This is a short review about CP violation and with some comments about the complex scenario of time reversal. The goal is to remember ‘mature’ readers what they have heard before; for the ‘young’ ones it should show the ‘roads’ where one can learn from references in details. Furthermore we have to use tools based on local gauge symmetries. One can see the difference about local vs. discrete symmetries in real world on the Fig. (2), namely scenarios of physics vs. chemistry.

There is a very long history on our planet. It has been the goal to understand fundamental dynamics: first about ‘elements’ and then ‘elementary particles’ in more & more refined versions. Afterwards we have used the practical words of ‘high energy physics’ (HEP) instead. Somewhat recently our community realized that we might barking at the wrong tree; instead we have to think about ‘symmetries’ like local ones starting with QED and later about weak and strong forces.

There is another class of symmetries, namely discrete ones; there are subclasses:

• (i) Parity (P), charge conjugation (C) and time reversal (T);
• (ii) chiral symmetry;
• (iii) flavor symmetry.

Those discrete symmetries are correlated due to dynamics in important ways as discussed below in some details.

Fig. (1). “ To be honest, I never would have invented the wheel if not for Urg's groundbreaking theoretical work with the circle.” [A long time ago I had found this cartoon on an in-flight journal.]

Fig. (2). Running with discrete symmetry.

I focus on CP violation; on the other hand (broken) chiral and flavor symmetries have great impact on CP asymmetries. I assume that the reader knows about basic tools for quantum mechanics & quantum field theories including non-abelian ones, Einstein-Podolsky-Rosen (EPR) correlations from quantum mechanics³ and the impact of symmetries like P/C/T⁴.

Operators P and C are unitary, namely P^†[C^†] = P⁻¹[C⁻¹]. However the situation for the anti-unitary operator T is more subtle; one of the reasons changes initial final states:

(1)

CPT invariance has been assumed as usual for excellent reasons in this article.

On the other hand it can help to understand the underlying dynamics, namely Kramers’ degeneracy:

(2)

i.e., T² = −1 applies to a system with an odd number of the same fermions and thus ‘usable’. It also helps to understand the rules of ‘detailed balance’.

There are more general comments about CP & T asymmetries:

• This article focuses on the weak decays of kaon, charm & beauty mesons and charged lepton τ
• I mention weak decays of top quarks, but also their production in the connection with other states at very high energy collisions like Higgs states [2, 3].
• We have to continue probing CP & T asymmetries in flavor independent transitions like in electric dipole moments (EDM); non-zero number has not been found yet.
• With neutrino oscillation having been established we have a subtle, but wide landscape for New Dynamics (ND) to probe. It needs long time to achieve the goal – but there is a needed ‘price’ to reach the ‘prize’.
• History shows us there was a difference between nuclear and high energy physics; however I think that is not the best road now. In particular in Europe there are groups working between nuclear forces and HEP called Hadrodynamics. We can see the connections between the tools produced in one section and applied to others. It says it with different words: there are excellent reasons to probe fundamental forces at much higher energies – but also to go from accuracy to precision at lower energies with different tools. It seems to me that we are still at the beginning of this latter road.

It is possible that we live on one of many, many universes (or multi-verses). However our one is very special not only about the huge asymmetry of matter vs. anti-matter. The data tell us that baryonic (or known) matter produces around ~ 4.5%, dark matter ~ 26.5 % and vacuum (or dark) energy ~ 69 % in our universe. Those ratios are not very close zero or 100% as you might guess, but sizable; therefore we have to deal with surprizing landscape. Furthermore we have candidates for dark matter (like several versions of Super-Symmetry (=SUSY)). On the other hand we have hardly ‘realistic’ candidates for dark energy; at least I am too old to spend daytime to think about vacuum energy.

