The Demon-Haunted World

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/The_Demon-Haunted_World 

 

The Demon-Haunted World: Science as a Candle in the Dark

Cover of the first edition
Author	Carl Sagan
Language	English
Subjects	Scientific skepticism Science Philosophy
Publisher	Random House
Publication date	1995/1997
Publication place	United States
Media type	Print (hardcover and paperback)
Pages	457
ISBN	0-345-40946-9
OCLC	32855551
Dewey Decimal	001.9 20
LC Class	Q175 .S215 1995
Preceded by	Pale Blue Dot
Followed by	Billions and Billions

The Demon-Haunted World: Science as a Candle in the Dark is a 1995 book by the astrophysicist Carl Sagan. (Four of the 25 chapters were written with Ann Druyan). In it, Sagan aims to explain the scientific method to laypeople and to encourage people to learn critical and skeptical thinking. He explains methods to help distinguish between ideas that are considered valid science and those that can be considered pseudoscience. Sagan states that when new ideas are offered for consideration, they should be tested by means of skeptical thinking and should stand up to rigorous questioning.

Themes

Sagan explains that science is not just a body of knowledge, but is a way of thinking. Sagan shows how scientific thinking is both imaginative and disciplined, bringing humans to an understanding of how the universe is, rather than how they wish to perceive it. He says that science works much better than any other system because it has a "built-in error-correcting machine". Superstition and pseudoscience get in the way of the ability of many laypersons to appreciate the beauty and benefits of science. Skeptical thinking allows people to construct, understand, reason, and recognize valid and invalid arguments. Wherever possible, there must be independent validation of the concepts whose truth should be proved. He states that reason and logic would succeed once the truth were known. Conclusions emerge from premises, and the acceptability of the premises should not be discounted or accepted because of bias.

Dragon in my garage

See also: Falsifiability

As an example of skeptical thinking, Sagan offers a story concerning a fire-breathing dragon who lives in his garage. When he persuades a rational, open-minded visitor to meet the dragon, the visitor remarks that they are unable to see the creature. Sagan replies that he "neglected to mention that she's an invisible dragon". The visitor suggests spreading flour on the floor so that the creature's footprints might be seen, which Sagan says is a good idea, "but this dragon floats in the air". When the visitor considers using an infrared camera to view the creature's invisible fire, Sagan explains that her fire is heatless. He continues to counter every proposed physical test with a reason why the test will not work.

Sagan concludes by asking: "Now what's the difference between an invisible, incorporeal, floating dragon who spits heatless fire and no dragon at all? If there's no way to disprove my contention, no conceivable experiment that would count against it, what does it mean to say that my dragon exists? Your inability to invalidate my hypothesis is not at all the same thing as proving it true."

Continuing with concepts relevant to the 'dragon in my garage' story, Sagan writes about a patient of John Mack who claimed to have scars on her body which were from encounters with aliens. Sagan writes that if the patient is asked what her scars look like, she is unable to show them because, unfortunately, they are located in the private areas of her body.

Baloney detection kit

Sagan presents a set of tools for skeptical thinking that he calls the "baloney detection kit". Skeptical thinking consists both of constructing a reasoned argument and recognizing a fallacious or fraudulent one. In order to identify a fallacious argument, Sagan suggests employing such tools as independent confirmation of facts, debate, development of different hypotheses, quantification, the use of Occam's razor, and the possibility of falsification. Sagan's "baloney detection kit" also provides tools for detecting "the most common fallacies of logic and rhetoric", such as argument from authority and statistics of small numbers. Through these tools, Sagan argues the benefits of a critical mind and the self-correcting nature of science can take place.

Sagan provides nine tools as the first part of this kit.

1. There must be independent confirmation of the facts given when possible.
2. Encourage debate on the evidence from all points of view.
3. Realize that an argument from authority is not always reliable. Sagan supports this by telling us that 'authorities" have made mistakes in the past and they will again in the future.
4. Consider more than one hypothesis. Sagan adds to this by telling us that we must think of the argument from all angles and think all the ways it can be explained or disproved. The hypothesis that then still hasn't been disproved has a much higher chance of being correct.
5. Try your best to not purely stick to a hypothesis that is your own and become biased. Sagan tells us to compare our own hypothesis with others to see if we can find reasons to reject our own hypothesis.
6. Quantify. Sagan tells us that if whatever we are trying to explain has numerical value or quantitative data related to it, then we'll be much more able to compete against other hypotheses.
7. If there is a chain of argument, every link in that chain must be correct.
8. The use of Occam's razor, which tells us to choose the hypothesis that is simpler and requires the least amount of assumptions.
9. Ask if a given hypothesis can be falsified. Sagan tells us that if a hypothesis cannot be tested or falsified then it is not worth considering.

Sagan suggests that with the use of this "baloney detection kit" it is easier to critically think and find the truth.

Logical fallacies

There is a second part to the kit. This consists of twenty logical fallacies that one must not commit when offering up a new claim.

1. Ad hominem. An arguer attacks the opposing arguer and not the actual argument.
2. Argument from authority. Someone expects another to immediately believe that a person of authority or higher knowledge is correct.
3. Argument from adverse consequences. Someone says that something must be done a certain way or else there will be adverse consequences.
4. Appeal to ignorance. One argues a claim in that whatever has not been proved false must be true, and vice versa.
5. Special pleading. An arguer responds to a deeply complex or rhetorical question or statement by, usually, saying "oh you don't understand how so and so works."
6. Begging the question. An arguer assumes the answer and makes a claim such as, this happened because of that, or, this needs to happen in order for that to happen.
7. Observational selection. Someone talks about how great something is by explaining all of the positive aspects of it while purposely not mentioning any of the negative aspects.
8. Statistics of small numbers. Someone argues something by giving the statistics in small numbers, which isn't very reliable.
9. Misunderstanding of the nature of statistics. Someone misinterprets statistics given to them.
10. Fallacy of inconsistency. An arguer is very inconsistent in their claims.
11. Non sequitur. This is Latin for "it doesn't follow." A claim is made that doesn't make much sense, such as "Our nation will prevail because God is great."
12. Post hoc ergo propter hoc. Latin for "it happened after, so it was caused by." An arguer claims that something happened because of a past event when really it probably didn't.
13. Meaningless question. Someone asks a question that has no real meaning or doesn't add to the argument at all.
14. The excluded middle. An arguer only considers or mentions the two opposite extremes of the conversation and excludes the aspects in between the two extremes.
15. Short-term vs. long-term. A subset of the excluded middle, but so important it was pulled it out for special attention.
16. Slippery slope, related to excluded middle (e.g., If we allow abortion in the first weeks of pregnancy, it will be impossible to prevent the killing of a full-term infant. Or, conversely: If the state prohibits…)
17. Confusion of correlation and causation. The latter causes the former
18. Straw man. Caricaturing a position to make it easier to attack. This is also a short-term/long-term fallacy
19. Suppressed evidence, or half-truth.
20. Weasel word. Talleyrand said: "An important art of politicians is to find new names for institutions which under old names have become odious to the public." 

Sagan provides a skeptical analysis of several examples of what he refers to as superstition, fraud, and pseudoscience such as witches, UFOs, ESP, and faith healing. He is critical of organized religion.

