open_problems:strong_cp_puzzle
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open_problems:strong_cp_puzzle [2018/03/16 09:20] jakobadmin [Layman] |
open_problems:strong_cp_puzzle [2020/04/10 19:51] (current) 71.46.88.3 [Intuitive] |
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| - | ====== The Strong CP Problem ====== | + | ====== Strong CP Puzzle ====== |
| - | <tabbox Why is it interesting?> | ||
| - | Nothing in QCD forbids the so-called $\theta$-term, which violates CP-symmetry. The strength of the CP violation is proportional to some parameter, called $\theta$. However, so far no CP-violation was ever observed in strong interactions. Hence, the strong CP problem is, why this $\theta$ parameter is so small, or even zero. | ||
| - | <tabbox Layman> | ||
| - | * http://web.mit.edu/physics/news/physicsatmit/physicsatmit_06_sciollafeature.pdf | + | <tabbox Intuitive> |
| - | <tabbox Student> | + | |
| + | The strong CP puzzle is the observation that in the [[models:standard_model|standard model]] nothing forbids that [[models:standard_model:qcd|strong interactions]] violate [[advanced_notions:cp_symmetry|CP symmetry]] but so far such a CP violation by strong interactions has never been observed. | ||
| + | |||
| + | The puzzle is regarded as deep and interesting since upon closer inspection there are possibly two sources of how strong interactions could violate CP symmetry. These two sources come from completely different sectors and thus it's somewhat a miracle that they cancel exactly. | ||
| + | |||
| + | One possible source comes from the nontrivial structure of the [[advanced_notions:quantum_field_theory:qcd_vacuum|ground state of the theory of strong interactions]] (QCD). | ||
| + | |||
| + | The other possible source comes from the [[advanced_notions:quantum_field_theory:anomalies|chiral anomaly]]. Thus contribution to the total CP violation of strong interactions depends on the couplings of the quark fields to the [[advanced_notions:symmetry_breaking:higgs_mechanism|Higgs field]]. The Higgs sector and the ground state of the theory of strong interactions are, in the standard model, completely different parts of the theory and there is no connection between them besides that they both are possible sources for CP violation in strong interactions. | ||
| + | |||
| + | |||
| + | |||
| + | <tabbox Concrete> | ||
| + | Nothing forbids that we add a new term to the Lagrangian | ||
| + | |||
| + | $$\mathcal{L}_{QCD} = \ldots + | ||
| + | \theta \frac{g^2}{32 \pi^2} G_{\mu \nu}^a \tilde{G}^{a, \mu \nu}, | ||
| + | $$ | ||
| + | |||
| + | $$\bar\theta=\theta_{\rm QCD}-\theta_{\rm F}<10^{-10} ,$$ | ||
| + | |||
| + | where | ||
| + | |||
| + | * $\theta_{\rm QCD}$ is the coefficient of the term $\alpha_s^2/ 8 \pi \, G\tilde G $ in the Lagrangian that we get when we consider [[models:standard_model:qcd|QCD]] alone. | ||
| + | * $\theta_{\rm F} = \arg \det M_u M_d$ is an additional contribution to the effective complete theta parameter $\theta$ that results when we consider QCD in the presence of fermions. The term enters since we have to diagonalize the mass matrices to switch to the mass basis and this diagonalization process necessarily involves a chiral rotation. | ||
| + | * The experimental bound $\bar\theta<10^{-10}$ comes from the observation that the dipole moment of the neutron is tiny: $|d_n| \le 3.6 \times 10^{-26} e \, {\rm cm}$ ([[https://arxiv.org/abs/1509.04411|Source]]). | ||
| + | |||
| + | Take note that | ||
| + | $$ \tilde{F}^{\mu\nu} F_{\mu\nu} \propto \vec E \cdot \vec B \propto \partial_t \vec A\cdot \vec B $$ | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | For a gentle introduction see the series "Demystifying the QCD vacuum": | ||
| - http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-1/ | - http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-1/ | ||