There is excellent evidence about the asymmetry of matter vs. anti-matter – namely ‘our existence’ on our earth. It was a real surprise to find that parity P and charge conjugation C are broken in charged weak forces. Our community quickly recovered that ‘τ ’ and ‘θ’ – called then – are the same state: K^± mesons decay to both parity even and odd final states. Furthermore we have neutral ones – and K⁰ – producing two mesons that are differentiated by their lifetimes: K_S and K_L. Therefore K_L was seen as parity odd mesons. Not only P & C violations were found, but also in maximal ways, namely charged weak mesons coupling only to left-chiral quarks. Also ν_L & were found , but not ν_R & for massless neutrinos. It is fine in a simple realization of CPT invariance.

A true revolution happened in 1964: it was found that K_L that usually decays into three pions, can also – rarely – to two pions ⁵. At first it was suggested to introduce non-linear terms into the Schrödinger equation with novel unobserved neutral particle U with K_L → K_S + U → π + U rather than giving up on CP symmetry. More data and more thinking showed we have found CPV in the data. Then Wolfenstein gave a paper about what is called super-weak CPV. In my view it is not even a model; it is a classification for models of CPV. In 1973 Kobayashi & Maskawa gave a published paper [1], where CPV can come from three classes: three (or more) families of quarks or/and charged Higgs states or/and more weak bosons with spin-one couplings to right-chiral bosons. At first some colleagues suggested the source of CPV comes from charged Higgs; afterwards we knew that we need (at least) three families of quarks for hadronic dynamics. Now we know that the CKM matrix produces at least the leading source of the measured CP asymmetries in the decays of kaons and B meson; that is a tested part of the SM.

No CPV has been established (yet) in the dynamics of charm hadrons, baryons in general (except our existence) and in the productions & decays of top quarks (before they can produce top hadrons [2]).

In the SM CP landscape is simple for leptons: when the three neutrinos are massless, one defines their leptonic flavor numbers by their couplings with charged leptons; furthermore e, µ and τ have not shown (yet) CP asymmetries in their decays – except in ... piggyback riding on – K ⁰ oscillations [4].

Very short summary about past experience and prediction for the future:

(1) In the SM CP violation is not always tiny as we found out first in neutral kaons. In the transitions of B mesons they are large, even somewhat close to 100 % [3]. The SM produces at least the leading source for those, but our understanding of the impact of non-perturbative QCD still is limited quantitatively.

(2) CPV was found and established in neutral kaon decays, namely indirect and direct ones in ϵ_K and ϵ^′/ϵ_K, respectively. However the ‘job’ has not been finished yet about fundamental dynamics. When one looks at the triangles from the CKM matrix, one seems the impact of our understanding of ϵ_K (like in Fig. 5 below). I was told there is ‘soon’ a chance that progress in lattice QCD will show the impact also on ϵ^′.

(3) On the other hand the CKM dynamics has nothing to do with huge asymmetries in matter vs. anti-matter. Therefore we still have to think & work about this source.

(4) Measurable CP asymmetries need interferences of at least two amplitudes. Never mind that we have failed to understand matter vs. anti-matter in our universe. The interference can linearly depend on the amplitude of New Dynamics (ND) and thus allows with much more sensitivity.

(5) Asymmetric beams of e⁺e⁻ collisions and new technologies for detectors with precision had entered a new era with the experiments BaBar at Stanford (U.S.A.), Belle at KEK (Japan) and now LHCb at CERN; it will continue with the experiment Belle-II at KEK. It is a real challenge to analyze huge amount of data. It is crucial to measure correlations between different final states – including multi-body FS in charm & beauty decays.

(6) On the theoretical side new tools with more accuracy probe fundamental dynamics including operator product expansion, heavy quark expansion and lattice QCD. While the source of CPV is weak forces, their impact depends on strong forces – i.e., nonperturbative QCD.

(7) Flavor independent CPV have been probed, in particular for EDMs in very different landscapes from elementary leptons to very complex states like nuclei or molecules – and we have to continue.

(8) Based on CPT invariance CPV & T are well connected. Of course one wants to probe CPT invariance. Usually it is assumed that EPR correlations are perfect.

