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Strong CP problem

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The strong CP problem is a question in <a href="Particle_physics" title="Particle physics">particle physics</a>, which brings up the following quandary: why does <a href="Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a> (QCD) seem to preserve <a href="CP_violation#CP-symmetry" title="CP violation">CP-symmetry</a>?

In particle physics, CP stands for the combination of <a href="C-symmetry" title="C-symmetry">C-symmetry</a> (charge conjugation symmetry) and <a href="Parity_(physics)" title="Parity (physics)">P-symmetry</a> (parity symmetry). According to the current mathematical formulation of quantum chromodynamics, a <a href="CP_violation" title="CP violation">violation of CP-symmetry</a> in <a href="Strong_interaction" title="Strong interaction">strong interactions</a> could occur. However, no violation of the CP-symmetry has ever been seen in any experiment involving only the strong interaction. As there is no known reason in QCD for it to necessarily be conserved, this is a "<a href="Fine-tuning_(physics)" title="Fine-tuning (physics)">fine tuning</a>" problem known as the strong CP problem.

The strong CP problem is sometimes regarded as an <a href="List_of_unsolved_problems_in_physics" title="List of unsolved problems in physics">unsolved problem in physics</a>, and has been referred to as "the most underrated puzzle in all of physics."<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a> There are several proposed solutions to solve the strong CP problem. The most well-known is <a href="Peccei%E2%80%93Quinn_theory" title="Peccei–Quinn theory">Peccei–Quinn theory</a>,<a href="#cite_note-3">[3]</a> involving new <a href="Pseudoscalar" title="Pseudoscalar">pseudoscalar</a> particles called <a href="Axion" title="Axion">axions</a>.

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

CP-symmetry states that physics should be unchanged if particles were swapped with their antiparticles and then left-handed and right-handed particles were also interchanged. This corresponds to performing a charge conjugation transformation and then a parity transformation. The symmetry is known to be broken in the <a href="Standard_Model" title="Standard Model">Standard Model</a> through <a href="Weak_interaction" title="Weak interaction">weak interactions</a>, but it is also expected to be broken through <a href="Strong_interaction" title="Strong interaction">strong interactions</a> which govern <a href="Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a> (QCD), something that has not yet been observed.

To illustrate how the CP violation can come about in QCD, consider a <a href="Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a> with a single massive <a href="Quark" title="Quark">quark</a>.<a href="#cite_note-4">[4]</a> The most general mass term possible for the quark is a complex mass written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle me^{i\theta '\gamma _{5}}}">

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<annotation encoding="application/x-tex">{\displaystyle me^{i\theta '\gamma _{5}}}</annotation>

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</math><img src="./b946fb8f4284e2b017b64fb9924e27c5885e9889.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.909ex; height:3.009ex;" alt="{\displaystyle me^{i\theta '\gamma _{5}}}" loading="lazy"> for some arbitrary phase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta '}">

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<annotation encoding="application/x-tex">{\displaystyle \theta '}</annotation>

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</math><img src="./c02a68dba2972e9cf4faf9656f58da0df8be6ae9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.775ex; height:2.509ex;" alt="{\displaystyle \theta '}" loading="lazy">. In that case the <a href="Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> describing the theory consists of four terms:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+\theta {\frac {g^{2}}{32\pi ^{2}}}F_{\mu \nu }{\tilde {F}}^{\mu \nu }+{\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-me^{i\theta '\gamma _{5}})\psi .}</annotation>

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</math><img src="./49ca3880ac6239a7563c291445056e346b7709c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:57.585ex; height:6.009ex;" alt="{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+\theta {\frac {g^{2}}{32\pi ^{2}}}F_{\mu \nu }{\tilde {F}}^{\mu \nu }+{\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-me^{i\theta '\gamma _{5}})\psi .}" loading="lazy"></dd></dl>

The first and third terms are the CP-symmetric <a href="Kinetic_term" title="Kinetic term">kinetic terms</a> of the <a href="Gauge_theory" title="Gauge theory">gauge</a> and quark fields. The fourth term is the quark mass term which is CP violating for non-zero phases <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta '\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \theta '\neq 0}</annotation>

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</math><img src="./f2330d4e05ed2860c1b719e536fe9176a8a22ae4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.036ex; height:3.009ex;" alt="{\displaystyle \theta '\neq 0}" loading="lazy"> while the second term is the so-called <a href="Theta_vacuum" title="Theta vacuum">θ-term</a> or “vacuum angle”, which also violates CP-symmetry.

