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The strong CP problem is a puzzling question in particle physics: Why does quantum chromodynamics (QCD) seem to preserve CP-symmetry?

In particle physics, CP stands for Charge+Parity or Charge-conjugation Parity symmetry: the combination of charge conjugation symmetry (C) and parity symmetry (P). According to the current mathematical formulation of quantum chromodynamics, a violation of CP-symmetry in strong interactions 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 "fine tuning" problem known as the strong CP problem.

The strong CP problem is sometimes regarded as an unsolved problem in physics, and has been referred to as "the most underrated puzzle in all of physics."^[1]^[2] There are several proposed solutions to solve the strong CP problem. The most well-known is Peccei–Quinn theory,^[3] involving new pseudoscalar particles called axions.

Theory

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 Standard Model through weak interactions, but it is generically also expect it to be broken through strong interactions which govern quantum chromodynamics (QCD), something that has not been found.

To illustrate how the CP violation can come about in QCD, consider a Yang-Mills theory with a single massive quark.^[4] The most general mass term possible for the quark is a complex mass written as $me^{i\theta '\gamma _{5}}$ for some arbitrary phase $\theta '$ . In that case the Lagrangian describing the theory consists of four terms

{\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 .

The first and third terms are the CP-symmetric kinetic terms of the gauge and quark fields. The fourth term is the quark mass term which is CP violating for non-zero phases $\theta '\neq 0$ while the second term is the so-called θ-term, which also violates CP symmetry.

Quark fields can always be redefined by performing a chiral transformation by some angle $\alpha$ as

\psi '=e^{i\alpha \gamma _{5}/2}\psi ,\ \ \ \ \ \ {\bar {\psi }}'={\bar {\psi }}e^{-i\alpha \gamma _{5}/2},

which changes the complex mass phase by $\theta '\rightarrow \theta '-\alpha$ while leaving the kinetic terms unchanged. The transformation also changes the θ-term as $\theta \rightarrow \theta +\alpha$ due to a change in the path integral measure, an effect closely connected to the chiral anomaly.

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 $\theta =-\theta '$ . This is because even under such field redefinitions, the combination $\theta '+\theta \rightarrow (\theta '-\alpha )+(\theta +\alpha )=\theta '+\theta$ remains unchanged. For example, the CP violation due to the mass term can be eliminated by picking $\alpha =\theta '$ , but then all the CP violation goes to the θ-term which is now proportional to ${\bar {\theta }}$ . If instead the 𝜃-term is eliminated through a chiral transformation, then there will be a CP violating complex mass with a phase ${\bar {\theta }}$ . 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 Yukawa matrices $Y_{u}$ and $Y_{d}$ , the physical CP violating angle is ${\bar {\theta }}=\theta -\arg \det(Y_{u}Y_{d})$ . 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 an neutron electric dipole moment^[5]

d_{N}=(5.2\times 10^{-16}{\text{e}}\cdot {\text{cm}}){\bar {\theta }}.

Current experimental upper bounds on the dipole moment give an upper bound of $d_{N}<10^{-26}{\text{e}}\cdot$ cm, which requires ${\bar {\theta }}<10^{-10}$ . The angle ${\bar {\theta }}$ can take any value between zero and $2\pi$ , so it taking on such a particularly small value is a fine-tuning problem called the strong CP problem.

Proposed Solution

The strong CP problem is solved automatically if one of the quarks is massless.^[6] In that case one can perform a set of chiral transformations on all the massive quark field 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 quark masses are known to be massive from experimental matching with lattice calculations. 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.^[7] This introduces a new global anomalous symmetry which is then spontaneously broken at low energies, giving rise to a pseudo-Goldstone boson called an axion. The axion ground state dynamically forces the theory to be CP symmetric by setting ${\bar {\theta }}=0$ . Axions are also considered viable candidates for dark matter and axion-like particles are also predicted string theory.

Other less popular proposed solutions exist such as Nelson-Barr models.^[8]^[9] These set ${\bar {\theta }}=0$ at some high energy scale where CP symmetry is exact but the symmetry is then spontaneously broken at low energies. The tricky part of the models is to account for why ${\bar {\theta }}$ remains small at low energies while the CP breaking phase in the CKM matrix becomes large.