(9) There are two classes of CP asymmetries:

(i) ‘Indirect’ CPV that can happen only on neutral mesons and need oscillations; these CP asymmetries depend on the time of decay; observables are defined by the initial final state (FS): K ⁰ (or K_L), D ⁰, B_d & B_s⁶.

(ii) ‘Direct’ CPV can be seen in the decays of hadrons (and possibly also in some them in production connected with other states). Its impact depends on the FS and does not depend on the time of decay.

(iii) In neutral mesons one sees the interferences with both classes of CPV. Their impact depends on strong final state interaction (FSI) or re-scattering based on quantum theory amplitudes. It can be described in the worlds of hadrons or quarks.

(10) CPT invariance tells us that CPV is described by complex phases. However, it does not mean that all of these produce observables like about fermion fields and in particular about quark ones. One can change the phase of a quark field given CKM matrix element and rotate it away; it will re-appear in other matrix elements; for example: . Kobayashi & Maskawa showed we need three families of quarks to produce CPV and describe with a single complex phase [1] ⁷. In other words: one describes CPV in six triangles with very different patterns; however they give the same area. You might say that ‘maximal’ CPV means a phase is 90°. However such a statement is fallacious as said above.

(11) Penguin diagrams are described in the world of quarks, gluons and weak bosons. Fig. (3) (a) sees Feynman diagrams with gluon & W gauge bosons and also b quarks in the initial state; part (b) describes wave lines for gauge bosons; it is assumed that non-perturbative QCD completes the FS. Sometimes art helps somewhere. There is a real challenge, namely to connect amplitudes in the world of quarks with those in the world of hadrons that are and can be measured.

This is a complex one on several levels. In the collisions of (anti-)baryons at low energies one hardly care about them being bound states of three constituent (anti-)quarks – unless one describes their EDMs, where one discusses the difference between current vs. constituent ones. When one talks about non-leptonic decays of hadrons, it is crucial to use current quarks. We know how to describe inclusive decays; however probing CPT violation the landscapes are much more complex, and we need more subtle tools to describe also multi-body FS. We cannot focus only on two-body FS.

There is a general comment: it is one thing to draw Feynman diagrams, but understanding the underlying forces is another thing; one needs more thinking and uses correlations with other transitions. One shows the impact of penguin diagrams in K → π decays, although loop diagrams are usually suppressed. On the other hand their impact is enhanced by chiral symmetry for two pions FS and somewhat for three pions. However this does not work for many-body FS in the decays of charm or beauty hadrons.

Fig. (3). (a) Diagrams of penguin amplitudes; the picture of the (b) diagram was reproduced from Parity by permission of T. Muta & T. Morozumi.

(12) Usually one compares the predictions from models with the information gotten by fitting best the data. There is a good reason to say that the analyses are model insensitive. However it is only the first (and second) step; in particular when one has a good candidate for a real theory, one has to focus whether these predictions come around within two sigma or so and think & probe correlations with other data. Theoretical uncertainties are systemetic at best; often ‘predictions’ follow the fashion.

It shows the connection of CPV with the violation of T reversal (TV). One might say it is obvious in e^iϕt to reach the same goal going down a different road. On the other hand, the landscape of T reversal is very complex. It depends on its definition. For example, we know that it happens already in classical physics: it is much easier to get ‘down’ than ‘up’ – i.e., the different scenarios of initial and final states. In this article I will discuss TV in fundamental forces.

(13) There is a short comment about ‘oscillations’ vs. ‘mixing’. Of course ‘mixing’ covers more items in dynamics than ‘oscillation’. However I see no reason to use the same word for different regions of dynamics.

· ‘Oscillations’ needs forces that can change the ‘flavor’ by two units. Their impact depends on the time of decay in well-know and measurable way. It depends on the initial neutral decaying hadrons like K ⁰ or B_d & B_s. Oscillation is a much more narrow meaning by focus on indirect CPV; oscillation is a crucial step to probe CPV, but it can happen without CPV – as we know so far about D ⁰ decays.

