"Dark Matter" was first shown to exist by Zwicky studying rotation curves of galaxies 23. The motion could only be explained if there was massive cloud of matter that was not luminous. We still do not know what composes this dark matter, though hopes are it will be discovered at the LHC. An even more mysterious phenomena called "Dark Energy" may also have a connection to particle physics experiments 24, perhaps via "Extra Dimensions" 25.

When the Universe began with the Big Bang, there was an equal amount of matter and antimatter. Now we have mostly matter. How did it happen? Sakharov gave three necessary conditions: Baryon (ℬ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@) number violation, departure from thermal equilibrium, and C and CP violation 26. (The operation of Charge Conjugation (C) takes particle to anti-particle and Parity (P) takes a vector r→ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkhaYzaalaaaaa@2C76@ to - r→ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkhaYzaalaaaaa@2C76@.)

These criteria are all satisfied by the Standard Model. ℬ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ is violated in Electroweak theory at high temperature, though baryon minus lepton number is conserved; in addition we need quantum tunneling, which is powerfully suppressed at the low temperatures that we now have. Non-thermal equilibrium is provided by the electroweak phase transition. C and CP are violated by weak interactions. However the violation is too small. The ratio of the number of baryons to the entropy in the observed part of the Universe needs to be ~5 × 10^-11, while the SM provides many orders of magnitude less. Therefore, there must be new physics 27.

Our worry is why the Planck scale at ~10¹⁹ GeV is so much higher than the scale at which we expect to find the Higgs Boson, ~100 GeV. As Lisa Randall said 28 "The gist of it is that the universe seems to have two entirely different mass scales, and we don't understand why they are so different. There's what's called the Planck scale, which is associated with gravitational interactions. It's a huge mass scale, but because gravitational forces are proportional to one over the mass squared, that means gravity is a very weak interaction. In units of GeV, which is how we measure masses, the Planck scale is 10¹⁹ GeV. Then there's the electroweak scale, which sets the masses for the W and Z bosons. These are particles that are similar to the photons of electromagnetism and which we have observed and studied well. They have a mass of about 100 GeV. So the hierarchy problem, in its simplest manifestation, is how can you have these particles be so light when the other scale is so big." We expect the explanation lies in physics beyond the Standard Model 29.

Consider the operations of Charge Conjugation, C, and Parity, P:

C | B ( p → ) 〉 = | B ¯ ( p → ) 〉 , C | B ¯ ( p → ) 〉 = | B ( p → ) 〉 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaem4qamKaeiiFaWNaemOqaiKaeiikaGIafmiCaaNbaSaacqGGPaqkcqGHQms8cqGH9aqpcqGG8baFcuWGcbGqgaqeaiabcIcaOiqbdchaWzaalaGaeiykaKIaeyOkJeVaeiilaWcabaGaem4qamKaeiiFaWNafmOqaiKbaebacqGGOaakcuWGWbaCgaWcaiabcMcaPiabgQYiXlabg2da9iabcYha8jabdkeacjabcIcaOiqbdchaWzaalaGaeiykaKIaeyOkJepaaaaa@4DAD@

P | B ( p → ) 〉 = − | B ( − p → ) 〉 , P | B ¯ ( p → ) 〉 = − | B ¯ ( − p → ) 〉 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaemiuaaLaeiiFaWNaemOqaiKaeiikaGIafmiCaaNbaSaacqGGPaqkcqGHQms8cqGH9aqpcqGHsislcqGG8baFcqWGcbGqcqGGOaakcqGHsislcuWGWbaCgaWcaiabcMcaPiabgQYiXlabcYcaSaqaaiabdcfaqjabcYha8jqbdkeaczaaraGaeiikaGIafmiCaaNbaSaacqGGPaqkcqGHQms8cqGH9aqpcqGHsislcqGG8baFcuWGcbGqgaqeaiabcIcaOiabgkHiTiqbdchaWzaalaGaeiykaKIaeyOkJepaaaaa@5195@

