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<title>Ultraviolet fixed point</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Ultraviolet fixed point</span></span>

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<p>In a <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, one may calculate an effective or <a href="Running_coupling_constant" class="mw-redirect" title="Running coupling constant">running coupling constant</a> that defines the coupling of the theory measured at a given momentum scale. One example of such a coupling constant is the <a href="Electric_charge" title="Electric charge">electric charge</a>.

</p><p>In approximate calculations in several quantum field theories, notably <a href="Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> and theories of the <a href="Higgs_particle" class="mw-redirect" title="Higgs particle">Higgs particle</a>, the running coupling appears to become infinite at a finite momentum scale. This is sometimes called the <i><a href="Landau_pole" title="Landau pole">Landau pole</a> problem</i>.

</p><p>It is not known whether the appearance of these inconsistencies is an artifact of the approximation, or a real fundamental problem in the theory. However, the problem can be avoided if an ultraviolet or <b>UV fixed point</b> appears in the theory. A quantum field theory has a UV fixed point if its <a href="Renormalization_group_flow" class="mw-redirect" title="Renormalization group flow">renormalization group flow</a> approaches a <a href="Renormalization_group" title="Renormalization group">fixed point</a> in the ultraviolet (i.e. short length scale/large energy) limit.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This is related to zeroes of the <a href="Beta_function_(physics)" title="Beta function (physics)">beta-function</a> appearing in the <a href="Callan%E2%80%93Symanzik_equation" title="Callan–Symanzik equation">Callan–Symanzik equation</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The large length scale/small energy limit counterpart is the <a href="Infrared_fixed_point" title="Infrared fixed point">infrared fixed point</a>.

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<div class="mw-heading mw-heading2"><h2 id="Specific_cases_and_details">Specific cases and details</h2></div>

<p>Among other things, it means that a theory possessing a UV fixed point may not be an <a href="Effective_field_theory" title="Effective field theory">effective field theory</a>, because it is well-defined at arbitrarily small distance scales. At the UV fixed point itself, the theory can behave as a <a href="Conformal_field_theory" title="Conformal field theory">conformal field theory</a>.

</p><p>The converse statement, that any <a href="Quantum_field_theory" title="Quantum field theory">QFT</a> which is valid at all distance scales (i.e. isn't an effective field theory) has a UV fixed point is false. See, for example, <a href="Cascading_gauge_theory" title="Cascading gauge theory">cascading gauge theory</a>.

</p><p><a href="Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative quantum field theories</a> have a UV cutoff even though they are not effective field theories.

</p><p>Physicists distinguish between trivial and nontrivial fixed points. If a UV fixed point is <a href="Trivial_fixed_point" class="mw-redirect" title="Trivial fixed point">trivial</a> (generally known as Gaussian fixed point), the theory is said to be <a href="Asymptotic_freedom" title="Asymptotic freedom">asymptotically free</a>. On the other hand, a scenario, where a non-Gaussian (i.e. nontrivial) fixed point is approached in the UV limit, is referred to as <a href="Asymptotic_safety" class="mw-redirect" title="Asymptotic safety">asymptotic safety</a>.<sup id="cite_ref-NR06_3-0" class="reference"><a href="#cite_note-NR06-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Asymptotically safe theories may be well defined at all scales despite being <i><a href="Nonrenormalizable" class="mw-redirect" title="Nonrenormalizable">nonrenormalizable</a></i> in perturbative sense (according to the <a href="Classical_scaling_dimension" class="mw-redirect" title="Classical scaling dimension">classical scaling dimensions</a>).

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<div class="mw-heading mw-heading2"><h2 id="Asymptotic_safety_scenario_in_quantum_gravity">Asymptotic safety scenario in quantum gravity</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Asymptotic_safety_in_quantum_gravity" title="Asymptotic safety in quantum gravity">Asymptotic safety in quantum gravity</a></div>

<p><a href="Steven_Weinberg" title="Steven Weinberg">Steven Weinberg</a> has proposed that the problematic <a href="Ultraviolet_divergence" title="Ultraviolet divergence">UV divergences</a> appearing in <a href="Quantum_gravity" title="Quantum gravity">quantum theories of gravity</a> may be cured by means of a nontrivial UV fixed point.<sup id="cite_ref-Weinberg_4-0" class="reference"><a href="#cite_note-Weinberg-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Such an <a href="Asymptotic_safety_in_quantum_gravity" title="Asymptotic safety in quantum gravity">asymptotically safe</a> theory is renormalizable in a nonperturbative sense, and due to the fixed point physical quantities are free from divergences. As yet, a general proof for the existence of the fixed point is still lacking, but there is mounting evidence for this scenario.<sup id="cite_ref-NR06_3-1" class="reference"><a href="#cite_note-NR06-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<ul><li><a href="Ultraviolet_divergence" title="Ultraviolet divergence">Ultraviolet divergence</a></li>

<li><a href="Landau_pole" title="Landau pole">Landau pole</a></li>

<li><a href="Quantum_triviality" title="Quantum triviality">Quantum triviality</a></li>

<li><a href="Asymptotic_safety_in_quantum_gravity" title="Asymptotic safety in quantum gravity">Asymptotic safety in quantum gravity</a></li>

<li><a href="Asymptotic_freedom" title="Asymptotic freedom">Asymptotic freedom</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFWilsonKogut,_John_B.1974" class="citation journal cs1">Wilson, Kenneth G.; Kogut, John B. (1974). "The renormalization group and the ε expansion". <i>Physics Reports</i>. <b>12</b> (2): <span class="nowrap">75–</span>199. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974PhR....12...75W">1974PhR....12...75W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-1573%2874%2990023-4">10.1016/0370-1573(74)90023-4</a>.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFZinn-Justin2002" class="citation book cs1">Zinn-Justin, Jean (2002). <i>Quantum Field Theory and Critical Phenomena</i>. Oxford University Press.</cite></span>

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<li id="cite_note-NR06-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-NR06_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-NR06_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFNiedermaierReuter2006" class="citation journal cs1">Niedermaier, Max; Reuter, Martin (2006). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5256001">"The Asymptotic Safety Scenario in Quantum Gravity"</a>. <i>Living Rev. Relativ</i>. <b>9</b> (1): 5. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006LRR.....9....5N">2006LRR.....9....5N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.12942%2Flrr-2006-5">10.12942/lrr-2006-5</a></span>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5256001">5256001</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28179875">28179875</a>.</cite></span>

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<li id="cite_note-Weinberg-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Weinberg_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFWeinberg1979" class="citation book cs1">Weinberg, Steven (1979). "Ultraviolet divergences in quantum theories of gravitation". In Hawking, S.W.; Israel, W. (eds.). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/generalrelativit00isra"><i>General Relativity: An Einstein centenary survey</i></a></span>. Cambridge University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/generalrelativit00isra/page/n805">790</a>–831. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521222853</bdi>.</cite></span>

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