References

^ Mannel, Thomas (2–8 July 2006). "Theory and Phenomenology of CP Violation" (PDF). Nuclear Physics B. The 7th International Conference on Hyperons, Charm, and Beauty Hadrons (BEACH 2006). Vol. 167. Lancaster: Elsevier. pp. 170–174. Bibcode:2007NuPhS.167..170M. doi:10.1016/j.nuclphysbps.2006.12.083. Retrieved 15 Aug 2015.
^ "The 'Strong CP Problem' is the Most Underrated Puzzle in All of Physics".
^ Peccei, Roberto D.; Quinn, Helen R. (1977). "CP conservation in the presence of pseudoparticles". Physical Review Letters. 38 (25): 1440–1443. Bibcode:1977PhRvL..38.1440P. doi:10.1103/PhysRevLett.38.1440.
^ Wu, Dan-di (1991). A Brief Introduction to the Strong CP Problem. Austin, Texas, United States. SSCL-548.
^ Schwartz, Matthew D. "29". Quantum Field Theory and the Standard Model (9 ed.). Cambridge University Press. p. 612. ISBN 9781107034730.
^ Hook, Anson (2019-07-22). "TASI Lectures on the Strong CP Problem and Axions". Proceedings of Science. 333. arXiv:1812.02669. doi:10.22323/1.333.0004. Retrieved 2021-12-02.{{cite journal}}: CS1 maint: unflagged free DOI (link)
^ Peccei, R. D. (2008). "The Strong CP Problem and Axions". In Kuster, Markus; Raffelt, Georg; Beltrán, Berta (eds.). Axions: Theory, Cosmology, and Experimental Searches. Lecture Notes in Physics. Vol. 741. pp. 3–17. arXiv:hep-ph/0607268. doi:10.1007/978-3-540-73518-2_1. ISBN 978-3-540-73517-5. S2CID 119482294.
^ Nelson, Ann (1984-03-15). "Naturally weak CP violation". Physics Letters B. 136 (5, 6): 387–391. doi:10.1016/0370-2693(84)92025-2. Retrieved 2021-12-02.
^ Barr, S. M. (1984-04-18). "Solving the Strong CP Problem without the Peccei-Quinn Symmetry". Phys. Rev. Lett. 53 (4): 329–332. doi:10.1103/PhysRevLett.53.329. Retrieved 2021-12-02.

[1] Mannel, Thomas (2–8 July 2006). "Theory and Phenomenology of CP Violation" (PDF). Nuclear Physics B. The 7th International Conference on Hyperons, Charm, and Beauty Hadrons (BEACH 2006). Vol. 167. Lancaster: Elsevier. pp. 170–174. Bibcode:2007NuPhS.167..170M. doi:10.1016/j.nuclphysbps.2006.12.083. Retrieved 15 Aug 2015.

[2] "The 'Strong CP Problem' is the Most Underrated Puzzle in All of Physics".

[3] Peccei, Roberto D.; Quinn, Helen R. (1977). "CP conservation in the presence of pseudoparticles". Physical Review Letters. 38 (25): 1440–1443. Bibcode:1977PhRvL..38.1440P. doi:10.1103/PhysRevLett.38.1440.

[4] Wu, Dan-di (1991). A Brief Introduction to the Strong CP Problem. Austin, Texas, United States. SSCL-548.

[5] Schwartz, Matthew D. "29". Quantum Field Theory and the Standard Model (9 ed.). Cambridge University Press. p. 612. ISBN 9781107034730.

[6] Hook, Anson (2019-07-22). "TASI Lectures on the Strong CP Problem and Axions". Proceedings of Science. 333. arXiv:1812.02669. doi:10.22323/1.333.0004. Retrieved 2021-12-02.{{cite journal}}: CS1 maint: unflagged free DOI (link)

[7] Peccei, R. D. (2008). "The Strong CP Problem and Axions". In Kuster, Markus; Raffelt, Georg; Beltrán, Berta (eds.). Axions: Theory, Cosmology, and Experimental Searches. Lecture Notes in Physics. Vol. 741. pp. 3–17. arXiv:hep-ph/0607268. doi:10.1007/978-3-540-73518-2_1. ISBN 978-3-540-73517-5. S2CID 119482294.