In a 2020 interview for Skeptical Inquirer, when Sagan's wife Ann Druyan was asked about the origin of the phrase "baloney detection kit", she said that

It didn't really come from Carl. It actually came from a friend of mine named Arthur Felberbaum who died about forty years ago. He and Carl and I once sat down for dinner together. His politics were very left wing, so Carl and Arthur and I were trying to find common ground so that we could have a really good dinner together. And at one point, Arthur said, "Carl, it's just that I dream that every one of us would have a baloney detection kit in our head." And that's where that idea came from.

Misuse of science

Sagan indicates that science can be misused. Thus, he is highly critical of Edward Teller, the "father of the hydrogen bomb", and Teller's influence on politics, and contrasts his stance to that of Linus Pauling and other scientists who took moral positions.

Sagan also discusses the misuse of science in representation. He relates to the depiction of the mad scientist character in children's TV shows and is critical of this occurrence. Sagan suggests an addition of scientific television programs, many of which would take a look at believed hoaxes of the past and encourage viewers to engage in critical thinking to better represent science on popular television.

Misuse of psychiatric authority

Sagan indicates that therapists can contribute to the growth of pseudoscience or the infusion of "false stories". He is critical of John Mack and his support of abduction cases, which were represented in his patients.

Sagan writes about the story of Paul Ingram. Ingram's daughter reported that her father had sexually abused her. He was told that "sex offenders often repressed memories of their crimes." Ingram was eventually able to have a foggy visualization of the claimed events, and he suggested that perhaps "a demon might be responsible." Sagan describes how once Ingram started remembering events, so did several other individuals and family members. A "memory recovery" technique was performed on Ingram, and he confessed to the crimes. A medical examination was done on his daughter, where none of the scars she described were actually found. Sagan writes that Ingram later tried to plead innocence once "away from his daughters, his police colleagues, and his pastor."

Hoaxes

Hoaxes have played a valuable role in the history of science by revealing the flaws in our thinking and helping us advance our critical thinking skills. One of Sagan’s examples is the "Carlos hoax" by James Randi that revealed flaws in reporting by news media. Carlos was described as an ancient spirit that supposedly possessed José Alvarez and provided Alvarez with advanced knowledge about the universe. Many news outlets assumed this was true and reported it as such, which spread misinformation.

Sagan also cites crop circles as hoaxes.

Reception and legacy

The book was a New York Times bestseller. The contemporary skeptical movement considers it an important book. The Demon-Haunted World has been criticized (in Smithsonian magazine and The New York Times) for not incorporating certain information relevant to the items he discusses in his book. The Smithsonian article by Paul Trachtman argues that Sagan relates issues of government choices and declining scientific thinking skills to pseudoscience topics like astrology and faith healing but ignores other issues that may be causing governmental bodies and other individuals to turn away from science. One such issue is consequences of pouring governmental money into cancer research. Trachtman writes, "it is not because of such beliefs that Congress now approaches the NIH budget with an ax. In fact, billions of dollars spent on years of research in the war on cancer have spawned growing professional bureaucracies and diminishing medical benefits." Trachtman argues that Sagan does not include problems like growing bureaucracies and diminishing medical benefits as reasons for a lack of scientific attention. In his review for the New York Times, James Gorman also argues for an unaddressed issue in Sagan's book, saying Sagan fails to emphasize the idea that scientists should take a more active role in reaching science to the public, while he does mention the failures of the education system to do so.

The review in the Smithsonian magazine and a review in the New York Review of Books provide a range of opinions on Sagan's attitude towards religious ideas. Per the New York Review article, "when it comes to the Supreme Extraterrestrial he is rather circumspect." The Smithsonian article suggests Sagan was very clear about his religious beliefs in the book, for he "splits his universe in two, into science and irrationality." The Smithsonian goes on to say that Sagan's defined religious views fall within the area of an untestable claim, a type of claim he argues against in The Demon-Haunted World.

The article in the New York Review also claims that Sagan includes something in The Demon-Haunted World which he also is arguing against in that same text. The article mentions how Sagan discusses a natural predisposition people have towards science; but, the article says, "He does not tell us how he used the scientific method to discover the "embedded" human proclivity for science." Sagan heavily discusses the importance of using the scientific method in his book, and this article claims he strays away from his own message by not including a description of his use of the scientific method on this topic.

An article in the Los Angeles Times describe Sagan's book positively. The Los Angeles Times describes Sagan's book as "a manifesto for clear thought", with the main issue being the length of eight chapters. This lengthy discussion is also addressed in the archived New York Times article, which relates to the book as having areas which repeat themselves.

The Demon-Haunted World has been defined in more current sources as still relevant. An article in The Guardian, 2012, suggests the still current relevance of The Demon-Haunted World. Another article from The Verge in 2017 also supports this relevance. The latter article mentions a disconnect between what is proven by science to be the best answer and what is chosen to be done by governmental bodies. Carl Sagan covers this concept as a prominent issue in his book (1995), and this article outlines it as a problem still occurring in 2017.

Relevance

The Demon-Haunted World provides many arguments about our society that will always remain relevant. Sagan mainly argues for increased education among younger populations regarding critical thinking and scientific literacy. First, he explains that scientific literacy and critical thinking are essential for democracy, especially in the United States. Democracy allows people to share their beliefs, but with this come many different perspectives and arguments. Critical thinking is required to determine which are true or false. Scientific literacy provides similar tools and benefits to critical thinking; Sagan asserts science allows us to participate in the most significant method of understanding our Universe. Science is constantly evolving and is intertwined with our democracy, so to fully participate in democracy, you must be able to understand science. Sagan explains that science can be used to escape poverty, aid us in determining how to preserve our world, and to understand our origin and place in the Universe, “Without science and democracy, we risk being a nation of suckers.” Science can also be a valuable tool in examining a community’s social, political, and economic systems to reveal flaws and provide methods of improvement. Sagan asserts all individuals should have the ability to recognize errors.

Sagan insists that "there are no stupid questions" because asking questions is how we advance scientific knowledge. Denying that any questions are stupid also encourages children to develop critical thinking skills. Some examples Sagan provides include “Why is the grass green?” and “Why do we have toes?” Both of these help answer questions about biology, evolution, and the human body. Discouraging anyone from asking these questions and instead just accepting that these are facts of life may contribute to flaws in thinking patterns in the future. Being used to accepting the way things are just because is exactly what Sagan is attempting to fix, encouraging everyone to remain skeptical and question everything. He also places emphasis on using the “Baloney Detection Kit” to spot flaws (see above).

In the decade or so prior to the publication of The Demon-Haunted World, alien abduction accounts were common. Sagan provides concrete examples for why we shouldn’t assume things are true and instead use tools such as Occam's razor to seek more reasonable, science-based explanations.

Authoritarian personality

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Authoritarian_personality

The authoritarian personality is a personality type characterized by a disposition to treat authority figures with unquestioning obedience and respect. Conceptually, the term authoritarian personality originated from the writings of Erich Fromm, and usually is applied to people who exhibit a strict and oppressive personality towards their subordinates. Regardless of whether authoritarianism is more of a personality, attitude, ideology or disposition, scholars find it has significant influence on public opinion and political behavior.

Historical origins

In his 1941 book Fear of Freedom, a psychological exploration of modern politics, Erich Fromm described authoritarianism as a defence mechanism.