Quark fields can always be redefined by performing a chiral transformation by some angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \psi '=e^{i\alpha \gamma _{5}/2}\psi ,\ \ \ \ \ \ {\bar {\psi }}'={\bar {\psi }}e^{i\alpha \gamma _{5}/2},}</annotation>

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</math><img src="./caed939549e85fa731462515aad5a4488c40fbfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.473ex; height:3.176ex;" alt="{\displaystyle \psi '=e^{i\alpha \gamma _{5}/2}\psi ,\ \ \ \ \ \ {\bar {\psi }}'={\bar {\psi }}e^{i\alpha \gamma _{5}/2},}" loading="lazy"></dd></dl>

which changes the complex mass phase by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta '\rightarrow \theta '-\alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \theta '\rightarrow \theta '-\alpha }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \theta \rightarrow \theta +\alpha }</annotation>

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The theory would be CP invariant if one could eliminate both sources of CP violation through such a field redefinition. But this cannot be done unless <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =-\theta '}">

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</math><img src="./7ad77d34a9b15a01f79fcd0a5654e9e3b0a19d5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.772ex; height:2.676ex;" alt="{\displaystyle \theta =-\theta '}" loading="lazy">. This is because even under such field redefinitions, the combination <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta '+\theta \rightarrow (\theta '-\alpha )+(\theta +\alpha )=\theta '+\theta }">

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<annotation encoding="application/x-tex">{\displaystyle \theta '+\theta \rightarrow (\theta '-\alpha )+(\theta +\alpha )=\theta '+\theta }</annotation>

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</math><img src="./b7de9a3d5aed6e5d077751e434fc8aef6e4cb66c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.105ex; height:3.009ex;" alt="{\displaystyle \theta '+\theta \rightarrow (\theta '-\alpha )+(\theta +\alpha )=\theta '+\theta }" loading="lazy"> remains unchanged. For example, the CP violation due to the mass term can be eliminated by picking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\theta '}">

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</math><img src="./4b7f2f26ad07020eb0c5a97e877547ef288cdf63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.361ex; height:2.509ex;" alt="{\displaystyle \alpha =\theta '}" loading="lazy">, but then all the CP violation goes to the θ-term which is now proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\theta }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\theta }}}</annotation>

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</math><img src="./9c601696fb36006323b35d998005eee37a587d9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.676ex;" alt="{\displaystyle {\bar {\theta }}}" loading="lazy">. If instead the θ-term is eliminated through a chiral transformation, then there will be a CP violating complex mass with a phase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\theta }}}">

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</math><img src="./9c601696fb36006323b35d998005eee37a587d9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.676ex;" alt="{\displaystyle {\bar {\theta }}}" loading="lazy">. Practically, it is usually useful to put all the CP violation into the θ-term and thus only deal with real masses.

In the Standard Model where one deals with six quarks whose masses are described by the <a href="Yukawa_interaction" class="mw-redirect" title="Yukawa interaction">Yukawa matrices</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{u}}">

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</math><img src="./8fc2226bb687f3673328384a50aaa34584b3c531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.523ex; height:2.509ex;" alt="{\displaystyle Y_{u}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{d}}">

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</math><img src="./50d7700bea96b0b3fb4f8f95af4883ffc129cfdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.443ex; height:2.509ex;" alt="{\displaystyle Y_{d}}" loading="lazy">, the physical CP violating angle is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\theta }}=\theta -\arg \det(Y_{u}Y_{d})}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\theta }}=\theta -\arg \det(Y_{u}Y_{d})}</annotation>

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</math><img src="./c96b2e6938f0c3232af81792a2f8395bf5d738f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.013ex; height:3.176ex;" alt="{\displaystyle {\bar {\theta }}=\theta -\arg \det(Y_{u}Y_{d})}" loading="lazy">. Since the θ-term has no contributions to perturbation theory, all effects from strong CP violation is entirely non-perturbative. Notably, it gives rise to a <a href="Neutron_electric_dipole_moment" title="Neutron electric dipole moment">neutron electric dipole moment</a><a href="#cite_note-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle d_{N}=(5.2\times 10^{-16}{\text{e}}\cdot {\text{cm}}){\bar {\theta }}.}</annotation>

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Current experimental upper bounds on the dipole moment give an upper bound of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{N}<10^{-26}{\text{e}}\cdot }">

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</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">, so it taking on such a particularly small value is a fine-tuning problem called the strong CP problem.

<div class="mw-heading mw-heading2"><h2 id="Proposed_solutions">Proposed solutions</h2></div>

The strong CP problem is solved automatically if one of the quarks is massless.<a href="#cite_note-7">[7]</a> In that case one can perform a set of chiral transformations on all the massive quark fields to get rid of their complex mass phases and then perform another chiral transformation on the massless quark field to eliminate the residual θ-term without also introducing a complex mass term for that field. This then gets rid of all CP violating terms in the theory. The problem with this solution is that all quarks are known to be massive from experimental matching with <a href="Lattice_QCD" title="Lattice QCD">lattice calculations</a>. Even if one of the quarks was essentially massless to solve the problem, this would in itself just be another fine-tuning problem since there is nothing requiring a quark mass to take on such a small value.