· I prefer to use the word of ‘mixing’ in narrow situations like s ↔ d about the Cabibbo angle or in general. It shows the connection of quarks with mass states that couple to weak charged bosons leading to the CKM matrix. Likewise for ‘massive’ neutrinos: they couple with charged leptons leading to the Pontecorvo–Maki–Nakagawa–Sakata matrix [5] (and maybe with Dark Matter).

CPV in neutral kaons and B mesons have been established; it depends on our quantitative understanding quark flavor dynamics including non-perturbative QCD. None have been found yet in charm hadrons; so far we have not enough rate to probe for top quarks [3].

One first focuses on the transitions of neutral mesons with richer landscapes. CPV was found in decays. We have many examples, and the future data will show more. We use tools based on quantum mechanics & quantum field theories and measure the correlations between different FS and differentiate the impact of ND and its features sometimes in subtle ways. It needs much more work & analysis. However I will emphasize the informations we get from many-body FS about direct CPV with accuracy and will discuss that about baryon and charged mesons decays below.

If ∆P ≠ 0 forces connect neutral flavor mesons P ⁰ (P⁰ = K ⁰, B_d, B_s, D⁰) and

Therefore mass eigenstates described by linear amplitudes based on CPT [3].

(3)

(4)

are mass & width eigenstates with eigenvalues & their differences ⁸:

(5)

(6)

(7)

(8)

q/p itself is not an observable. One can change the phase of anti-particles: will modify the off-diagonal elements of M & Γ and thus q/p → e^−iξ q/p. However both are invariant and observable in different ways:

· P₁ and P₂ states in general are not orthogonal to each other: .

· This situation can be measured in semi-leptonic rates using CPT invariance with :

(9)

(10)

One probes CPV based on oscillations, however it is independent of time:

(11)

It depends on the initial state with only indirect CPV with ∆P = 2, namely K⁰, B_d, B_s and D⁰ transitions.

· Basically quantum mechanics tell us about the two mass eigenstates P₁ & P₂ using the Schwartz inequality [3, 6] arrive at:

(12)

This inequality is numerically relevant for kaons due to Γ_L ˂ Γ_S 2ΔM_K

(13)

as a very conservative bound: K_L and K_S are close to be odd and even CP states. Does it mean that we are just being lucky with 3m_π < M_K < 4m_π or is a deep reason about these limits?

The landscape is more complex with non-leptonic decays about indirect and direct CPV even with based on quantum field theories with CPT invariance [3]:

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

It is important that and do not depend on the definition of the phases and therefore are observables, while (q/p) and by itself are not.

Here you can see the general situation. In our world we have |q/p| ~ 1 and ∆Γ/Γ ~ 0 (except for ∆Γ(K_L)/(Γ(K_L) + Γ(K_S)), ∆Γ(K_L)/Γ(K_S) = 0.49). ∆Γ = 0 can happen only due to a miracle; however, ∆Γ can be smaller or larger than expected from SM values; the impact of ND can hide in the experimental and/or theoretical uncertainties.

When the FS are even/odd CP eigenstates, one gets:

(25)

For charged P mesons the landscapes look much simpler ⁹ – but not very much in reality:

(26)

(27)

There are several important statements, although they are not always obvious:

· Time depending rates show the impact of indirect vs. direct CPV in neutral mesons. The amplitudes of indirect ones depend in the initial states: K⁰, B_d, B_s and D⁰. These can be probed in two-body FS.

· Direct CPV affect differently FS in the decays of hadrons. It is not enough to understand the dynamics with two-body FS; it is crucial to measure three- and four-body FS with accuracy, but not only as a back-up information.

· The impact of strong re-scattering is crucial as discussed below. It happens in the world of hadrons (and of quarks) as indicated in f_a vs. ; however it is a true challenge to describe them quantitatively with subtle theoretical tools.

