrelevant parameter. However, the crucial point is, that the [[Vacuum expectation value|<math>\mathrm{VEV}</math>]]

:<math>\langle\phi\rangle=\left(\frac{a\alpha}{b\beta}\right)^{\frac{1}{\beta-\alpha}}=\left(\frac{a\alpha}{\beta}\right)^{\frac{1}{\beta-\alpha}}b^{-\frac{1}{\beta-\alpha}}</math>.

depends very sensitively upon <math>''b</math>'', at least for small values of <math>''b</math>''. Because the nature of the IR physics also depends upon the <math>\mathrm{VEV}</math>, the IR physics looks very different even for a tiny change in <math>''b</math>'' not because the physics in the vicinity of <math>\phi=0</math> changes much — it hardly changes at all — but because the <math>\mathrm{VEV}</math> we are expanding about has changed enormously.

In [[supersymmetry|supersymmetric]] models with a [[moduli space|modulus]], we can often have dangerously irrelevant parameters.

{{DEFAULTSORT:Dangerously Irrelevant Operator}}