In The Authoritarian Personality (1950), Theodor W. Adorno, Else Frenkel-Brunswik, Daniel Levinson, and Nevitt Sanford proposed a personality type that involved the "potentially fascistic individual". The historical background that influenced the theoretical development of the authoritarian personality included the rise of fascism in the 1930s, World War II (1939–1945), and The Holocaust, which indicated that the fascistic individual was psychologically susceptible to the ideology of antisemitism and to the emotional appeal of anti-democratic politics. Known as the Berkeley studies, the researches of Adorno and Frenkel-Brunswik, and of Levinson and Sanford concentrated upon prejudice, which they studied within psychoanalytic and psychosocial frameworks of Freudian and Frommian theories. The book was described as a landmark work in social science that generated significant criticism of certain methods and results but also confirmation of many of the findings in independent studies. Following its publication was an extensive debate on the merits of the work, with many of the themes of this debate persisting in authoritarianism research today. The authoritarian person also presents a cynical and disdainful view of humanity, and a need to wield power and be tough, which arise from the anxieties produced by the perceived lapses of people who do not abide by the conventions and social norms of society (destructiveness and cynicism); a general tendency to focus upon people who violate the value system, and to act oppressively against them (authoritarian aggression); anti-intellectualism, a general opposition to the subjective and imaginative tendencies of the mind (anti-intraception); and an exaggerated concern with sexual promiscuity, especially when concerning women. The f-scale fell into disrepute as being unreliable after about 10 years. Other criticisms of the sociologic theory presented in The Authoritarian Personality are the validity of the psychoanalytic interpretation of personality and bias that authoritarianism exists only in the right wing of the political spectrum.

In human psychological development, the formation of the authoritarian personality occurs within the first years of a child's life, strongly influenced and shaped by the parents' personalities and the organizational structure of the child's family; thus, parent-child relations that are "hierarchical, authoritarian, [and] exploitative" can result in a child developing an authoritarian personality. Authoritarian-personality characteristics are fostered by parents who have a psychological need for domination, and who harshly threaten their child to compel obedience to conventional behaviors. Moreover, such domineering parents also are preoccupied with social status, a concern they communicate by having the child follow rigid, external rules. In consequence of such domination, the child suffers emotionally from the suppression of his or her feelings of aggression and resentment towards the domineering parents, whom the child reverently idealizes, but does not criticize. Such personalities may also be related to studies in preschool children of personality and political views as reported by scientists in 2006 which concluded that some children described as being "somewhat dominating" were later found, as adults, to be "relatively liberal", and those described as "relatively over-controlled" were later found, as adults, to be "relatively conservative"; in the words of the researchers,

Preschool children who 20 years later were relatively liberal were characterized as: developing close relationships, self-reliant, energetic, somewhat dominating, relatively under-controlled, and resilient. Preschool children subsequently relatively conservative at age 23 were described as: feeling easily victimized, easily offended, indecisive, fearful, rigid, inhibited, and relatively over-controlled and vulnerable.

Perceived threat

See also: Fear of crime

Hetherington and Weiler argue that perceived threat is a crucial variable in activating an authoritarian disposition. They suggest this helps to explain higher rates of authoritarianism in developing countries, after social or economic crises, as well as security crises like September 11th. The authors believe that some people, who tend to be more pessimistic and experience higher levels of stress and must rely more on instinct than cognition for decision-making, while those who under normal circumstances are more optimistic and at-ease, will also respond in a more authoritarian way when feeling threatened.

Links to gender inequality

According to a study by Brandt and Henry, there is a direct correlation between the rates of gender inequality and the levels of authoritarian ideas in the male and female populations. It was found that in countries with less gender equality where individualism was encouraged and men occupied the dominant societal roles, women were more likely to support traits such as obedience which would allow them to survive in an authoritarian environment and less likely to encourage ideas such as independence and imagination. In countries with higher levels of gender equality, men held less authoritarian views. It is theorized that this occurs due to the stigma attached to individuals who question the cultural norms set by the dominant individuals and establishments in an authoritarian society as a way to prevent the psychological stress caused by the active ostracizing of the stigmatized individuals.

Modern models

C.G. Sibley and J. Duckitt reported that more recent research has produced two more effective scales of measurement for predicting prejudice and other characteristics associated with authoritative personalities. The first scale is called the Right-wing authoritarianism (RWA) and the second is called the social dominance orientation (SDO).

Bob Altemeyer used the right-wing authoritarianism (RWA) scale, to identify, measure, and quantify the personality traits of authoritarian people. The political personality type identified with the RWA scale indicates the existence of three psychological tendencies and attitudinal clusters characteristic of the authoritarian personality: (i) Submission to legitimate authorities; (ii) Aggression towards minority groups whom authorities identified as targets for sanctioned political violence; and (iii) Adherence to cultural values and political beliefs endorsed by the authorities. As measured with the NEO-PI-R Openness scale, the research indicates a negative correlation (r = 0.57) between the personality trait of "openness to experience", of the Five Factor Model of the human personality.

The research of Jost, Glaser, Arie W. Kruglanski, and Sulloway (2003) indicates that authoritarianism and right-wing authoritarianism are ideological constructs for social cognition, by which political conservatives view people who are the Other who is not the Self. That the authoritarian personality and the conservative personality share two, core traits: (i) resistance to change (social, political. economic), and (ii) justification for social inequality among the members of society. Conservatives have a psychological need to manage existential uncertainty and threats with situational motives (striving for dominance in social hierarchies) and with dispositional motives (self-esteem and the management of fear).

The research on ideology, politics, and racist prejudice, by John Duckitt and Chris Sibley, identified two types of authoritarian worldview: (i) that the social world is dangerous, which leads to right-wing authoritarianism; and (ii) that the world is a ruthlessly competitive jungle, which leads to social dominance orientation. In a meta-analysis of the research, Sibley and Duckitt explained that the social-dominance orientation scale helps to measure the generalization of prejudice and other authoritarian attitudes that can exist within social groups. Although both the right-wing authoritarianism scale and the social-dominance orientation scale can accurately measure authoritarian personalities, the scales usually are not correlated.

Hetherington and Weiler describe the authoritarian personality as one that has a greater need for order, and less willingness to tolerate ambiguity as well as a tendency to rely on established authorities to provide that order. They acknowledge that while everyone seeks to bring some semblence of order to their world, non-authoritarian personalities are more likely to use concepts like fairness and equality, instead of the time-honored texts, conventions or leaders that are more common among authoritarian personalities. They also note that almost everyone becomes more authoritarian when they feel threat, anxiety or fatigue, as the emotional, reactive parts of the brain crowd out cognitive abilities. They also assert that scholars do not know whether to consider authoritarianism a personality trait, an attitude or an ideology.

Prevalence

Western countries

In 2021, Morning Consult (an American data intelligence company) published the results of a survey measuring the levels of authoritarianism in adults in America and seven other Western countries. The study used Bob Altemeyer's right-wing authoritarianism scale, but they omitted the following two statements from Altemeyer's scale: (1) "The established authorities generally turn out to be right about things, while the radicals and protestors are usually just "loud mouths" showing off their ignorance"; and (2) "Women should have to promise to obey their husbands when they get married." Morning Consult's scale thus had just 20 items, with a score range of 20 to 180 points. Morning Consult found that 25.6% of American adults qualify as "high RWA" (scoring between 111 and 180 points), while 13.4% of American adults qualify as "low RWA" (scoring 20 to 63 points).

