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class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D1%81%D0%B8%D0%BC%D0%BF%D1%82%D0%BE%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D1%81%D0%B2%D0%BE%D0%B1%D0%BE%D0%B4%D0%B0" title="Асимптотична свобода – Bulgarian" lang="bg" hreflang="bg" data-title="Асимптотична свобода" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Llibertat_asimpt%C3%B2tica" title="Llibertat asimptòtica – Catalan" lang="ca" hreflang="ca" data-title="Llibertat asimptòtica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Asymptotick%C3%A1_volnost" title="Asymptotická volnost – Czech" lang="cs" hreflang="cs" data-title="Asymptotická volnost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Asymptotische_Freiheit" title="Asymptotische Freiheit – German" lang="de" hreflang="de" data-title="Asymptotische Freiheit" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Libertad_asint%C3%B3tica" title="Libertad asintótica – Spanish" lang="es" hreflang="es" data-title="Libertad asintótica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a 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class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Asimptotska_sloboda" title="Asimptotska sloboda – Croatian" lang="hr" hreflang="hr" data-title="Asimptotska sloboda" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Libert%C3%A0_asintotica" title="Libertà asintotica – Italian" lang="it" hreflang="it" data-title="Libertà asintotica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Asymptotische_vrijheid" title="Asymptotische vrijheid – Dutch" lang="nl" hreflang="nl" data-title="Asymptotische vrijheid" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%BC%B8%E8%BF%91%E7%9A%84%E8%87%AA%E7%94%B1%E6%80%A7" title="漸近的自由性 – Japanese" lang="ja" hreflang="ja" data-title="漸近的自由性" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Asymptotisk_frihet" title="Asymptotisk frihet – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Asymptotisk frihet" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a 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data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D1%81%D0%B8%D0%BC%D0%BF%D1%82%D0%BE%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%81%D0%B2%D0%BE%D0%B1%D0%BE%D0%B4%D0%B0" title="Асимптотическая свобода – Russian" lang="ru" hreflang="ru" data-title="Асимптотическая свобода" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Asimptotik_%C3%B6zg%C3%BCrl%C3%BCk" title="Asimptotik özgürlük – Turkish" lang="tr" hreflang="tr" data-title="Asimptotik özgürlük" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk 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searchaux" style="display:none">Property of gauge theories in particle physics</div> <p>In <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, <b>asymptotic freedom</b> is a property of some <a href="/wiki/Gauge_theory" title="Gauge theory">gauge theories</a> that causes interactions between particles to become <a href="/wiki/Asymptotically" class="mw-redirect" title="Asymptotically">asymptotically</a> weaker as the energy scale increases and the corresponding length scale decreases. (Alternatively, and perhaps contrarily, in applying an <a href="/wiki/S-matrix" title="S-matrix">S-matrix</a>, asymptotically free refers to free particles states in the distant past or the distant future.) </p><p>Asymptotic freedom is a feature of <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a> (QCD), the <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> of the <a href="/wiki/Strong_interaction" title="Strong interaction">strong interaction</a> between <a href="/wiki/Quark" title="Quark">quarks</a> and <a href="/wiki/Gluon" title="Gluon">gluons</a>, the fundamental constituents of nuclear matter. Quarks interact weakly at high energies, allowing <a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbative calculations</a>. At low energies, the interaction becomes strong, leading to the <a href="/wiki/Color_confinement" title="Color confinement">confinement</a> of quarks and gluons within composite <a href="/wiki/Hadron" title="Hadron">hadrons</a>. </p><p>The asymptotic freedom of QCD was discovered in 1973 by <a href="/wiki/David_Gross" title="David Gross">David Gross</a> and <a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Frank Wilczek</a>,<sup id="cite_ref-GrossWilczek_1-0" class="reference"><a href="#cite_note-GrossWilczek-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and independently by <a href="/wiki/David_Politzer" class="mw-redirect" title="David Politzer">David Politzer</a> in the same year.<sup id="cite_ref-Politzer_2-0" class="reference"><a href="#cite_note-Politzer-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For this work all three shared the 2004 <a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a>.<sup id="cite_ref-Nobel_3-0" class="reference"><a href="#cite_note-Nobel-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Discovery">Discovery</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asymptotic_freedom&action=edit&section=1" title="Edit section: Discovery"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Asymptotic freedom in QCD was discovered in 1973 by David Gross and Frank Wilczek,<sup id="cite_ref-GrossWilczek_1-1" class="reference"><a href="#cite_note-GrossWilczek-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and independently by David Politzer in the same year.