C P | B ( p → ) 〉 = − | B ¯ ( − p → ) 〉 , C P | B ¯ ( p → ) 〉 = − | B ( − p → ) 〉 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaem4qamKaemiuaaLaeiiFaWNaemOqaiKaeiikaGIafmiCaaNbaSaacqGGPaqkcqGHQms8cqGH9aqpcqGHsislcqGG8baFcuWGcbGqgaqeaiabcIcaOiabgkHiTiqbdchaWzaalaGaeiykaKIaeyOkJeVaeiilaWcabaGaem4qamKaemiuaaLaeiiFaWNafmOqaiKbaebacqGGOaakcuWGWbaCgaWcaiabcMcaPiabgQYiXlabg2da9iabgkHiTiabcYha8jabdkeacjabcIcaOiabgkHiTiqbdchaWzaalaGaeiykaKIaeyOkJeVaeiOla4caaaaa@5497@

For neutral mesons we can construct the CP eigenstates

| B 1 0 〉 = 1 2 ( | B 0 〉 − | B ¯ 0 〉 ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqGG8baFcqWGcbGqdaqhaaWcbaGaeGymaedabaGaeGimaadaaOGaeyOkJeVaeyypa0tcfa4aaSaaaeaacqaIXaqmaeaadaGcaaqaaiabikdaYaqabaaaaOWaaeWaaeaacqGG8baFcqWGcbGqdaahaaWcbeqaaiabicdaWaaakiabgQYiXlabgkHiTiabcYha8jqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaGccqGHQms8aiaawIcacaGLPaaacqGGSaalaaa@42C8@

| B 2 0 〉 = 1 2 ( | B 0 〉 + | B ¯ 0 〉 ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqGG8baFcqWGcbGqdaqhaaWcbaGaeGOmaidabaGaeGimaadaaOGaeyOkJeVaeyypa0tcfa4aaSaaaeaacqaIXaqmaeaadaGcaaqaaiabikdaYaqabaaaaOWaaeWaaeaacqGG8baFcqWGcbGqdaahaaWcbeqaaiabicdaWaaakiabgQYiXlabgUcaRiabcYha8jqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaGccqGHQms8aiaawIcacaGLPaaacqGGSaalaaa@42BF@

where

C P | B 1 0 〉 = | B 1 0 〉 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqadaaabaGaem4qamKaemiuaaLaeiiFaWNaemOqai0aa0baaSqaaiabigdaXaqaaiabicdaWaaakiabgQYiXdqaaiabg2da9aqaaiabcYha8jabdkeacnaaDaaaleaacqaIXaqmaeaacqaIWaamaaGccqGHQms8cqGGSaalaaaaaa@3B5B@

C P | B 2 0 〉 = − | B 2 0 〉 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqadaaabaGaem4qamKaemiuaaLaeiiFaWNaemOqai0aa0baaSqaaiabikdaYaqaaiabicdaWaaakiabgQYiXdqaaiabg2da9aqaaiabgkHiTiabcYha8jabdkeacnaaDaaaleaacqaIYaGmaeaacqaIWaamaaGccqGHQms8cqGGUaGlaaaaaa@3C50@

Since B⁰ and B¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ can mix, the mass eigenstates are a superposition of a|B⁰⟩ + b|B¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@⟩ which obey the Schrodinger equation

i d d t ( a b ) = H ( a b ) = ( M − i 2 Γ ) ( a b ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGPbqAjuaGdaWcaaqaaiabdsgaKbqaaiabdsgaKjabdsha0baakmaabmaabaqbaeqabiqaaaqaaiabdggaHbqaaiabdkgaIbaaaiaawIcacaGLPaaacqGH9aqpt0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaqadaqaauaabeqaceaaaeaacqWGHbqyaeaacqWGIbGyaaaacaGLOaGaayzkaaGaeyypa0ZaaeWaaeaacqWGnbqtcqGHsisljuaGdaWcaaqaaiabdMgaPbqaaiabikdaYaaakiabfo5ahbGaayjkaiaawMcaamaabmaabaqbaeqabiqaaaqaaiabdggaHbqaaiabdkgaIbaaaiaawIcacaGLPaaacqGGUaGlaaa@5292@