| - http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-2/ | - http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-2/ | ||
| Line 13: | Line 43: | ||
| - http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-5-anomalies-and-the-strong-cp-problem/ | - http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-5-anomalies-and-the-strong-cp-problem/ | ||
| - | <note tip><blockquote>Since one can show that no physical | + | ---- |
| + | |||
| + | * A great introduction to and review of the strong CP problem is [[http://inspirehep.net/record/244939?ln=en|The Strong CP Problem Revisited]] by Hai-Yang Cheng | ||
| + | * Another good summary can be found in chapter 4 of [[https://link.springer.com/content/pdf/10.1007/3-540-12301-6_24.pdf|GAUGE THEORIES IN THREE DIMENSIONS (= AT FINITE TEMPERATURE)]] by R. Jackiw. Especially he discusses the two possible ways to see how $\theta$ emerges. Either via the functional approach or via the Hamiltonian approach. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | <blockquote>Since one can show that no physical | ||
| operator can connect states with different values of | operator can connect states with different values of | ||
| 6), it follows that $\theta$ labels completely disjoint sectors of | 6), it follows that $\theta$ labels completely disjoint sectors of | ||
| Line 23: | Line 60: | ||
| values of observables in a pure state, not in | values of observables in a pure state, not in | ||
| a mixture, one must project onto a single $\theta$ sector.<cite>[[http://einrichtungen.ph.tum.de/T30f/lec/TopicQFT/groospisarskiyaffe.pdf|QCD and instantons at finite temperature]] by David J. Gross et. | a mixture, one must project onto a single $\theta$ sector.<cite>[[http://einrichtungen.ph.tum.de/T30f/lec/TopicQFT/groospisarskiyaffe.pdf|QCD and instantons at finite temperature]] by David J. Gross et. | ||
| - | al.</cite></blockquote></note> | + | al.</cite></blockquote> |
| - | * A great introduction to and review of the strong CP problem is [[http://inspirehep.net/record/244939?ln=en|The Strong CP Problem Revisited]] by Hai-Yang Cheng | ||
| - | * Another good summary can be found in chapter 4 of [[https://link.springer.com/content/pdf/10.1007/3-540-12301-6_24.pdf|GAUGE THEORIES IN THREE DIMENSIONS (= AT FINITE TEMPERATURE)]] by R. Jackiw. Especially he discusses the two possible ways to see how $\theta$ emerges. Either via the functional approach or via the Hamiltonian approach. | ||
| - | $$ \tilde{F}^{\mu\nu} F_{\mu\nu} \propto \vec E \cdot \vec B \propto \partial_t \vec A\cdot \vec B $$ | ||
| + | <tabbox Abstract> | ||
| - | + | <blockquote>Unfortunately the relation between $d_n$, | |
| - | <tabbox Researcher> | + | $\bar{\theta}$ is not known to better than an order of magnitude |
| - | + | (see e.g. [5][6] and references therein), so the constraint on $\bar{\theta}$ quoted in (1.1) — obtained | |
| - | <blockquote> | + | using naive dimensional analysis (NDA) — suffers from a large uncertainty.<cite>https://arxiv.org/pdf/1412.3805.pdf</cite></blockquote> |
| - | + | ||
| - | **It may strike the reader (and has struck the author) that | + | |
| - | enormous theoretical superstructures are here being erected upon | + | |
| - | a very narrow foundation. | + | |
| - | The whole superstructure of axions could | + | |
| - | be made obsolete if a good alternative approach to the problem of | + | |
| - | strong CP invariance were found.** | + | |
| - | Even if this does happen, I am | + | |
| - | confident that techniques for dealing with approximate Nambu- | + | |
| - | Goldstone bosons and their phenomenological (including cosmological) | + | |
| - | implications will be of enduring interest -- so I won't be completely wasting your time.<cite>THE U(1) PROBLEM: | + | |
| - | INSTANTONS, AXIONS, AND FAMILONS by | + | |
| - | Frank Wilczek </cite></blockquote> | + | |