Prevalence among adults in Western countries
2021 Morning Consult survey

	Low RWA	High RWA
US	13.4%	25.6%
UK	13.6%	10.4%
Germany	17.4%	6.7%
France	10.2%	10.7%
Spain	17.9%	9.2%
Italy	17.9%	12.9%
Australia	17.1%	12.9%
Canada	21.3%	13.4%

United States

In a 2009 book, Marc J. Hetherington and Jonathan D. Weiler identified evangelical Protestants as the most authoritarian of voting blocs in the United States. Furthermore, the former Confederate states (i.e. "the South") showed higher levels of authoritarianism than the rest. Rural populations tend to be more authoritarian than urban ones. People who preferred simple problems to complex ones, had less formal education, and those scoring lower in political knowledge also tended to score higher in authoritarianism. The authoritarianism levels of these demographics were assessed with four items that appeared in the 2004 American National Election Studies survey:

1. Please tell me which one you think is more important for a child to have: INDEPENDENCE or RESPECT FOR ELDERS
2. Please tell me which one you think is more important for a child to have: CURIOSITY or GOOD MANNERS
3. Please tell me which one you think is more important for a child to have: OBEDIENCE or SELF-RELIANCE
4. Please tell me which one you think is more important for a child to have: BEING CONSIDERATE or WELL BEHAVED

These questions were designed to force a choice, not unlike in politics, when voters are forced to choose between competing values. Some respondents chose both responses to some of the questions, and the four questions are averaged together, with a score of 1 meaning they answered all four questions with a more authoritarian response and 0 with a less authoritarian response. Half of the americans surveyed scored .75 or higher, indicating that the average American had a more authoritarian disposition in 2004.

Average authoritarianism by relevant party coalition groups

Group	Mean authoritarianism (2004 data)
Religion
Evangelical Protestant	0.709
Catholic	0.571
Mainline Protestant	0.530
Secular	0.481
Jewish	0.383
Church Attendance
Weekly or More	0.689
Less than Weekly	0.549
Region
South	0.657
Non-south	0.547
Population density
Rural	0.603
Small town	0.584
Suburb	0.524
Large City	0.502
Inner City	0.549
Education
Less than High School	0.754
High School Degree	0.657
Some College	0.590
College Degree	0.505
Graduate Degree	0.373

CP violation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/CP_violation

Beyond the Standard Model
Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge conjugation symmetry) and P-symmetry (parity symmetry). CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle (C-symmetry) while its spatial coordinates are inverted ("mirror" or P-symmetry). The discovery of CP violation in 1964 in the decays of neutral kaons resulted in the Nobel Prize in Physics in 1980 for its discoverers James Cronin and Val Fitch.

It plays an important role both in the attempts of cosmology to explain the dominance of matter over antimatter in the present universe, and in the study of weak interactions in particle physics.

Overview

Until the 1950s, parity conservation was believed to be one of the fundamental geometric conservation laws (along with conservation of energy and conservation of momentum). After the discovery of parity violation in 1956, CP-symmetry was proposed to restore order. However, while the strong interaction and electromagnetic interaction seem to be invariant under the combined CP transformation operation, further experiments showed that this symmetry is slightly violated during certain types of weak decay.

Only a weaker version of the symmetry could be preserved by physical phenomena, which was CPT symmetry. Besides C and P, there is a third operation, time reversal T, which corresponds to reversal of motion. Invariance under time reversal implies that whenever a motion is allowed by the laws of physics, the reversed motion is also an allowed one and occurs at the same rate forwards and backwards.

The combination of CPT is thought to constitute an exact symmetry of all types of fundamental interactions. Because of the long-held CPT symmetry theorem, provided that it is valid, a violation of the CP-symmetry is equivalent to a violation of the T-symmetry. In this theorem, regarded as one of the basic principles of quantum field theory, charge conjugation, parity, and time reversal are applied together. Direct observation of the time reversal symmetry violation without any assumption of CPT theorem was done in 1998 by two groups, CPLEAR and KTeV collaborations, at CERN and Fermilab, respectively. Already in 1970 Klaus Schubert observed T violation independent of assuming CPT symmetry by using the Bell–Steinberger unitarity relation.

History

P-symmetry

The idea behind parity symmetry was that the equations of particle physics are invariant under mirror inversion. This led to the prediction that the mirror image of a reaction (such as a chemical reaction or radioactive decay) occurs at the same rate as the original reaction. However, in 1956 a careful critical review of the existing experimental data by theoretical physicists Tsung-Dao Lee and Chen-Ning Yang revealed that while parity conservation had been verified in decays by the strong or electromagnetic interactions, it was untested in the weak interaction. They proposed several possible direct experimental tests.

The first test based on beta decay of cobalt-60 nuclei was carried out in 1956 by a group led by Chien-Shiung Wu, and demonstrated conclusively that weak interactions violate the P-symmetry or, as the analogy goes, some reactions did not occur as often as their mirror image. However, parity symmetry still appears to be valid for all reactions involving electromagnetism and strong interactions.

CP-symmetry

Overall, the symmetry of a quantum mechanical system can be restored if another approximate symmetry S can be found such that the combined symmetry PS remains unbroken. This rather subtle point about the structure of Hilbert space was realized shortly after the discovery of P violation, and it was proposed that charge conjugation, C, which transforms a particle into its antiparticle, was the suitable symmetry to restore order.

In 1956 Reinhard Oehme in a letter to Chen-Ning Yang and shortly after, Boris L. Ioffe, Lev Okun and A. P. Rudik showed that the parity violation meant that charge conjugation invariance must also be violated in weak decays. Charge violation was confirmed in the Wu experiment and in experiments performed by Valentine Telegdi and Jerome Friedman and Garwin and Lederman who observed parity non-conservation in pion and muon decay and found that C is also violated. Charge violation was more explicitly shown in experiments done by John Riley Holt at the University of Liverpool.

Oehme then wrote a paper with Lee and Yang in which they discussed the interplay of non-invariance under P, C and T. The same result was also independently obtained by Ioffe, Okun and Rudik. Both groups also discussed possible CP violations in neutral kaon decays.

Lev Landau proposed in 1957 CP-symmetry, often called just CP as the true symmetry between matter and antimatter. CP-symmetry is the product of two transformations: C for charge conjugation and P for parity. In other words, a process in which all particles are exchanged with their antiparticles was assumed to be equivalent to the mirror image of the original process and so the combined CP-symmetry would be conserved in the weak interaction.

In 1962, a group of experimentalists at Dubna, on Okun's insistence, unsuccessfully searched for CP-violating kaon decay.

Experimental status

Indirect CP violation

In 1964, James Cronin, Val Fitch and coworkers provided clear evidence from kaon decay that CP-symmetry could be broken. This work won them the 1980 Nobel Prize. This discovery showed that weak interactions violate not only the charge-conjugation symmetry C between particles and antiparticles and the P or parity symmetry, but also their combination. The discovery shocked particle physics and opened the door to questions still at the core of particle physics and of cosmology today. The lack of an exact CP-symmetry, but also the fact that it is so close to a symmetry, introduced a great puzzle.

The kind of CP violation discovered in 1964 was linked to the fact that neutral kaons can transform into their antiparticles (in which each quark is replaced with the other's antiquark) and vice versa, but such transformation does not occur with exactly the same probability in both directions; this is called indirect CP violation.