<sup id="cite_ref-Politzer_2-1" class="reference"><a href="#cite_note-Politzer-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The same phenomenon had previously been observed (in <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> with a charged vector field, by V.S. Vanyashin and M.V. Terent'ev in 1965;<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a> by <a href="/wiki/Iosif_Khriplovich" title="Iosif Khriplovich">Iosif Khriplovich</a> in 1969<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">Gerard 't Hooft</a> in 1972<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-tHooft_7-0" class="reference"><a href="#cite_note-tHooft-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>), but its physical significance was not realized until the work of Gross, Wilczek and Politzer, which was recognized by the 2004 Nobel Prize in Physics.<sup id="cite_ref-Nobel_3-1" class="reference"><a href="#cite_note-Nobel-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Experiments at the <a href="/wiki/Stanford_Linear_Accelerator" class="mw-redirect" title="Stanford Linear Accelerator">Stanford Linear Accelerator</a> showed that inside protons, quarks behaved as if they were free. This was a great surprise, as many believed quarks to be tightly bound by the strong interaction, and so they should rapidly dissipate their motion by strong interaction radiation when they got violently accelerated, much like how electrons emit electromagnetic radiation when accelerated.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>The discovery was instrumental in "rehabilitating" quantum field theory.<sup id="cite_ref-tHooft_7-1" class="reference"><a href="#cite_note-tHooft-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Prior to 1973, many theorists suspected that field theory was fundamentally inconsistent because the interactions become infinitely strong at short distances. This phenomenon is usually called a <a href="/wiki/Landau_pole" title="Landau pole">Landau pole</a>, and it defines the smallest length scale that a theory can describe. This problem was discovered in field theories of interacting scalars and <a href="/wiki/Spinor" title="Spinor">spinors</a>, including quantum electrodynamics (QED), and <a href="/wiki/K%C3%A4ll%C3%A9n%E2%80%93Lehmann_spectral_representation" title="Källén–Lehmann spectral representation">Lehmann positivity</a> led many to suspect that it is unavoidable.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Asymptotically free theories become weak at short distances, there is no Landau pole, and these quantum field theories are believed to be completely consistent down to any length scale. </p><p>Electroweak theory within the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> is not asymptotically free. So a Landau pole exists in the Standard Model. With the Landau pole a problem arises when <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a> is being considered. <a href="/wiki/Quantum_triviality" title="Quantum triviality">Quantum triviality</a> can be used to bound or predict parameters such as the Higgs boson mass. This leads to a predictable Higgs mass in <a href="/wiki/Physics_applications_of_asymptotically_safe_gravity#The_mass_of_the_Higgs_boson" title="Physics applications of asymptotically safe gravity">asymptotic safety</a> scenarios. In other scenarios, interactions are weak so that any inconsistency arises at distances shorter than the <a href="/wiki/Planck_length" class="mw-redirect" title="Planck length">Planck length</a>.<sup id="cite_ref-TrivPurs_10-0" class="reference"><a href="#cite_note-TrivPurs-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Screening_and_antiscreening">Screening and antiscreening</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asymptotic_freedom&action=edit&section=2" title="Edit section: Screening and antiscreening"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Vacuum_polarization.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Vacuum_polarization.svg/200px-Vacuum_polarization.svg.png" decoding="async" width="200" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Vacuum_polarization.svg/300px-Vacuum_polarization.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Vacuum_polarization.svg/400px-Vacuum_polarization.svg.png 2x" data-file-width="144" data-file-height="75" /></a><figcaption>Charge screening in QED</figcaption></figure> <p>The variation in a physical coupling constant under changes of scale can be understood qualitatively as coming from the action of the field on <a href="/wiki/Virtual_particle" title="Virtual particle">virtual particles</a> carrying the relevant charge. The Landau pole behavior of QED (related to <a href="/wiki/Quantum_triviality" title="Quantum triviality">quantum triviality</a>) is a consequence of <i>screening</i> by virtual charged particle–<a href="/wiki/Antiparticle" title="Antiparticle">antiparticle</a> pairs, such as <a href="/wiki/Electron" title="Electron">electron</a>–<a href="/wiki/Positron" title="Positron">positron</a> pairs, in the vacuum. In the vicinity of a charge, the vacuum becomes <i>polarized</i>: virtual particles of opposing charge are attracted to the charge, and