[8] Nelson, Ann (1984-03-15). "Naturally weak CP violation". Physics Letters B. 136 (5, 6): 387–391. doi:10.1016/0370-2693(84)92025-2. Retrieved 2021-12-02.

[9] Barr, S. M. (1984-04-18). "Solving the Strong CP Problem without the Peccei-Quinn Symmetry". Phys. Rev. Lett. 53 (4): 329–332. doi:10.1103/PhysRevLett.53.329. Retrieved 2021-12-02.

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@@ Line 8: / Line 8: @@
 |volume=167 |pages=170–174 |publisher=Elsevier |conference=The 7th International Conference on Hyperons, Charm, and Beauty Hadrons (BEACH 2006) |date=2–8 July 2006 |location=Lancaster |url=https://indico.cern.ch/event/427023/session/6/contribution/43/attachments/912026/1288208/Lancester-Mannel-Proc.pdf |doi=10.1016/j.nuclphysbps.2006.12.083 |access-date=15 Aug 2015 |bibcode=2007NuPhS.167..170M}}</ref><ref>{{Cite web | url=https://www.forbes.com/sites/startswithabang/2019/11/19/the-strong-cp-problem-is-the-most-underrated-puzzle-in-all-of-physics |title = The 'Strong CP Problem' is the Most Underrated Puzzle in All of Physics}}</ref> There are several proposed solutions to solve the strong CP problem. The most well-known is [[Peccei–Quinn theory]],<ref>{{Cite journal|last1=Peccei|first1=Roberto D.|last2=Quinn|first2=Helen R.|author-link2=Helen Quinn|year=1977|title=''CP'' conservation in the presence of pseudoparticles|url=https://www.researchgate.net/publication/248549883|journal=[[Physical Review Letters]]|volume=38|issue=25|pages=1440–1443|bibcode=1977PhRvL..38.1440P|doi=10.1103/PhysRevLett.38.1440|author-link1=Roberto Peccei}}</ref> involving new [[pseudoscalar]] particles called [[axion]]s.
-==CP violation==
+==Theory==
-{{Main|CP violation}}
+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 [[Standard Model]] through [[weak interaction|weak interactions]], but it is generically also expect it to be broken through [[strong interaction|strong interactions]] which govern [[quantum chromodynamics]] (QCD), something that has not been found.
-CP-symmetry states that the laws of physics should be the same if a particle were interchanged with its antiparticle (C symmetry, as charges of antiparticles are the negative of the corresponding particle), and then the left and right were swapped (P symmetry). P is the parity symmetry, and C is the chiral symmetry.
+To illustrate how the CP violation can come about in QCD, consider a [[Yang-Mills theory]] with a single massive [[quark]].<ref>{{cite conference|url=https://www.osti.gov/servlets/purl/6260191|title=A Brief Introduction to the Strong CP Problem|last1=Wu|first1=Dan-di|date=1991|publisher=|location=Austin, Texas, United States|id=SSCL-548}}</ref> The most general mass term possible for the quark is a complex mass written as <math>m e^{i\theta' \gamma_5}</math> for some arbitrary phase <math>\theta'</math>. In that case the [[Lagrangian]] describing the theory consists of four terms
-Experiments do not indicate any CP violation in the QCD sector. For example, a generic CP violation in the strongly interacting sector would create an [[electric dipole moment]] of the [[neutron]] which would be comparable to 10<sup>−18</sup>&nbsp;[[Elementary charge|e]]·[[Metre|m]] while [[neutron electric dipole moment|the current experimental upper bound]] is roughly one billionth that size.<ref>{{Cite journal |last1=Baker |first1=C.A. |last2=Doyle |first2=D.D. |last3=Geltenbort |first3=P. |last4=Green |first4=K. |last5=van&nbsp;der&nbsp;Grinten |first5=M.G.D. |last6=Harris |first6=P.G. |last7=Iaydjiev |first7=P. |last8=Ivanov |first8=S.N. |last9=May|first9=D.J.R. |date=2006-09-27 |df=dmy-all |title=Improved experimental limit on the electric dipole moment of the neutron |journal=Physical Review Letters |volume=97 |issue=13 |pages=131801 |doi=10.1103/PhysRevLett.97.131801 |pmid=17026025 |arxiv=hep-ex/0602020|s2cid=119431442 }}</ref>