Direct CP violation

Kaon oscillation box diagram

The two box diagrams above are the Feynman diagrams providing the leading contributions to the amplitude of
K⁰
-
K⁰
oscillation

Despite many searches, no other manifestation of CP violation was discovered until the 1990s, when the NA31 experiment at CERN suggested evidence for CP violation in the decay process of the very same neutral kaons (direct CP violation). The observation was somewhat controversial, and final proof for it came in 1999 from the KTeV experiment at Fermilab and the NA48 experiment at CERN.

Starting in 2001, a new generation of experiments, including the BaBar experiment at the Stanford Linear Accelerator Center (SLAC) and the Belle Experiment at the High Energy Accelerator Research Organisation (KEK) in Japan, observed direct CP violation in a different system, namely in decays of the B mesons. A large number of CP violation processes in B meson decays have now been discovered. Before these "B-factory" experiments, there was a logical possibility that all CP violation was confined to kaon physics. However, this raised the question of why CP violation did not extend to the strong force, and furthermore, why this was not predicted by the unextended Standard Model, despite the model's accuracy for "normal" phenomena.

In 2011, a hint of CP violation in decays of neutral D mesons was reported by the LHCb experiment at CERN using 0.6 fb⁻¹ of Run 1 data. However, the same measurement using the full 3.0 fb⁻¹ Run 1 sample was consistent with CP-symmetry.

In 2013 LHCb announced discovery of CP violation in strange B meson decays.

In March 2019, LHCb announced discovery of CP violation in charmed ${\displaystyle D^0}$ decays with a deviation from zero of 5.3 standard deviations.

In 2020, the T2K Collaboration reported some indications of CP violation in leptons for the first time. In this experiment, beams of muon neutrinos (
ν
_μ) and muon antineutrinos (
ν
_μ) were alternately produced by an accelerator. By the time they got to the detector, a significantly higher proportion of electron neutrinos (
ν
_e) were detected from the
ν
_μ beams, than electron antineutrinos (
ν
_e) were from the
ν
_μ beams. The results were not yet precise enough to determine the size of the CP violation, relative to that seen in quarks. In addition, another similar experiment, NOvA sees no evidence of CP violation in neutrino oscillations and is in slight tension with T2K.

CP violation in the Standard Model

"Direct" CP violation is allowed in the Standard Model if a complex phase appears in the CKM matrix describing quark mixing, or the PMNS matrix describing neutrino mixing. A necessary condition for the appearance of the complex phase is the presence of at least three generations of fermions. If fewer generations are present, the complex phase parameter can be absorbed into redefinitions of the fermion fields.

A popular rephasing invariant whose vanishing signals absence of CP violation and occurs in most CP violating amplitudes is the Jarlskog invariant:

$${\displaystyle J=c_{12}{c}_{13}^2{c}_{23}{s}_{12}{s}_{13}{s}_{23}\sin \delta \approx 0.00003,}$$

for quarks, which is ${\displaystyle 0.0003}$ times the maximum value of ${\displaystyle J_{max}=\frac{1}{6}\sqrt{3}\approx 0.1.}$ For leptons, only an upper limit exists: ${\displaystyle |J|<0.03.}$

The reason why such a complex phase causes CP violation is not immediately obvious, but can be seen as follows. Consider any given particles (or sets of particles) ${\displaystyle a}$ and ${\displaystyle b,}$ and their antiparticles ${\displaystyle \overline{a}}$ and ${\displaystyle \overline{b}.}$ Now consider the processes ${\displaystyle a\to b}$ and the corresponding antiparticle process ${\displaystyle \overline{a}\to \overline{b},}$ and denote their amplitudes ${\displaystyle M}$ and ${\displaystyle \overline{M}}$ respectively. Before CP violation, these terms must be the same complex number. We can separate the magnitude and phase by writing ${\displaystyle M=|M|e^{i\theta }.}$ If a phase term is introduced from (e.g.) the CKM matrix, denote it ${\displaystyle e^{i\varphi }.}$ Note that ${\displaystyle \overline{M}}$ contains the conjugate matrix to ${\displaystyle M,}$ so it picks up a phase term ${\displaystyle e^{-i\varphi }.}$

Now the formula becomes:

$${\displaystyle \begin{array}{rl}M& =|M|e^{i\theta }{e}^{+i\varphi }\\ \overline{M}& =|M|e^{i\theta }{e}^{-i\varphi }\end{array}}$$

Physically measurable reaction rates are proportional to ${\displaystyle |M{|}^2,}$ thus so far nothing is different. However, consider that there are two different routes: ${\displaystyle a\stackrel{1}{⟶}b}$ and ${\displaystyle a\stackrel{2}{⟶}b}$ or equivalently, two unrelated intermediate states: ${\displaystyle a\to 1\to b}$ and ${\displaystyle a\to 2\to b.}$ Now we have:

$${\displaystyle \begin{array}{rlrl}M& =|M_1|e^{i{\theta }_1}{e}^{i{\varphi }_1}& & +|M_2|e^{i{\theta }_2}{e}^{i{\varphi }_2}\\ \overline{M}& =|M_1|e^{i{\theta }_1}{e}^{-i{\varphi }_1}& & +|M_2|e^{i{\theta }_2}{e}^{-i{\varphi }_2}.\end{array}}$$

Some further calculation gives:

$${\displaystyle |M{|}^2-|\overline{M}{|}^2=-4|M_1||M_2|\sin ({\theta }_1-{\theta }_2)\sin ({\varphi }_1-{\varphi }_2).}$$

Thus, we see that a complex phase gives rise to processes that proceed at different rates for particles and antiparticles, and CP is violated.

From the theoretical end, the CKM matrix is defined as ${\displaystyle {\mathrm{V}}_{\mathsf{C}\mathsf{K}\mathsf{M}}={\mathrm{U}}_{\mathsf{u}}{\mathrm{U}}_{\mathsf{d}}^{†},}$ where ${\displaystyle {\mathrm{U}}_{\mathsf{u}}}$ and ${\displaystyle {\mathrm{U}}_{\mathsf{d}}}$ are unitary transformation matrices which diagonalize the fermion mass matrices ${\displaystyle M_{\mathsf{u}}}$ and ${\displaystyle M_{\mathsf{d}},}$ respectively.

Thus, there are two necessary conditions for getting a complex CKM matrix:

1. At least one of ${\displaystyle U_u}$ and ${\displaystyle U_d}$ is complex, or the CKM matrix will be purely real.
2. If both of them are complex, ${\displaystyle U_u}$ and ${\displaystyle U_d}$ must be different, i.e., ${\displaystyle U_u\ne U_d}$, or the CKM matrix will be an identity matrix, which is also purely real.

