Dangerously irrelevant operator: Difference between revisions

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:<math>V\left(\phi\right)=-a \phi^\alpha + b\phi^\beta</math>
 
Let us also suppose that ''a'' is positive and nonzero and <math>\beta</math> > <math>\alpha</math>. If ''b'' is zero, there is no stable equilibrium. If the [[scaling dimension]] of <math>\phi</math> is ''c'', then the scaling dimension of ''b'' is <math>d-\beta c</math> where ''d'' is the number of dimensions. It is clear that if the scaling dimension of ''b'' is negative, ''b'' is an irrelevant parameter. However, the crucial point is, that the VEV
 
:<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>.