It is a rich landscape that shows the connection of HEP, Hadrodynamics, nuclear, atomic & molecule physics.

· We have found large CPV flavor dynamics, but it has nothing to do with the truly huge observed asymmetry in known matter vs. anti-matter of baryons.

· EDMs have been probed in many different situations [27], but none has been found yet despite hard work both on the experimental & theoretical side. Still I see no good reason to give up: future work might tell us the direction for the dynamics to produce that asymmetry.

· QCD faces the challenge to solve the problem with basically zero contributions of the operator and the gateway for ‘traditional’ and ‘novel’ axions.

· There might be a connection of known vs. dark matter.

· A very general statement: to understand fundamental dynamics it needs a lot of time, new tools – and thinking & new ideas.

Direct test of T invariance comes for single particle static transitions. The energy shift of a system due to external small electric field can be described in powers of :

(69)

The linear vector d_i is called the EDM. A non-zero value of d_i show the violation both discrete P and T symmetries. The crucial point is not ‘elementary’, but ‘non-degeneracy’ of the impact of the dynamics. It is a well-known example to compare the neutron d_N with water molecules or ‘dumb-bells’ based on classical forces [3].

In quantum field theory EDMs are described by an operator in the Lagrangian:

(70)

In the SM one gets EDM values for neutrons, deuterons, molecules and also for e, µ & τ ones that are clearly beyond what one can reach. Therefore it is a rich landscape for the existence of ND and its features, if you are patient enough to make the efforts that are needed with thinking about & working on ideas.

The situation is more subtle: QCD can produce CPV in flavor independent transitions, namely EDMs in hadrons. It was realized long time ago that QCD with vector bosons have a problem used ‘U (1)_A problem’ [3]. Let us look at QCD with only one family with u & d quarks. With massless quarks – which is very close to – one might think that QCD possesses a global U (2)_L × U (2)_R symmetry. Indeed the vectorial component U (2)_L+R is conserved even after QM corrections and axial SU (2)_L-R also in subtle ways to give masses to W_µ^± and Z⁰. However about U (1)_L-R? Axial currents are conserved in the classical symmetry due to chiral invariance for massless quarks; however they are not conserved called ‘quantum anomaly’ (or ‘triangle anomaly’) due to one-loop corrections with internal quarks:

(71)

The resolution of the U (1)_L-R due to complex structure of the QCD ‘vacuum’ comes with a price, namely the ‘Strong CP Problem’. The U (1)_L-R & Strong CP is actually intertwined, when one includes the weak dynamics ¹⁶:

(72)

with the observable arg detM. It describes the mixing matrix of U=(t,c,u) and D=(b,s,d) quarks. Photon can couple neutrons with internal virtual protons & pions. One of the two effective pion nucleon operators couple by ordinary QCD, while to other one are due . A guess tells us:

(73)

The limit from the data gives:

(74)

While it is possible or worse ‘accidentally’, but very ‘un-natural’.

4.1. Traditional & Novel Axion Scenarios

Most members of our community agree that one needs organizing principle to produce the required cancellations. The best known tool is some kind of symmetry. A real intriguing ansatz is to assume that a physical quantity usually used as a constant is re-interpret as a dynamics degree of freedom. In this case it was suggested by Peccei & Quinn [28], namely to add the SM a global U (1)_PQ as a Nambu-Goldstone boson which is axial with following properties:

· it is a classical symmetry;

· it is subject to an axial anomaly;

· it is broken spontaneously as well and

· possesses a huge vacuum expectation value (VEV) υ_PQ > υ_EW.

Previously we thought there are two classes, namely (A) ‘visible’ axion with m_a ~ O(1 MeV), and (B) ‘invisible’ axion with m_a < 1 MeV. It seems there is no chance that class (A) axion can exist. ‘Invisible’ axion might be found using coupling of axion with two photons. The name ‘invisible’ is obvious, namely due to the tiny axion mass the lifetime is larger than the age of our universe, and couplings to other fields are so minute that they would not betray their presence under ‘ordinary’ circumstances. The best tool might be by conversion an axion into a photon in a strong magnetic field B [29]:

(75)

It is still probed in our present world (including solar system due to astrophysical indirect information).