For a standard model with three fermion generations, the most general non-Hermitian pattern of its mass matrices can be given by

${\displaystyle M=\left[\begin{array}{ccc}{A}_1+i{D}_1& B_1+i{C}_1& B_2+i{C}_2\\ B_4+i{C}_4& A_2+i{D}_2& B_3+i{C}_3\\ B_5+i{C}_5& B_6+i{C}_6& A_3+i{D}_6\end{array}\right].}$

This M matrix contains 9 elements and 18 parameters, 9 from the real coefficients and 9 from the imaginary coefficients. Obviously, a 3x3 matrix with 18 parameters is too difficult to diagonalize analytically. However, a naturally Hermitian ${\displaystyle {\mathbf{M}}^{\mathbf{2}}=M\cdot M^{+}}$ can be given by

${\displaystyle {\mathbf{M}}^{\mathbf{2}}=\left[\begin{array}{ccc}{\mathbf{A}}_{\mathbf{1}}& {\mathbf{B}}_{\mathbf{1}}+i{\mathbf{C}}_{\mathbf{1}}& {\mathbf{B}}_{\mathbf{2}}+i{\mathbf{C}}_{\mathbf{2}}\\ {\mathbf{B}}_{\mathbf{1}}-i{\mathbf{C}}_{\mathbf{1}}& {\mathbf{A}}_{\mathbf{2}}& {\mathbf{B}}_{\mathbf{3}}+i{\mathbf{C}}_{\mathbf{3}}\\ {\mathbf{B}}_{\mathbf{2}}-i{\mathbf{C}}_{\mathbf{2}}& {\mathbf{B}}_{\mathbf{3}}-i{\mathbf{C}}_{\mathbf{3}}& {\mathbf{A}}_{\mathbf{3}}\end{array}\right],}$

and it has the same unitary transformation matrix U with M. Besides, parameters in ${\displaystyle {\mathbf{M}}^{\mathbf{2}}}$ are correlated to those in M directly in the ways shown below

${\displaystyle {\mathbf{A}}_{\mathbf{1}}=A_1^2+D_1^2+B_1^2+C_1^2+B_2^2+C_2^2,}$

${\displaystyle {\mathbf{A}}_{\mathbf{2}}=A_2^2+D_2^2+B_3^2+C_3^2+B_4^2+C_4^2,}$

${\displaystyle {\mathbf{A}}_{\mathbf{3}}=A_3^2+D_3^2+B_5^2+C_5^2+B_6^2+C_6^2,}$

${\displaystyle {\mathbf{B}}_{\mathbf{1}}=A_1{B}_4+D_1{C}_4+B_1{A}_2+C_1{D}_2+B_2{B}_3+C_2{C}_3,}$

${\displaystyle {\mathbf{B}}_{\mathbf{2}}=A_1{B}_5+D_1{C}_5+B_1{B}_6+C_1{C}_6+B_2{A}_3+C_2{D}_3,}$

${\displaystyle {\mathbf{B}}_{\mathbf{3}}=B_4{B}_5+C_4{C}_5+B_6{A}_2+C_6{D}_2+A_3{B}_3+D_3{C}_3,}$

${\displaystyle {\mathbf{C}}_{\mathbf{1}}=D_1{B}_4-A_1{C}_4+A_2{C}_1-B_1{D}_2+B_3{C}_2-B_2{C}_3,}$

${\displaystyle {\mathbf{C}}_{\mathbf{2}}=D_1{B}_5-A_1{C}_5+B_6{C}_1-B_1{C}_6+A_3{C}_2-B_2{D}_3,}$

${\displaystyle {\mathbf{C}}_{\mathbf{3}}=C_4{B}_5-B_4{C}_5+D_2{B}_6-A_2{C}_6+A_3{C}_3-B_3{D}_3.}$

That means if we diagonalize an ${\displaystyle {\mathbf{M}}^{\mathbf{2}}}$ matrix with 9 parameters, it has the same effect as diagonalizing an M matrix with 18 parameters. Therefore, diagonalizing the ${\displaystyle {\mathbf{M}}^{\mathbf{2}}}$ matrix is certainly the most reasonable choice.

The M and ${\displaystyle {\mathbf{M}}^{\mathbf{2}}}$ matrix patterns given above are the most general ones. The perfect way to solve the CPV problem in the standard model is to diagonalize such matrices analytically and to achieve a U matrix which applies to both. Unfortunately, even though the ${\displaystyle {\mathbf{M}}^{\mathbf{2}}}$ matrix has only 9 parameters, it is still too complicated to be diagonalized directly. Thus, an assumption

${\displaystyle {\mathbf{M}}_{\mathbf{R}}^{\mathbf{2}}}\cdot {\mathbf{M}}_{\mathbf{I}}^{\mathbf{2}\mathbf{+}}-{\mathbf{M}}_{\mathbf{I}}^{\mathbf{2}}\cdot {\mathbf{M}}_{\mathbf{R}}^{\mathbf{2}\mathbf{+}}\mathbf{=}\mathbf{0}$

was employed to simplify the pattern, where ${\displaystyle {\mathbf{M}}_{\mathbf{R}}^{\mathbf{2}}}$ is the real part of ${\displaystyle {\mathbf{M}}^{\mathbf{2}}}$ and ${\displaystyle {\mathbf{M}}_{\mathbf{I}}^{\mathbf{2}}}$ is the imaginary part.

Such an assumption could further reduce the parameter number from 9 to 5 and the reduced ${\displaystyle {\mathbf{M}}^{\mathbf{2}}}$ matrix can be given by

${\displaystyle {\mathbf{M}}^{\mathbf{2}}=\left[\begin{array}{ccc}\mathbf{A}+\mathbf{B}\mathbf{(}\mathbf{x}\mathbf{y}\mathbf{-}\frac{\mathbf{x}}{\mathbf{y}}\mathbf{)}& \mathbf{y}\mathbf{B}& \mathbf{x}\mathbf{B}\\ \mathbf{y}\mathbf{B}& \mathbf{A}\mathbf{+}\mathbf{B}\mathbf{(}\frac{\mathbf{y}}{\mathbf{x}}\mathbf{-}\frac{\mathbf{x}}{\mathbf{y}}\mathbf{)}& \mathbf{B}\\ \mathbf{x}\mathbf{B}& \mathbf{B}& \mathbf{A}\end{array}\right]}$ ${\displaystyle +i\left[\begin{array}{ccc}0& \frac{\mathbf{C}}{\mathbf{y}}& -\frac{\mathbf{C}}{\mathbf{x}}\\ -\frac{\mathbf{C}}{\mathbf{y}}& 0& \mathbf{C}\\ i\frac{\mathbf{C}}{\mathbf{x}}& -\mathbf{C}& 0\end{array}\right]\equiv {\mathbf{M}}_{\mathbf{R}}^{\mathbf{2}}+i{\mathbf{M}}_{\mathbf{I}}^{\mathbf{2}},}$

where ${\displaystyle \mathbf{A}\equiv {\mathbf{A}}_{\mathbf{3}},\mathbf{B}\equiv {\mathbf{B}}_{\mathbf{3}},\mathbf{C}\equiv {\mathbf{C}}_{\mathbf{3}},\mathbf{x}\equiv {\mathbf{B}}_{\mathbf{2}}\mathbf{/}{\mathbf{B}}_{\mathbf{3}},}$ and ${\displaystyle \mathbf{y}\equiv {\mathbf{B}}_{\mathbf{1}}\mathbf{/}{\mathbf{B}}_{\mathbf{3}}}$.