virtual particles of like charge are repelled. The net effect is to partially cancel out the field at any finite distance. Getting closer and closer to the central charge, one sees less and less of the effect of the vacuum, and the effective charge increases. </p><p>In QCD the same thing happens with virtual quark-antiquark pairs; they tend to screen the <a href="/wiki/Color_charge" title="Color charge">color charge</a>. However, QCD has an additional wrinkle: its force-carrying particles, the gluons, themselves carry color charge, and in a different manner. Each gluon carries both a color charge and an anti-color magnetic moment. The net effect of polarization of virtual gluons in the vacuum is not to screen the field but to <i>augment</i> it and change its color. This is sometimes called <i>antiscreening</i> (color paramagnetism<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>). Getting closer to a quark diminishes the antiscreening effect of the surrounding virtual gluons, so the contribution of this effect would be to weaken the effective charge with decreasing distance. </p><p>Since the virtual quarks and the virtual gluons contribute opposite effects, which effect wins out depends on the number of different kinds, or <a href="/wiki/Flavor_(particle_physics)" class="mw-redirect" title="Flavor (particle physics)">flavors</a>, of quark. For standard QCD with three colors, as long as there are no more than 16 flavors of quark (not counting the antiquarks separately), antiscreening prevails and the theory is asymptotically free. In fact, there are only 6 known quark flavors. </p> <div class="mw-heading mw-heading2"><h2 id="Calculating_asymptotic_freedom">Calculating asymptotic freedom</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asymptotic_freedom&action=edit&section=3" title="Edit section: Calculating asymptotic freedom"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Asymptotic freedom can be derived by calculating the <a href="/wiki/Beta_function" title="Beta function">beta function</a> describing the variation of the theory's <a href="/wiki/Coupling_constant" title="Coupling constant">coupling constant</a> under the <a href="/wiki/Renormalization_group" title="Renormalization group">renormalization group</a>. For sufficiently short distances or large exchanges of <a href="/wiki/Momentum" title="Momentum">momentum</a> (which probe short-distance behavior, roughly because of the inverse relationship between a quantum's momentum and <a href="/wiki/De_Broglie_wavelength" class="mw-redirect" title="De Broglie wavelength">De Broglie wavelength</a>), an asymptotically free theory is amenable to <a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation theory</a> calculations using <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagrams</a>. Such situations are therefore more theoretically tractable than the long-distance, strong-coupling behavior also often present in such theories, which is thought to produce <a href="/wiki/Color_confinement" title="Color confinement">confinement</a>. </p><p>Calculating the beta-function is a matter of evaluating Feynman diagrams contributing to the interaction of a quark emitting or absorbing a gluon. Essentially, the beta-function describes how the coupling constants vary as one scales the system <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\rightarrow bx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>b</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\rightarrow bx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c2e4172e8d40f58cb98ebf600f802a3d668bab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.271ex; height:2.176ex;" alt="{\displaystyle x\rightarrow bx}"></span>. The calculation can be done using rescaling in position space or momentum space (momentum shell integration). In <a href="/wiki/Non-abelian_gauge_theory" class="mw-redirect" title="Non-abelian gauge theory">non-abelian</a> gauge theories such as QCD, the existence of asymptotic freedom depends on the <a href="/wiki/Gauge_group" class="mw-redirect" title="Gauge group">gauge group</a> and number of <a href="/wiki/Flavor_(particle_physics)" class="mw-redirect" title="Flavor (particle physics)">flavors</a> of interacting particles. To lowest nontrivial order, the beta-function in an SU(N) gauge theory with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0172eb646140c291d0afb8fcd0d550f828d1affd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.531ex; height:2.343ex;" alt="{\displaystyle n_{f}}"></span> kinds of quark-like particle is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{1}(\alpha )={\alpha ^{2} \over \pi }\left(-{11N \over 6}+{n_{f} \over 3}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>π</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>11</mn> <mi>N</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{1}(\alpha )={\alpha ^{2} \over \pi }\left(-{11N \over 6}+{n_{f} \over 3}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa63c0b94cd67b4df4ee68fb4958dc10f109a285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.192ex; height:6.343ex;" alt="{\displaystyle \beta _{1}(\alpha )={\alpha ^{2} \over \pi }\left(-{11N \over 6}+{n_{f} \over 3}\right)}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is the theory's equivalent of the <a