| - | + | <tabbox Why is it interesting?> | |
| - | + | Nothing in QCD forbids the so-called $\theta$-term, which violates CP-symmetry. The strength of the CP violation is proportional to some parameter, called $\theta$. However, so far no CP-violation was ever observed in strong interactions. Hence, the strong CP problem is, why this $\theta$ parameter is so small, or even zero. | |
| - | <tabbox Examples> | + | |
| - | + | ||
| - | --> Example1# | + | |
| - | + | ||
| - | + | ||
| - | <-- | + | |
| - | + | ||
| - | --> Example2:# | + | |
| - | + | ||
| - | + | ||
| - | <-- | + | |
| <tabbox FAQ> | <tabbox FAQ> | ||
| Line 74: | Line 84: | ||
| If at least one quark were massless, the Lagrangian would have a chiral symmetry $u \to e^{i\alpha \gamma_5} \Psi$. Such a chiral rotation shifts the $\theta$ parameter: $\theta \to \theta + c \alpha$. When at least one quark is massless, we can perform one completely arbitrary chiral rotation and hence eliminate $\theta \to 0$. | If at least one quark were massless, the Lagrangian would have a chiral symmetry $u \to e^{i\alpha \gamma_5} \Psi$. Such a chiral rotation shifts the $\theta$ parameter: $\theta \to \theta + c \alpha$. When at least one quark is massless, we can perform one completely arbitrary chiral rotation and hence eliminate $\theta \to 0$. | ||
| - | However, when all quarks do have mass, we can't choose the phase $\alpha$ arbitrary. Instead, it is fixed by the requirement that we must get a real mass for the quarks. Hence $\alpha$ is not an arbitrary parameter and $\theta$ can not be shifted to zero. Of course, it is possible that $\alpha$ happens to be exactly the right value to set $\theta$ to zero. However, there is no reason why this should be the case and this is the strong CP problem. Why should this seemingly unrelated, but fixed parameter $\alpha$ exactly cancel the bare parameter $\theta$? | + | However, when all quarks do have mass, we can't choose the phase $\alpha$ arbitrarily. Instead, it is fixed by the requirement that we must get a real mass for the quarks. Hence $\alpha$ is not an arbitrary parameter and $\theta$ can not be shifted to zero. Of course, it is possible that $\alpha$ happens to be exactly the right value to set $\theta$ to zero. However, there is no reason why this should be the case and this is the strong CP problem. Why should this seemingly unrelated, but fixed parameter $\alpha$ exactly cancel the bare parameter $\theta$? |
| A chiral rotation shifts $\theta$, because a chiral rotation is basically a change of basis, i.e. a relabelling of the quark fields. Such changes of basis are accompanied by a Jacobian. In this case, the Jacobian is proportional to $F \tilde F$ and thus shifts $\theta$. | A chiral rotation shifts $\theta$, because a chiral rotation is basically a change of basis, i.e. a relabelling of the quark fields. Such changes of basis are accompanied by a Jacobian. In this case, the Jacobian is proportional to $F \tilde F$ and thus shifts $\theta$. | ||
| Line 95: | Line 105: | ||
| Source: https://physics.stackexchange.com/questions/27462/why-is-there-no-theta-angle-topological-term-for-the-weak-interactions/27463 | Source: https://physics.stackexchange.com/questions/27462/why-is-there-no-theta-angle-topological-term-for-the-weak-interactions/27463 | ||
| + | |||
| + | See also https://physics.stackexchange.com/questions/176424/why-is-the-strong-cp-term-theta-fracg232-pi2-g-mu-nua-tildeg?noredirect=1&lq=1 | ||
| <-- | <-- | ||
| Line 155: | Line 167: | ||
| by Alvarez-Gaume et. al.</cite></blockquote> | by Alvarez-Gaume et. al.</cite></blockquote> | ||
| <-- | <-- | ||
| - | <tabbox History> | + | |
| </tabbox> | </tabbox> | ||
open_problems/strong_cp_puzzle.1521188409.txt.gz · Last modified: 2018/03/16 08:20 (external edit)
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