Later ‘old’ cosmology enters the ‘scene’: it gave lower bound on the mass:

(76)

Then connections of dark matter suggest stronger bounds:

(77)

Does it mean that the ‘dawn’ of axions goes to the ‘dusk’ ? Maybe the landscape of axion dynamics is even more subtle; namely actually PQ symmetry can be broken not only in QCD anomaly, but also in the UV region in many ways (and ideas) due to connection with gravity, gravitational waves, string theoretical realization of the QCD axion etc. Axions produced in the very early universe, can be part of the Dark Matter (and maybe also in Dark Energy) in the present universe and can be tested experimentally and directly. For example, it was described with more observables in the Refs. [30-32] about the PLANCK & BICEP2 data.

I am not convinced (yet) by some comments; however even if those projects will not be realized, they show the active situation in fundamental physics, which is wonderful in my view: a true ‘Renaissance’ gets from an excellent idea about the impact of symmetries.

In the SM the landscape of CPV in hadrons and leptons are quite different. There the charged leptons and neutrinos are elementary with no original CPV. This century data showed us that neutrinos are not massless due to oscillations. Some of us think that CPV in neutrino oscillations can show the road to understand the huge difference between matter vs. antimatter. It also shows we need a very long time efforts to make progress there.

The SM landscape of leptonic dynamics about CPV is not very complex with massless neutrinos and e, µ and τ transitions. One can see it as not very interesting – or opposite, since there is hardly SM background on the theoretical side.

5.1. τ Cabibbo Suppressed Decays

Present data about CPV in SCS τ decays τ ⁻ → νKS (π...)⁻ show one can compare SM prediction due to well-known oscillation with a difference of 2.9 sigma:

(78)

(79)

one can note the sign. One can probe CPV decays like τ ⁻ → νK ⁻π ⁰, νK⁻π⁺π⁻etc. and think about correlations due to CPT. We have to probe CPV in several FS like τ ⁻ → νK ⁻π ⁰, νK ⁻π⁺π⁻, νK_S π⁻η.

Now available data probe only integrated CP asymmetries. It is important to probe regional CP asymmetries in τ − → ν [S = −1] FS; we have to wait for Belle II (and Super-Tau-Charm Factory if & when it exists). Furthermore one has to compare regional data from τ − → ν [S = 0] FS like τ − → ν π−π ⁰, ν π−η, ν π−π+π−, ν π−π ⁰π ⁰ etc. with accuracy. It is a test of experimental uncertainties; it would be a miracle to show CPV there.

It is important (as pointed out two years ago) to measure the correlations with D+ → K +π+π−/K +K +K − etc. [34]. Furthermore we have to look for regional asymmetries and spin correlations in the pairs of τ +τ −, in particular with polarized e+e− beams if we can use them.

Now we have entered the era where ‘accuracy’ has been changed into ‘precision’ with better tools including much better understanding of strong forces – and the possible connection with dark matter.

Up to now CPV basically have been measured in two-body FS in kaons and B mesons. It is crucial to probe many-body FS in kaons, D_u,d,s and B_u,d,s and in baryons in general. Furthermore we have to use CPT as a tool to connect informations from different FS and regional CPV. EDMs in nuclei & molecules show us a new road for ND even, when it is not connected with the asymmetry in matter vs. anti-matter.

No CPV or TV has been found in leptonic dynamics with small limits so far. However we have to continue with precision, not only to understand those, but have a chance to find the source of the huge asymmetry in baryons vs. anti-baryons. Finally we know that the SM is not enough in our universe due to dark matter existence & neutrino oscillations. Therefore we have to probe CPV in neutrino oscillations, although it is a true challenge where we need long time projects based on HEP, Hadrodynamics & Nuclear Physics and combine their informations.