href="/wiki/Fine-structure_constant" title="Fine-structure constant">fine-structure constant</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{2}/(4\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{2}/(4\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24688eb47fa3904fa0c956accffba03de135f37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.639ex; height:3.176ex;" alt="{\displaystyle g^{2}/(4\pi )}"></span> in the units favored by particle physicists. If this function is negative, the theory is asymptotically free. For SU(3), one has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=3,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4e9399e6c87b6265497457d9219d8aa812cbb5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.971ex; height:2.509ex;" alt="{\displaystyle N=3,}"></span> and the requirement that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{1}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{1}<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bb5e95ec7eaa52c132df47a87f232a25157e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.631ex; height:2.509ex;" alt="{\displaystyle \beta _{1}<0}"></span> gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{f}<{33 \over 2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>33</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{f}<{33 \over 2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69403c7ce4b934d5061acac9c2e55a349a824486" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.437ex; height:5.176ex;" alt="{\displaystyle n_{f}<{33 \over 2}.}"></span></dd></dl> <p>Thus for SU(3), the <a href="/wiki/Color_charge" title="Color charge">color charge</a> gauge group of QCD, the theory is asymptotically free if there are 16 or fewer flavors of quarks. </p><p>Besides QCD, asymptotic freedom can also be seen in other systems like the nonlinear <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span>-model in 2 dimensions, which has a structure similar to the <a href="/wiki/SU(N)" class="mw-redirect" title="SU(N)">SU(N)</a> invariant Yang–Mills theory in 4 dimensions. </p><p>Finally, one can find theories that are asymptotically free and reduce to the full Standard Model of electromagnetic, weak and strong forces at low enough energies.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asymptotic_freedom&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Asymptotic_safety" class="mw-redirect" title="Asymptotic safety">Asymptotic safety</a></li> <li><a href="/wiki/Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li> <li><a href="/wiki/Quantum_triviality" title="Quantum triviality">Quantum triviality</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asymptotic_freedom&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div 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"Softened Gravity and the Extension of the Standard Model up to Infinite Energy". <i><a href="/wiki/Journal_of_High_Energy_Physics" title="Journal of High Energy Physics">Journal of High Energy Physics</a></i>. <b>2015</b> (2): 137. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1412.2769">1412.2769</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015JHEP...02..137G">2015JHEP...02..137G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FJHEP02%282015%29137">10.1007/JHEP02(2015)137</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6129732">6129732</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+High+Energy+Physics&rft.atitle=Softened+Gravity+and+the+Extension+of+the+Standard+Model+up+to+Infinite+Energy&rft.volume=2015&rft.issue=2&rft.pages=137&rft.date=2015&rft_id=info%3Aarxiv%2F1412.2769&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6129732%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FJHEP02%282015%29137&rft_id=info%3Abibcode%2F2015JHEP...02..137G&rft.au=G.+F.+Giudice&rft.au=G.+Isidori&rft.au=A.+Salvio&rft.au=A.+Strumia&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAsymptotic+freedom" class="Z3988"></span></span> </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFS._Pokorski1987" class="citation book cs1">S. Pokorski (1987). <i>Gauge Field Theories</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-36846-4" title="Special:BookSources/0-521-36846-4"><bdi>0-521-36846-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauge+Field+Theories&rft.pub=Cambridge+University+Press&rft.date=1987&rft.isbn=0-521-36846-4&rft.au=S.+Pokorski&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAsymptotic+freedom" class="Z3988"></span></li></ul> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Asymptotic_freedom&oldid=1254552136">https://en.wikipedia.org/w/index.php?title=Asymptotic_freedom&oldid=1254552136</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Quantum_field_theory" title="Category:Quantum field theory">Quantum field theory</a></li><li><a href="/wiki/Category:Quantum_chromodynamics" title="Category:Quantum chromodynamics">Quantum chromodynamics</a></li><li><a href="/wiki/Category:Renormalization_group" title="Category:Renormalization group">Renormalization group</a></li><li><a href="/wiki/Category:Gauge_theories" title="Category:Gauge theories">Gauge theories</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 31 October 2024, at 15:19<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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