If CP is not conserved then the eigenvectors, the mass eigenstates |B_L⟩ and |B_H⟩, are not the CP eigenstates but are

| B L 〉 = p | B 0 〉 + q | B ¯ 0 〉 , | B H 〉 = p | B 0 〉 − q | B ¯ 0 〉 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaeiiFaWNaemOqai0aaSbaaSqaaiabdYeambqabaGccqGHQms8cqGH9aqpcqWGWbaCcqGG8baFcqWGcbGqdaahaaWcbeqaaiabicdaWaaakiabgQYiXlabgUcaRiabdghaXjabcYha8jqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaGccqGHQms8cqGGSaalaeaacqGG8baFcqWGcbGqdaWgaaWcbaGaemisaGeabeaakiabgQYiXlabg2da9iabdchaWjabcYha8jabdkeacnaaCaaaleqabaGaeGimaadaaOGaeyOkJeVaeyOeI0IaemyCaeNaeiiFaWNafmOqaiKbaebadaahaaWcbeqaaiabicdaWaaakiabgQYiXlabcYcaSaaaaaa@571A@

where

p = 1 2 1 + { B 1 + | { B | 2 , q = 1 2 1 − { B 1 + | { B | 2 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@5802@

CP is violated if ϵ_B ≠ 0, which occurs if |q/p| ≠ 1.

The time dependence of the mass eigenstates is

| B L ( t ) 〉 = e − Γ L t / 2 e i m L t / 2 | B L ( 0 ) 〉 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqGG8baFcqWGcbGqdaWgaaWcbaGaemitaWeabeaakiabcIcaOiabdsha0jabcMcaPiabgQYiXlabg2da9iabdwgaLnaaCaaaleqabaGaeyOeI0Iaeu4KdC0aaSbaaWqaaiabdYeambqabaWcdaWcgaqaaiabdsha0bqaaiabikdaYaaaaaGccqWGLbqzdaahaaWcbeqaaiabdMgaPjabd2gaTnaaBaaameaacqWGmbataeqaaSWaaSGbaeaacqWG0baDaeaacqaIYaGmaaaaaOGaeiiFaWNaemOqai0aaSbaaSqaaiabdYeambqabaGccqGGOaakcqaIWaamcqGGPaqkcqGHQms8aaa@4C46@

| B H ( t ) 〉 = e − Γ H t / 2 e i m H t / 2 | B H ( 0 ) 〉 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqGG8baFcqWGcbGqdaWgaaWcbaGaemisaGeabeaakiabcIcaOiabdsha0jabcMcaPiabgQYiXlabg2da9iabdwgaLnaaCaaaleqabaGaeyOeI0Iaeu4KdC0aaSbaaWqaaiabdIeaibqabaWcdaWcgaqaaiabdsha0bqaaiabikdaYaaaaaGccqWGLbqzdaahaaWcbeqaaiabdMgaPjabd2gaTnaaBaaameaacqWGibasaeqaaSWaaSGbaeaacqWG0baDaeaacqaIYaGmaaaaaOGaeiiFaWNaemOqai0aaSbaaSqaaiabdIeaibqabaGccqGGOaakcqaIWaamcqGGPaqkcqGHQms8cqGGSaalaaa@4D06@

leading to the time evolution of the flavor eigenstates as

| B 0 ( t ) 〉 = e − ( i m + Γ 2 ) t ( cos ⁡ Δ m t 2 | B 0 ( 0 ) 〉 + i q p sin ⁡ Δ m t 2 | B ¯ 0 ( 0 ) 〉 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6B9C@

| B ¯ 0 ( t ) 〉 = e − ( i m + Γ 2 ) t ( i P q sin ⁡ Δ m t 2 | B 0 ( 0 ) 〉 + cos ⁡ Δ m t 2 | B ¯ 0 ( 0 ) 〉 ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6C54@

where m = (m_L + m_H)/2, Δm = m_H - m_L and Γ = Γ_L ≈ Γ_H, and t is the decay time in the B⁰ rest frame, the so-called "proper time". Note that the probability of a B⁰ decay as a function of t is given by ⟨B⁰ (t)|B⁰(t)⟩*, and is a pure exponential, e^-Γt/2, in the absence of CP violation.

