Editing Renormalization group (section)

=== Conformal symmetry ===
Conformal symmetry is associated with the vanishing of the beta function. This can occur naturally if a coupling constant is attracted, by running, toward a ''fixed point'' at which ''β''(''g'') = 0. In QCD, the fixed point occurs at short distances where ''g'' → 0 and is called a ([[Quantum triviality|trivial]]) [[ultraviolet fixed point]]. For heavy quarks, such as the [[top quark]], the coupling to the mass-giving [[Higgs boson]] runs toward a fixed non-zero (non-trivial) [[infrared fixed point]], first predicted by Pendleton and Ross (1981),<ref>{{cite journal |first1=Brian |last1=Pendleton |first2=Graham |last2=Ross |title=Mass and mixing angle predictions from infrared fixed points |journal=Physics Letters B |volume=98 |issue=4 |year=1981 |pages=291–294 |doi=10.1016/0370-2693(81)90017-4
|bibcode=1981PhLB...98..291P }}</ref> and [[C. T. Hill]].<ref>{{cite journal |first=Christopher T. |last=Hill |author-link=C. T. Hill |title=Quark and lepton masses from renormalization group fixed points |journal=Physical Review D |volume=24 |issue=3 |year=1981 |pages=691–703 |doi=10.1103/PhysRevD.24.691|bibcode=1981PhRvD..24..691H }}</ref> 
The top quark Yukawa coupling lies slightly below the infrared fixed point of the Standard Model suggesting the possibility of additional new physics, such as sequential heavy Higgs bosons.{{citation needed|date=December 2022}}

In [[string theory]], conformal invariance of the string world-sheet is a fundamental symmetry: ''β'' = 0 is a requirement. Here, ''β'' is a function of the geometry of the space-time in which the string moves. This determines the space-time dimensionality of the string theory and enforces Einstein's equations of [[general relativity]] on the geometry. The RG is of fundamental importance to string theory and theories of [[grand unification]].

It is also the modern key idea underlying [[critical phenomena]] in condensed matter physics.<ref>{{Cite journal |last=Shankar |first=R. |doi=10.1103/RevModPhys.66.129 |title=Renormalization-group approach to interacting fermions |journal=Reviews of Modern Physics |volume=66 |issue=1 |pages=129–192 |year=1994 |arxiv=cond-mat/9307009 |bibcode=1994RvMP...66..129S}} (For nonsubscribers see {{cite journal |title= Renormalization-group approach to interacting fermions|arxiv = cond-mat/9307009|doi = 10.1103/RevModPhys.66.129|last =  Shankar|first = R. |journal = Reviews of Modern Physics|year = 1993|volume = 66|issue = 1|pages = 129–192|bibcode = 1994RvMP...66..129S}}.)</ref> Indeed, the RG has become one of the most important tools of modern physics.<ref>{{cite journal |first1=L.Ts. |last1=Adzhemyan |first2=T.L. |last2=Kim |first3=M.V. |last3=Kompaniets |first4=V.K. |last4=Sazonov |title=Renormalization group in the infinite-dimensional turbulence: determination of the RG-functions without renormalization constants |journal=Nanosystems: Physics, Chemistry, Mathematics |date=August 2015 |volume=6 |issue=4 |page=461|doi=10.17586/2220-8054-2015-6-4-461-469 |doi-access=free }}</ref> It is often used in combination with the [[Monte Carlo method]].<ref name="CallawayPetronzio1984">{{cite journal |last1=Callaway |first1=David J.E. |last2=Petronzio |first2=Roberto |title=Determination of critical points and flow diagrams by Monte Carlo renormalization group methods |journal=Physics Letters B |volume=139 |issue=3 |year=1984 |pages=189–194 |issn=0370-2693 |doi=10.1016/0370-2693(84)91242-5 |bibcode=1984PhLB..139..189C |url=https://cds.cern.ch/record/149868}}</ref>