Diagonalizing ${\displaystyle {\mathbf{M}}^{\mathbf{2}}}$ analytically, the eigenvalues are given by

${\displaystyle {\mathbf{m}}_{\mathbf{1}}^{\mathbf{2}}=\mathbf{A}\mathbf{-}\mathbf{B}\frac{\mathbf{x}}{\mathbf{y}}\mathbf{-}\mathbf{C}\sqrt{\frac{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}{\mathbf{x}\mathbf{y}}},}$

${\displaystyle {\mathbf{m}}_{\mathbf{2}}^{\mathbf{2}}=\mathbf{A}\mathbf{-}\mathbf{B}\frac{\mathbf{x}}{\mathbf{y}}\mathbf{+}\mathbf{C}\frac{\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}}{\mathbf{x}\mathbf{y}},}$

${\displaystyle {\mathbf{m}}_{\mathbf{3}}^{\mathbf{2}}=\mathbf{A}\mathbf{+}\mathbf{B}\frac{\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{y}}{\mathbf{x}},}$

and the U matrix for up-type quarks can then be given by

${\displaystyle {\mathbf{U}}^{\mathbf{u}}=\left[\begin{array}{ccc}\frac{-\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}}{\sqrt{\mathbf{2}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}\mathbf{)}}}& \frac{\mathbf{x}\mathbf{(}{\mathbf{y}}^{\mathbf{2}}\mathbf{-}\mathbf{i}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}\mathbf{)}}{\sqrt{2}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}}& \frac{\mathbf{y}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{i}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}\mathbf{)}}{\sqrt{2}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}}\\ \frac{-\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}}{\sqrt{\mathbf{2}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}\mathbf{)}}}& \frac{\mathbf{x}\mathbf{(}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}\mathbf{i}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}\mathbf{)}}{\sqrt{2}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}}}& \frac{\mathbf{y}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{i}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}\mathbf{)}}}{\sqrt{\mathbf{2}}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}}\\ \frac{\mathbf{x}\mathbf{y}}{\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}}& \frac{\mathbf{y}}{\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}}& \frac{\mathbf{x}}{\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}}}\end{array}\right].}$

However, the eigenvalues' order does not necessarily have to be ${\displaystyle ({\mathbf{m}}_{\mathbf{1}}^{\mathbf{2}},{\mathbf{m}}_{\mathbf{2}}^{\mathbf{2}},{\mathbf{m}}_{\mathbf{3}}^{\mathbf{2}})}$; they can also be any permutation of them.

After obtaining a general U matrix pattern, it can also be applied to down-type quarks by introducing primed parameters. To construct the CKM matrix, the U matrix for up-type quarks, denoted as ${\displaystyle U^u}$, can be multiplied with the conjugate transpose of the U matrix for down-type quarks, denoted as ${\displaystyle U^{d+}}$. As mentioned earlier, there are no inherent constraints that dictate the assignment of eigenvalues to specific quark flavors. Consequently, all 36 potential permutations of eigenvalues are listed in the provided reference. 

Among these 36 potential CKM matrices, 4 of them

${\displaystyle V[52]=V\left[\begin{array}{c}1\\ 3\\ 2\end{array}\right]\left[\begin{array}{ccc}2& 3& 1\end{array}\right]=V[52{]}^{\ast }=V^{\ast }\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right]\left[\begin{array}{ccc}1& 3& 2\end{array}\right]=\left[\begin{array}{ccc}s& p^{\ast }& r^{\ast }\\ p^{\mathrm{\prime }\ast }& q& p^{\mathrm{\prime }}\\ r& p& s^{\ast }\end{array}\right]}$ and

${\displaystyle V[22]=V\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right]\left[\begin{array}{ccc}2& 3& 1\end{array}\right]=V^{\ast }[55]=V^{\ast }\left[\begin{array}{c}1\\ 3\\ 2\end{array}\right]\left[\begin{array}{ccc}1& 3& 2\end{array}\right]=\left[\begin{array}{ccc}r& p& s^{\ast }\\ p^{\mathrm{\prime }\ast }& q& p^{\mathrm{\prime }}\\ s& p^{\ast }& r^{\ast }\end{array}\right],}$

fit experimental data to the order of ${\displaystyle {\lambda }^{1/2}}$ or better, at tree level, where ${\displaystyle \lambda }$ is one of the Wolfenstein parameters.

The full expressions of parameters ${\displaystyle p,q,r,s,}$ and ${\displaystyle p^{\mathrm{\prime }}}$ are given by

${\displaystyle r=\frac{(x^2+y^2)(x^{\prime 2}+y^{\prime 2})+(x{x}^{\prime }+y{y}^{\prime })(xy{x}^{\prime }{y}^{\prime }+\sqrt{x^2+y^2+x^2{y}^2}\sqrt{x^{\prime 2}+y^{\prime 2}+x^{\prime 2}{y}^{\prime 2}})}{2\sqrt{x^2+y^2}\sqrt{x^{\prime 2}+y^{\prime 2}}\sqrt{x^2+y^2+x^2{y}^2}\sqrt{x^{\prime 2}+y^{\prime 2}+x^{\prime 2}{y}^{\prime 2}}}}$

 ${\displaystyle +i\frac{(x{y}^{\prime }-x^{\prime }y)(x^{\prime }{y}^{\prime }\sqrt{x^2+y^2+x^2{y}^2}+xy\sqrt{x^{\prime 2}+y^{\prime 2}+x^{\prime 2}{y}^{\prime 2}})}{2\sqrt{x^2+y^2}\sqrt{x^{\prime 2}+y^{\prime 2}}\sqrt{x^2+y^2+x^2{y}^2}\sqrt{x^{\prime 2}+y^{\prime 2}+x^{\prime 2}{y}^{\prime 2}}},}$

${\displaystyle s=\frac{(x^2+y^2)(x^{\prime 2}+y^{\prime 2})+(x{x}^{\prime }+y{y}^{\prime })(xy{x}^{\prime }{y}^{\prime }-\sqrt{x^2+y^2+x^2{y}^2}\sqrt{x^{\prime 2}+y^{\prime 2}+x^{\prime 2}{y}^{\prime 2}})}{2\sqrt{x^2+y^2}\sqrt{x^{\prime 2}+y^{\prime 2}}\sqrt{x^2+y^2+x^2{y}^2}\sqrt{x^{\prime 2}+y^{\prime 2}+x^{\prime 2}{y}^{\prime 2}}}}$

${\displaystyle +i\frac{(x{y}^{\prime }-x^{\prime }y)(x^{\prime }{y}^{\prime }\sqrt{x^2+y^2+x^2{y}^2}-xy\sqrt{x^{\prime 2}+y^{\prime 2}+x^{\prime 2}{y}^{\prime 2}})}{2\sqrt{x^2+y^2}\sqrt{x^{\prime 2}+y^{\prime 2}}\sqrt{x^2+y^2+x^2{y}^2}\sqrt{x^{\prime 2}+y^{\prime 2}+x^{\prime 2}{y}^{\prime 2}}},}$

${\displaystyle p=\frac{[y^{\prime }{y}^2(x-x^{\prime })+x^{\prime }{x}^2(y-y^{\prime })]+i(x{y}^{\prime }-x^{\prime }y)\sqrt{x^2+y^2+x^2{y}^2}}{\sqrt{2}\sqrt{x^2+y^2}\sqrt{x^2+y^2+x^2{y}^2}\sqrt{x^{\prime 2}+y^{\prime 2}+x^{\prime 2}{y}^{\prime 2}}},}$

${\displaystyle p^{\mathrm{\prime }}=\frac{[y{y}^{\prime 2}(x^{\prime }-x)+x{x}^{\prime 2}(y^{\prime }-y)]+i(x{y}^{\prime }-x^{\prime }y)\sqrt{x^{\prime 2}+y^{\prime 2}+x^{\prime 2}{y}^{\prime 2}}}{\sqrt{2}\sqrt{x^2+y^2+x^2{y}^2}\sqrt{x^{\prime 2}+y^{\prime 2}}\sqrt{x^{\prime 2}+y^{\prime 2}+x^{\prime 2}{y}^{\prime 2}}},}$