Here we choose a final state f which is accessible to both B⁰ and B¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ decays 3. The second amplitude necessary for interference is provided by mixing. Figure 6 shows the decay into f either directly or indirectly via mixing. It is necessary only that f be accessible directly from either state; however if f is a CP eigenstate the situation is far simpler. For CP eigenstates

CP|f_CP⟩ = ± |f_CP⟩.

It is useful to define the amplitudes

A = 〈 f C P | H | B 0 〉 , A ¯ = 〈 f C P | H | B ¯ 0 〉 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaemyqaeKaeyypa0JaeyykJeUaemOzay2aaSbaaSqaaiabdoeadjabdcfaqbqabaGccqGG8baFt0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecscqGG8baFcqWGcbGqdaahaaWcbeqaaiabicdaWaaakiabgQYiXlabcYcaSaqaaiqbdgeabzaaraGaeyypa0JaeyykJeUaemOzay2aaSbaaSqaaiabdoeadjabdcfaqbqabaGccqGG8baFcqWFlecscqGG8baFcuWGcbGqgaqeamaaCaaaleqabaGaeGimaadaaOGaeyOkJeVaeiOla4caaaaa@5518@

If |A¯A| MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaaemaajuaGbaWaaSaaaeaacuWGbbqqgaqeaaqaaiabdgeabbaaaOGaay5bSlaawIa7aaaa@30EF@ ≠ 1, then we have "direct" CP violation in the decay amplitude, which we will discuss in detail later. Here CP can be violated by having

λ = q p ⋅ A ¯ A ≠ 1 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqaH7oaBcqGH9aqpjuaGdaWcaaqaaiabdghaXbqaaiabdchaWbaakiabgwSixNqbaoaalaaabaGafmyqaeKbaebaaeaacqWGbbqqaaGccqGHGjsUcqaIXaqmcqGGSaalaaa@3943@

which requires only that λ acquire a non-zero phase, i.e. |λ| could be unity and CP violation can occur.

Other useful variables, that are independent of any phase convention are

ϕ 12 = arg ⁡ ( − M 12 Γ 12 ) ,  and Im ⁡ Γ 12 M 12 = 1 − | q / p | 4 1 + | q / p | 4 ≡ A S L ( t ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakqaabeqaaiabew9aMnaaBaaaleaacqaIXaqmcqaIYaGmaeqaaOGaeyypa0JagiyyaeMaeiOCaiNaei4zaC2aaeWaaeaacqGHsisljuaGdaWcaaqaaiabd2eannaaBaaabaGaeGymaeJaeGOmaidabeaaaeaacqqHtoWrdaWgaaqaaiabigdaXiabikdaYaqabaaaaaGccaGLOaGaayzkaaGaeiilaWIaeeiiaaIaeeyyaeMaeeOBa4MaeeizaqgabaGagiysaKKaeiyBa0wcfa4aaSaaaeaacqqHtoWrdaWgaaqaaiabigdaXiabikdaYaqabaaabaGaemyta00aaSbaaeaacqaIXaqmcqaIYaGmaeqaaaaakiabg2da9KqbaoaalaaabaGaeGymaeJaeyOeI0IaeiiFaWNaemyCaeNaei4la8IaemiCaaNaeiiFaW3aaWbaaeqabaGaeGinaqdaaaqaaiabigdaXiabgUcaRiabcYha8jabdghaXjabc+caViabdchaWjabcYha8naaCaaabeqaaiabisda0aaaaaGccqGHHjIUcqWGbbqqdaWgaaWcbaGaem4uamLaemitaWeabeaakiabcIcaOiabdsha0jabcMcaPiabc6caUaaaaa@6ABB@

The first quantity can be related to CKM angles, while the second can be measured by the "semileptonic asymmetry," or for that matter in any flavor specific decay 46:

A S L ( t ) = Γ [ B ¯ 0 ( t ) → ℓ − X ] − Γ [ B 0 ( t ) → ℓ + X ] Γ [ B ¯ 0 ( t ) → ℓ − X ] + Γ [ B 0 ( t ) → ℓ + X ] = Δ Γ Δ M tan ⁡ ϕ 12 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@7EBE@

for either B⁰ or for B_s mesons, separately.