+:<math>
-===How CP can be violated in QCD===
+\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.
-QCD does not violate the CP-symmetry as easily as the [[electroweak theory]]; unlike the electroweak theory in which the gauge fields couple to parity-violating [[chirality (physics)|chiral]] currents, the gluons of QCD couple to vector currents. The absence of any observed violation of CP-symmetry is a problem because there are natural terms in the QCD [[Lagrangian (field theory)|Lagrangian]] that are able to break the CP-symmetry.
+</math>
+The first and third terms are the CP-symmetric [[kinetic term|kinetic terms]] of the [[gauge theory|gauge]] and quark fields. The fourth term is the quark mass term which is CP violating for non-zero phases <math>\theta' \neq 0</math> while the second term is the so-called θ-term, which also violates CP symmetry.
-:<math>{\mathcal L} = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu}-\frac{n_f g^2\theta}{32\pi^2}
-F_{\mu\nu}\tilde F^{\mu\nu}+\bar \psi(i\gamma^\mu D_\mu - m
-e^{i\theta^{\,\prime}\gamma_5})\psi</math> {{clarify|date=June 2019| reason=That looks mostly like terms of the electroweak Lagrangian, and certainly does not match any terms of the Lagrangian given in the QCD article, but F is a derived field, so maybe this F is not the F from the electroweak theory? Also, is that gamma matrix supposed to be in the exponent? If so, it probably warrants a comment.}}
+Quark fields can always be redefined by performing a chiral transformation by some angle <math>\alpha</math> as
-For a nonzero choice of the ''{{mvar|θ}}'' angle and the chiral quark mass phase ''{{mvar|θ′}}'' one expects the CP-symmetry to be violated. If the chiral quark mass phase ''{{mvar|θ′}}'' can be converted to a contribution to the total effective ''{{mvar|{{overline|θ}}}}'' angle, it will have to be explained why this effective angle is extremely small instead of being of order one; the particular value of the angle that must be very close to zero (in this case) is an example of a [[fine-tuning|fine-tuning problem]] in physics. If the phase ''{{mvar|θ′}}'' is absorbed in the gamma-matrices, one has to explain why ''{{mvar|θ}}'' is small, but it will not be unnatural to set it equal to zero.
+:<math>
-If at least one of the [[quarks]] of the standard model were massless, ''{{mvar|θ}}'' would become unobservable; i.e. it would vanish from the theory. However, empirical evidence strongly suggests that none of the quarks are massless and so this solution to the strong CP problem fails.
+\psi' = e^{i\alpha \gamma_5/2}\psi, \ \ \ \ \ \ \bar \psi' = \bar \psi e^{-i\alpha \gamma_5/2},
+</math>
+which changes the complex mass phase by <math>\theta' \rightarrow \theta'-\alpha</math> while leaving the kinetic terms unchanged. The transformation also changes the θ-term as <math>\theta \rightarrow \theta + \alpha</math> due to a change in the [[path integral formulation|path integral]] measure, an effect closely connected to the [[chiral anomaly]].
+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>\theta = -\theta'</math>. This is because even under such field redefinitions, the combination <math>\theta'+ \theta \rightarrow (\theta'-\alpha) + (\theta + \alpha) = \theta'+\theta</math> remains unchanged. For example, the CP violation due to the mass term can be eliminated by picking <math>\alpha = \theta'</math>, but then all the CP violation goes to the θ-term which is now proportional to <math>\bar \theta</math>. If instead the {{theta}}-term is eliminated through a chiral transformation, then there will be a CP violating complex mass with a phase <math>\bar \theta</math>. 