Dangerously irrelevant operator: Difference between revisions

Revision as of 08:12, 29 November 2008

Latest revision as of 11:57, 4 April 2024

(38 intermediate revisions by 26 users not shown)

Line 1:

{{short description|Class of operators in quantum field theory}}

In [[statistical mechanics]] and [[quantum field theory]], a '''dangerously irrelevant operator''' (or '''dangerous irrelevant operator''') is an [[operator (mathematics)|operator]] which is irrelevant, butat 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 dangerouslycoefficient irrelevantof this operator).

== ExampleCritical 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 (<ref name=":0">{{Cite book|last=Cardy|first=John|title=Scaling and Renormalization in Statistical Physics|publisher=Cambridge University Press|year=1996}}</ref> p.&nbsp;49).

The presence of dangerously irrelevant operators leads to the violation of the hyperscaling relation <math>\alpha=2-d\nu</math> between the critical exponents <math>\alpha</math> and <math>\nu</math> in <math>d</math> dimensions. The simplest example (<ref name=":0" /> p.&nbsp;93) is the critical point of the Ising ferromagnet in <math>d\ge4</math> dimensions, which is a gaussian theory (free massless scalar <math>\phi</math>), but the leading irrelevant perturbation <math>\phi^4</math> 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 (<ref name=":0" /> p.&nbsp;164).

==Quantum field theory==

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 VEV

:<math>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}}</math>.

:<math>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}}</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>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>VEV</math> we are expanding about has changed enormously.

depends very sensitively upon <math>b</math>, at least for small values of <math>b</math>. Because the nature of the IRinfrared physics also depends upon the <math>VEV</math>, the IR physicsit 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>VEV</math> we are expanding about has changed enormously.

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

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

[[Category:Renormalization group]]

== Other uses of the term ==

{{phys-stub}}

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<ref>{{Cite journal|last=Gukov|first=Sergei|date=2016-01-05|title=Counting RG flows|journal=Journal of High Energy Physics|language=en|volume=2016|issue=1|pages=20|arxiv=1503.01474|doi=10.1007/JHEP01(2016)020|bibcode=2016JHEP...01..020G |s2cid=23582290|issn=1029-8479}}</ref> calls dangerously irrelevant a UV operator whose scaling dimension, while irrelevant at short distances: <math>\Delta_{\rm UV}>d</math> , receives a negative correction along a renormalization group flow, so that the operator becomes relevant at long distances: <math>\Delta_{\rm IR}<d</math>. This usage of the term is different from the one originally introduced in statistical physics.<ref>{{Cite journal|last1=Amit|first1=Daniel J|last2=Peliti|first2=Luca|date=1982|title=On dangerous irrelevant operators|journal=Annals of Physics|language=en|volume=140|issue=2|pages=207–231|doi=10.1016/0003-4916(82)90159-2|bibcode=1982AnPhy.140..207A }}</ref>

==References==

{{Reflist}}

{{DEFAULTSORT:Dangerously Irrelevant Operator}}

[[Category:Renormalization group]]

{{physquantum-stub}}