A comment on neutral B production at e⁺ e^- colliders is in order. At the ϒ(4S) resonance there is coherent production of B⁰ B¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ pairs. This puts the B's in a C = -1 state. In hadron colliders, or at e⁺ e^- machines operating at the Z⁰, the B's are produced incoherently. For the rest of this article we will assume incoherent production except where explicitly noted.

The asymmetry, in the case of CP eigenstates, is defined as

a f C P = Γ ( B 0 ( t ) → f C P ) − Γ ( B ¯ 0 ( t ) → f C P ) Γ ( B 0 ( t ) → f C P ) + Γ ( B ¯ 0 ( t ) → f C P ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6B59@

which for |q/p| = 1 gives

a f C P = ( 1 − | λ | 2 ) cos ⁡ ( Δ m t ) − 2 Im ⁡ λ sin ⁡ ( Δ m t ) 1 + | λ | 2 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGHbqydaWgaaWcbaGaemOzay2aaSbaaWqaaiabdoeadjabdcfaqbqabaaaleqaaOGaeyypa0tcfa4aaSaaaeaadaqadaqaaiabigdaXiabgkHiTiabcYha8jabeU7aSjabcYha8naaCaaabeqaaiabikdaYaaaaiaawIcacaGLPaaacyGGJbWycqGGVbWBcqGGZbWCcqGGOaakcqqHuoarcqWGTbqBcqWG0baDcqGGPaqkcqGHsislcqaIYaGmcyGGjbqscqGGTbqBcqaH7oaBcyGGZbWCcqGGPbqAcqGGUbGBcqGGOaakcqqHuoarcqWGTbqBcqWG0baDcqGGPaqkaeaacqaIXaqmcqGHRaWkcqGG8baFcqaH7oaBcqGG8baFdaahaaqabeaacqaIYaGmaaaaaOGaeiOla4caaa@5D31@

For the cases where there is only one decay amplitude A, |λ| equals 1, and we have

a f C P = − Im ⁡ λ sin ⁡ ( Δ m t ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGHbqydaWgaaWcbaGaemOzay2aaSbaaWqaaiabdoeadjabdcfaqbqabaaaleqaaOGaeyypa0JaeyOeI0IagiysaKKaeiyBa0Maeq4UdWMagi4CamNaeiyAaKMaeiOBa4MaeiikaGIaeuiLdqKaemyBa0MaemiDaqNaeiykaKIaeiOla4caaa@40BF@

Only the factor -Imλ contains information about the level of CP violation, the sine term is determined by B mixing. In fact, the time integrated asymmetry is given by

a f C P = − x 1 + x 2 Im ⁡ λ = − 0.48 Im ⁡ λ . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGHbqydaWgaaWcbaGaemOzay2aaSbaaWqaaiabdoeadjabdcfaqbqabaaaleqaaOGaeyypa0JaeyOeI0scfa4aaSaaaeaacqWG4baEaeaacqaIXaqmcqGHRaWkcqWG4baEdaahaaqabeaacqaIYaGmaaaaaOGagiysaKKaeiyBa0Maeq4UdWMaeyypa0JaeyOeI0IaeGimaaJaeiOla4IaeGinaqJaeGioaGJagiysaKKaeiyBa0Maeq4UdWMaeiOla4caaa@4710@

This is quite lucky for the study of B_d mesons, as the maximum size of the coefficient for any x is -0.5, close to the measured value of x_d.

Let us now find out how Imλ relates to the CKM parameters. Recall λ=qp⋅A¯A MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeU7aSjabg2da9KqbaoaalaaabaGaemyCaehabaGaemiCaahaaOGaeyyXICDcfa4aaSaaaeaacuWGbbqqgaqeaaqaaiabdgeabbaaaaa@3643@. The first term is the part that comes from mixing:

q p = ( V t b * V t d ) 2 | V t b V t d | 2 = ( 1 − ρ − i η ) 2 ( 1 − ρ + i η ) ( 1 − ρ − i η ) = e − 2 i β  and MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@715C@