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 [[Yukawa interaction|Yukawa matrices]] <math>Y_u</math> and <math>Y_d</math>, the physical CP violating angle is <math>\bar \theta = \theta - \arg \det(Y_u Y_d)</math>. 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 an [[neutron electric dipole moment]]<ref>{{cite book|first=Matthew D.|last=Schwartz|title=Quantum Field Theory and the Standard Model|publisher=Cambridge University Press|chapter=29|edition=9|page=612|isbn=9781107034730}}</ref>
+:<math>
+d_N = (5.2 \times 10^{-16}\text{e}\cdot\text{cm}) \bar \theta.
+</math>
+Current experimental upper bounds on the dipole moment give an upper bound of <math>d_N < 10^{-26} \text{e}\cdot</math>cm, which requires <math>\bar \theta < 10^{-10}</math>. The angle <math>\bar \theta</math> can take any value between zero and <math>2\pi</math>, so it taking on such a particularly small value is a fine-tuning problem called the strong CP problem.
+==Proposed Solution==
+The strong CP problem is solved automatically if one of the quarks is massless.<ref>{{cite journal|last1=Hook|first1=Anson|date=2019-07-22|title=TASI Lectures on the Strong CP Problem and Axions|url=https://pos.sissa.it/333/004/pdf|journal=Proceedings of Science|volume=333|doi=10.22323/1.333.0004|arxiv=1812.02669|access-date=2021-12-02}}</ref> In that case one can perform a set of chiral transformations on all the massive quark field 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 quark masses are known to be massive from experimental matching with [[lattice QCD|lattice calculations]]. 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.<ref>{{Cite book|author=Peccei, R. D. |year=2008 |chapter=The Strong CP Problem and Axions |title=Axions: Theory, Cosmology, and Experimental Searches |editor1-last=Kuster |editor1-first=Markus |editor2-last=Raffelt |editor2-first=Georg |editor3-last=Beltrán |editor3-first=Berta |series=Lecture Notes in Physics |volume=741 |pages=3–17 |arxiv=hep-ph/0607268 |doi=10.1007/978-3-540-73518-2_1 |isbn=978-3-540-73517-5|s2cid=119482294 }}</ref> This introduces a new [[global]] [[anomaly (physics)|anomalous]] symmetry which is then [[spontaneous symmetry breaking|spontaneously broken]] at low energies, giving rise to a [[Goldstone boson|pseudo-Goldstone]] boson called an axion. The axion ground state dynamically forces the theory to be CP symmetric by setting <math>\bar \theta = 0</math>. Axions are also considered viable candidates for [[dark matter]] and axion-like particles are also predicted [[string theory]].
+Other less popular proposed solutions exist such as Nelson-Barr models.<ref>{{cite journal|last=Nelson|first=Ann|date=1984-03-15|title=Naturally weak CP violation|url=https://www.sciencedirect.com/science/article/pii/0370269384920252|journal=Physics Letters B|volume=136|issue=5,6|pages=387-391|doi=10.1016/0370-2693(84)92025-2|pmid=|arxiv=|s2cid=|access-date=2021-12-02}}</ref><ref>{{cite journal|last=Barr|first=S. M.|date=1984-04-18|title=Solving the Strong CP Problem without the Peccei-Quinn Symmetry|url=https://link.aps.org/doi/10.1103/PhysRevLett.53.329|journal=Phys. Rev. Lett.|volume=53|issue=4|pages=329-332|doi=10.1103/PhysRevLett.53.329|pmid=|arxiv=|s2cid=|access-date=2021-12-02}}</ref> These set <math>\bar \theta = 0</math> at some high energy scale where CP symmetry is exact but the symmetry is then spontaneously broken at low energies. The tricky part of the models is to account for why <math>\bar \theta</math> remains small at low energies while the CP breaking phase in the [[Cabibbo–Kobayashi–Maskawa matrix|CKM matrix]] becomes large.
 ==See also==