# Dangerously irrelevant operator

In statistical mechanics and quantum field theory, a dangerously irrelevant operator (or dangerous irrelevant operator) is an operator which is irrelevant at a renormalization group fixed point, yet affects the infrared (IR) physics significantly (e.g. because the vacuum expectation value (VEV) of some field depends sensitively upon the coefficient of this operator).

## Critical phenomena

In the theory of critical phenomena, free energy of a system near the critical point depends analytically on the coefficients of generic (not dangerous) irrelevant operators, while the dependence on the coefficients of dangerously irrelevant operators is non-analytic ( p. 49).
The presence of dangerously irrelevant operators leads to the violation of the hyperscaling relation $\alpha =2-d\nu$ between the critical exponents $\alpha$ and $\nu$ in $d$ dimensions. The simplest example ( p. 93) is the critical point of the Ising ferromagnet in $d\geq 4$ dimensions, which is a gaussian theory (free massless scalar $\phi$), but the leading irrelevant perturbation $\phi ^{4}$ is dangerously irrelevant. Another example occurs for the Ising model with random-field disorder, where the fixed point occurs at zero temperature, and the temperature perturbation is dangerously irrelevant ( p. 164).

## Quantum field theory

Let us suppose there is a field $\phi$ with a potential depending upon two parameters, $a$ and $b$.
    $V\left(\phi \right)=-a\phi ^{\alpha }+b\phi ^{\beta }$
Let us also suppose that $a$ is positive and nonzero and $\beta$ > $\alpha$. If $b$ is zero, there is no stable equilibrium. If the scaling dimension of $\phi$ is $c$, then the scaling dimension of $b$ is $d-\beta c$ 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
    $\langle \phi \rangle =\left({\frac {a\alpha }{b\beta }}\right)^{\frac {1}{\beta -\alpha }}=\left({\frac {a\alpha }{\beta }}\right)^{\frac {1}{\beta -\alpha }}b^{-{\frac {1}{\beta -\alpha }}}$.
depends very sensitively upon $b$, at least for small values of $b$. Because the nature of infrared physics also depends upon the VEV, it looks very different even for a tiny change in $b$ not because the physics in the vicinity of $\phi =0$ changes much — it hardly changes at all — but because the VEV we are expanding about has changed enormously.
Supersymmetric models with a modulus can often have dangerously irrelevant parameters.

## Other uses of the term

Consider a renormalization group (RG) flow triggered at short distances by a relevant perturbation of an ultra-violet (UV) fixed point, and flowing at long distances to an infra-red (IR) fixed point. It may be possible (e.g. in perturbation theory) to monitor how dimensions of UV operators change along the RG flow. In such a situation, one sometimes calls dangerously irrelevant a UV operator whose scaling dimension, while irrelevant at short distances: $\Delta _{\rm {UV}}>d$ , receives a negative correction along a renormalization group flow, so that the operator becomes relevant at long distances: $\Delta _{\rm {IR}}<d$. This usage of the term is different from the one originally introduced in statistical physics.