Im ⁡ q p = − 2 ( 1 − ρ ) η ( 1 − ρ ) 2 + η 2 = sin ⁡ ( 2 β ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaajuaGcyGGjbqscqGGTbqBdaWcaaqaaiabdghaXbqaaiabdchaWbaacqGH9aqpcqGHsisldaWcaaqaaiabikdaYiabcIcaOiabigdaXiabgkHiTiabeg8aYjabcMcaPiabeE7aObqaaiabcIcaOiabigdaXiabgkHiTiabeg8aYjabcMcaPmaaCaaabeqaaiabikdaYaaacqGHRaWkcqaH3oaAdaahaaqabeaacqaIYaGmaaaaaiabg2da9iGbcohaZjabcMgaPjabc6gaUjabcIcaOiabikdaYiabek7aIjabcMcaPiabc6caUaaa@4E98@

To evaluate the decay part we need to consider specific final states. For example, consider the final state J/ψ K_s. The decay diagram is shown in Figure 7. In this case we do not get a phase from the decay part because

A ¯ A = ( V c b V c s * ) 2 | V c b V c s | 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiqbdgeabzaaraaabaGaemyqaeeaaOGaeyypa0tcfa4aaSaaaeaadaqadaqaaiabdAfawnaaBaaabaGaem4yamMaemOyaigabeaacqWGwbGvdaqhaaqaaiabdogaJjabdohaZbqaaiabcQcaQaaaaiaawIcacaGLPaaadaahaaqabeaacqaIYaGmaaaabaGaeiiFaWNaemOvay1aaSbaaeaacqWGJbWycqWGIbGyaeqaaiabdAfawnaaBaaabaGaem4yamMaem4CamhabeaacqGG8baFdaahaaqabeaacqaIYaGmaaaaaaaa@466A@

is real to order 1/λ⁴.

In this case the final state is a state of negative CP, i.e. CP|J/ψ K_s⟩ = -|J/ψ K_s⟩. This introduces an additional minus sign in the result for Imλ. Before finishing discussion of this final state we need to consider in more detail the presence of the K_s in the final state. Since neutral kaons can mix, we pick up another mixing phase (similar diagrams as for B⁰, see Figure 3). This term creates a phase given by

( q p ) K = ( V c d * V c s ) 2 | V c d V c s | 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaadaqadaqcfayaamaalaaabaGaemyCaehabaGaemiCaahaaaGccaGLOaGaayzkaaWaaSbaaSqaaiabdUealbqabaGccqGH9aqpjuaGdaWcaaqaamaabmaabaGaemOvay1aa0baaeaacqWGJbWycqWGKbazaeaacqGGQaGkaaGaemOvay1aaSbaaeaacqWGJbWycqWGZbWCaeqaaaGaayjkaiaawMcaamaaCaaabeqaaiabikdaYaaaaeaacqGG8baFcqWGwbGvdaWgaaqaaiabdogaJjabdsgaKbqabaGaemOvay1aaSbaaeaacqWGJbWycqWGZbWCaeqaaiabcYha8naaCaaabeqaaiabikdaYaaaaaGaeiilaWcaaa@4AD6@

which is real to order λ⁴. It necessary to include this term, however, since there are other formulations of the CKM matrix than Wolfenstein, which have the phase in a different location. It is important that the physics predictions not depend on the CKM convention. (Here we do not include CP violation in the neutral kaon since it is much smaller than what is expected in the B decay.)

In summary, for the case of f = J/ψ K_s, Imλ = - sin(2β). The angle β is the best measured CP violating angle measured in B meson decays. The process used is B⁰ → J/ψ K_S, although there is some data used with the ψ (2S) or with K_L (where the phase is + sin(2β)).

Although it is normally thought that only one decay amplitude contributes here, in fact one can look for the presence of another source of CP violation, presumably in the decay amplitude, by not assuming |λ| equals one in Eq. 36. Then the time dependence of the decay rate is given by

a f C P = 2 Im ⁡ λ 1 + | λ | 2 sin ⁡ ( Δ m t ) − 1 − | λ | 2 1 + | λ | 2 cos ⁡ ( Δ m t ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@63EA@

Thus afCP MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdggaHnaaBaaaleaacqWGMbGzdaWgaaadbaGaem4qamKaemiuaafabeaaaSqabaaaaa@3033@ (t) has both sin(Δmt) and cos(Δmt) terms, and the coefficients of these terms can be measured. Let us assign the labels