The most popular solution to the problem is through the Peccei–Quinn mechanism.<a href="#cite_note-8">[8]</a> This introduces a new global <a href="Anomaly_(physics)" title="Anomaly (physics)">anomalous</a> symmetry which is then <a href="Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneously broken</a> at low energies, giving rise to a <a href="Goldstone_boson" title="Goldstone boson">pseudo-Goldstone</a> boson called an axion. The axion ground state dynamically forces the theory to be CP-symmetric by setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\theta }}=0}">

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Other less popular proposed solutions exist such as Nelson–Barr models.<a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a> These set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\theta }}=0}">

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</math><img src="./2b7a73ba60497312b6aeadaf7b4265c4591fcaad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.617ex; height:2.676ex;" alt="{\displaystyle {\bar {\theta }}=0}" loading="lazy"> at some high energy scale where CP-symmetry is exact but the symmetry is then spontaneously broken. The Nelson–Barr mechanism is a way of explaining why <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\theta }}}">

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<ul><li><a href="Axion" title="Axion">Axion</a></li>

<li><a href="CP_violation" title="CP violation">CP violation</a></li></ul>

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

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</style><cite id="CITEREFMannel2006" class="citation conference cs1">Mannel, T. (2–8 July 2006). <a rel="nofollow" class="external text" href="https://indico.cern.ch/event/427023/session/6/contribution/43/attachments/912026/1288208/Lancester-Mannel-Proc.pdf">"Theory and Phenomenology of CP Violation"</a> (PDF). Nuclear Physics B. The 7th International Conference on Hyperons, Charm, and Beauty Hadrons (BEACH 2006). Vol. 167. Lancaster: Elsevier. pp. 170–174. <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/2007NuPhS.167..170M">2007NuPhS.167..170M</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%2Fj.nuclphysbps.2006.12.083">10.1016/j.nuclphysbps.2006.12.083</a>. Retrieved 15 Aug 2015.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFPeccei,_R.D.Quinn,_H.R.1977" class="citation journal cs1"><a href="Roberto_Peccei" title="Roberto Peccei">Peccei, R.D.</a>; <a href="Helen_Quinn" title="Helen Quinn">Quinn, H.R.</a> (1977). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/248549883">"CP conservation in the presence of pseudoparticles"</a>. <a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a>. 38 (25): 1440–1443. <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/1977PhRvL..38.1440P">1977PhRvL..38.1440P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.38.1440">10.1103/PhysRevLett.38.1440</a>.</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFWu1991" class="citation conference cs1">Wu, D. (1991). <a rel="nofollow" class="external text" href="https://www.osti.gov/servlets/purl/6260191">A Brief Introduction to the Strong CP Problem</a>. Austin, Texas, United States. SSCL-548.</cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFSchwartz2014" class="citation book cs1">Schwartz, M.D. (2014). "29". Quantum Field Theory and the Standard Model. Cambridge University Press. p. 612. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781107034730</bdi>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFBakerDoyleGeltenbortGreen2006" class="citation journal cs1">Baker, C.A.; Doyle, D.D.; Geltenbort, P.; Green, K.; van der Grinten, M.G.D.; Harris, P.G.; Iaydjiev, P.; Ivanov, S.N.; May, D.J.R. (27 September 2006). "Improved experimental limit on the electric dipole moment of the neutron". Physical Review Letters. 97 (13): 131801. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-ex/0602020">hep-ex/0602020</a>. <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/2006PhRvL..97m1801B">2006PhRvL..97m1801B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.97.131801">10.1103/PhysRevLett.97.131801</a>. <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/17026025">17026025</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119431442">119431442</a>.</cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFHook2019" class="citation journal cs1">Hook, A. (2019-07-22). <a rel="nofollow" class="external text" href="https://pos.sissa.it/333/004/pdf">"TASI Lectures on the Strong CP Problem and Axions"</a>. Proceedings of Science. 333: 004. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1812.02669">1812.02669</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.22323%2F1.333.0004">10.22323/1.333.0004</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119073163">119073163</a>. Retrieved 2021-12-02.</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFPeccei,_R._D.2008" class="citation book cs1">Peccei, R. D. (2008). "The Strong CP Problem and Axions". In Kuster, M.; Raffelt, G.; Beltrán, B. (eds.). Axions: Theory, Cosmology, and Experimental Searches. Lecture Notes in Physics. Vol. 741. pp. 3–17. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-ph/0607268">hep-ph/0607268</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-73518-2_1">10.1007/978-3-540-73518-2_1</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-73517-5</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119482294">119482294</a>.</cite>

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFNelson1984" class="citation journal cs1">Nelson, A. (1984-03-15). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016/0370-2693%2884%2992025-2">"Naturally weak CP violation"</a>. Physics Letters B. 136 (5, 6): 387–391. <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/1984PhLB..136..387N">1984PhLB..136..387N</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%2F0370-2693%2884%2992025-2">10.1016/0370-2693(84)92025-2</a>. Retrieved 2021-12-02.</cite>

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFBarr1984" class="citation journal cs1">Barr, S. M. (1984-04-18). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevLett.53.329">"Solving the Strong CP Problem without the Peccei–Quinn Symmetry"</a>. Phys. Rev. Lett. 53 (4): 329–332. <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/1984PhRvL..53..329B">1984PhRvL..53..329B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.53.329">10.1103/PhysRevLett.53.329</a>. Retrieved 2021-12-02.</cite>

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