${\displaystyle q=\frac{x{x}^{\prime }+y{y}^{\prime }+xy{x}^{\prime }{y}^{\prime }}{\sqrt{x^2+y^2+x^2{y}^2}\sqrt{x^{\prime 2}+y^{\prime 2}+x^{\prime 2}{y}^{\prime 2}}},}$

The best fit of the CKM elements are

${\displaystyle |V_{ud}|=|V_{tb}|\sim 0.9925,}$

${\displaystyle |V_{ub}|=|V_{td}|\sim 0.0075,}$

${\displaystyle |V_{us}|=|V_{ts}|=|V_{cd}|=|V_{cb}|\sim 0.122023,}$ and

${\displaystyle |V_{cs}|\sim \mathrm{0.9845.}}$

Since the discovery of CP violation in 1964, physicists have believed that in theory, within the framework of the Standard Model, it is sufficient to search for appropriate Yukawa couplings (equivalent to a mass matrix) in order to generate a complex phase in the CKM matrix, thus automatically breaking CP symmetry. However, the specific matrix pattern has remained elusive. The above derivation provides the first evidence for this idea and offers some explicit examples to support it.

Strong CP problem

Main article: Strong CP problem

Unsolved problem in physics:

Why is the strong nuclear interaction force CP-invariant?

(more unsolved problems in physics)

There is no experimentally known violation of the CP-symmetry in quantum chromodynamics. As there is no known reason for it to be conserved in QCD specifically, this is a "fine tuning" problem known as the strong CP problem.

QCD does not violate the CP-symmetry as easily as the electroweak theory; unlike the electroweak theory in which the gauge fields couple to chiral currents constructed from the fermionic fields, the gluons couple to vector currents. Experiments do not indicate any CP violation in the QCD sector. For example, a generic CP violation in the strongly interacting sector would create the electric dipole moment of the neutron which would be comparable to 10⁻¹⁸ e·m while the experimental upper bound is roughly one trillionth that size.

This is a problem because at the end, there are natural terms in the QCD Lagrangian that are able to break the CP-symmetry.

$${\displaystyle \mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\frac{n_f{g}^2\theta }{32{\pi }^2}{F}_{\mu \nu }{\tilde{F}}^{\mu \nu }+\overline{\psi }\left(i{\gamma }^{\mu }{D}_{\mu }-m{e}^{i{\theta }^{\prime }{\gamma }_5}\right)\psi }$$

For a nonzero choice of the θ angle and the chiral phase of the quark mass θ′ one expects the CP-symmetry to be violated. One usually assumes that the chiral quark mass phase can be converted to a contribution to the total effective ${\displaystyle \tilde{\theta }}$ angle, but it remains to be explained why this angle is extremely small instead of being of order one; the particular value of the θ angle that must be very close to zero (in this case) is an example of a fine-tuning problem in physics, and is typically solved by physics beyond the Standard Model.

There are several proposed solutions to solve the strong CP problem. The most well-known is Peccei–Quinn theory, involving new scalar particles called axions. A newer, more radical approach not requiring the axion is a theory involving two time dimensions first proposed in 1998 by Bars, Deliduman, and Andreev.

Matter–antimatter imbalance

Main articles: Baryon asymmetry and Baryogenesis

See also: T-symmetry, Arrow of time, and Lorentz transformation

 

Unsolved problem in physics:

Why does the universe have so much more matter than antimatter?

The non-dark matter universe is made chiefly of matter, rather than consisting of equal parts of matter and antimatter as might be expected. It can be demonstrated that, to create an imbalance in matter and antimatter from an initial condition of balance, the Sakharov conditions must be satisfied, one of which is the existence of CP violation during the extreme conditions of the first seconds after the Big Bang. Explanations which do not involve CP violation are less plausible, since they rely on the assumption that the matter–antimatter imbalance was present at the beginning, or on other admittedly exotic assumptions.

The Big Bang should have produced equal amounts of matter and antimatter if CP-symmetry was preserved; as such, there should have been total cancellation of both—protons should have cancelled with antiprotons, electrons with positrons, neutrons with antineutrons, and so on. This would have resulted in a sea of radiation in the universe with no matter. Since this is not the case, after the Big Bang, physical laws must have acted differently for matter and antimatter, i.e. violating CP-symmetry.

The Standard Model contains at least three sources of CP violation. The first of these, involving the Cabibbo–Kobayashi–Maskawa matrix in the quark sector, has been observed experimentally and can only account for a small portion of the CP violation required to explain the matter-antimatter asymmetry. The strong interaction should also violate CP, in principle, but the failure to observe the electric dipole moment of the neutron in experiments suggests that any CP violation in the strong sector is also too small to account for the necessary CP violation in the early universe. The third source of CP violation is the Pontecorvo–Maki–Nakagawa–Sakata matrix in the lepton sector. The current long-baseline neutrino oscillation experiments, T2K and NOνA, may be able to find evidence of CP violation over a small fraction of possible values of the CP violating Dirac phase while the proposed next-generation experiments, Hyper-Kamiokande and DUNE, will be sensitive enough to definitively observe CP violation over a relatively large fraction of possible values of the Dirac phase. Further into the future, a neutrino factory could be sensitive to nearly all possible values of the CP violating Dirac phase. If neutrinos are Majorana fermions, the PMNS matrix could have two additional CP violating Majorana phases, leading to a fourth source of CP violation within the Standard Model. The experimental evidence for Majorana neutrinos would be the observation of neutrinoless double-beta decay. The best limits come from the GERDA experiment. CP violation in the lepton sector generates a matter-antimatter asymmetry through a process called leptogenesis. This could become the preferred explanation in the Standard Model for the matter-antimatter asymmetry of the universe if CP violation is experimentally confirmed in the lepton sector.

If CP violation in the lepton sector is experimentally determined to be too small to account for matter-antimatter asymmetry, some new physics beyond the Standard Model would be required to explain additional sources of CP violation. Adding new particles and/or interactions to the Standard Model generally introduces new sources of CP violation since CP is not a symmetry of nature.

Sakharov proposed a way to restore CP-symmetry using T-symmetry, extending spacetime before the Big Bang. He described complete CPT reflections of events on each side of what he called the "initial singularity". Because of this, phenomena with an opposite arrow of time at t < 0 would undergo an opposite CP violation, so the CP-symmetry would be preserved as a whole. The anomalous excess of matter over antimatter after the Big Bang in the orthochronous (or positive) sector, becomes an excess of antimatter before the Big Bang (antichronous or negative sector) as both charge conjugation, parity and arrow of time are reversed due to CPT reflections of all phenomena occurring over the initial singularity:

We can visualize that neutral spinless maximons (or photons) are produced at t < 0 from contracting matter having an excess of antiquarks, that they pass "one through the other" at the instant t = 0 when the density is infinite, and decay with an excess of quarks when t > 0, realizing total CPT symmetry of the universe. All the phenomena at t < 0 are assumed in this hypothesis to be CPT reflections of the phenomena at t > 0.

— Andrei Sakharov, in Collected Scientific Works (1982).