S = 2 Im ⁡ λ 1 + | λ | 2 , C = 1 − | λ | 2 1 + | λ | 2 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NeXpLaeyypa0tcfa4aaSaaaeaacqaIYaGmcyGGjbqscqGGTbqBcqaH7oaBaeaacqaIXaqmcqGHRaWkcqGG8baFcqaH7oaBcqGG8baFdaahaaqabeaacqaIYaGmaaaaaOGaeiilaWcabaGae8NaXpKaeyypa0tcfa4aaSaaaeaacqaIXaqmcqGHsislcqGG8baFcqaH7oaBcqGG8baFdaahaaqabeaacqaIYaGmaaaabaGaeGymaeJaey4kaSIaeiiFaWNaeq4UdWMaeiiFaW3aaWbaaeqabaGaeGOmaidaaaaakiabc6caUaaaaaa@58D3@

(Note that the sign of the S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jr8tbaa@368F@ term changes depending on the CP of the final state, but not the C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ term.)

The most precise measurements of sin(2β) have been made by the BaBar and Belle experiments 4748. These measurements are made at the ϒ(4S) resonance using e⁺ e^- → ϒ(4S) → B⁰ B¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@, and with one of the neutral B's decaying into J/ψ K⁰. Because the ϒ(4S) is a definite quantum state with C = -1, the formulae given above have to modified somewhat. One important change is the definition of t. The time used here is the difference between the decay time of the J/ψ K⁰ and the other B⁰, also called the "tagging" B because we need to determine its flavor, whether B⁰ or B¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@, in order to make the asymmetry measurement.

While we will not discuss flavor tagging in general, it is an important part of CP violation measurements. At the ϒ(4S) once such very useful tag is that of high momentum lepton as a b-quark, part of a B¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ meson decays semileptonically into an ℓ^-, while a b¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkgaIzaaraaaaa@2C5C@-quark decays into an e⁺. Other flavor tagging observables include the charge of kaons and fast pions. Modern experiments combine these in a "neural network," to maximize their tagging efficiencies 49 (see also 50). The BaBar data are shown in Figure 8, Belle has similar results.

The average value as determined by the Heavy Flavor Averaging Group is sin(2β) = 0.671 ± 0.024, where the dominant part of the error is statistical. No evidence is found for a non-zero C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ term with the measured average given as 0.005 ± 0.020.

Determining the sine of any angle gives a four-fold ambiguity in the angle. The decay mode B⁰ → J/ψ K*⁰, K*⁰ → K_s π⁰ offers a way of measuring cos 2β and resolving the ambiguities.

This is a subtle analysis that we will not go into detail on 5152. The result is that β=42.14−1.83+1.88 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabek7aIjabg2da9iabisda0iabikdaYiabc6caUiabigdaXiabisda0maaDaaaleaacqGHsislcqaIXaqmcqGGUaGlcqaI4aaocqaIZaWmaeaacqGHRaWkcqaIXaqmcqGGUaGlcqaI4aaocqaI4aaoaaaaaa@3BE2@ degrees.

The next state to discuss is the CP + eigenstate f ≡ π⁺ π^-. The simple spectator decay diagram is shown in Figure 9(a). For the moment we will assume that this is the only diagram, though the Penguin diagram shown in Figure 9(b) could also contribute; its presence can be inferred because it would induce a non-zero value for C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@, the coefficient of the cosine term in Eq. 43. For this b → uu¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdwha1zaaraaaaa@2C82@d process we have

A ¯ A = ( V u d * V u b ) 2 | V u d V u b | 2 = ( ρ − i η ) 2 ( ρ − i η ) ( ρ + i η ) = e − 2 i γ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6718@

and

Im (λ) = Im (e^{-2i β} e^{-2i γ}) = Im (e^{2i α}) = - sin (2α)

Time dependent CP violation measurements have been made by both the BaBar and Belle collaborations 5354. Both groups find a non-zero value of both C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ and S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jr8tbaa@368F@, though their values are not in particularly good agreement. The HFAG average is shown in Figure 10 along with the experimental results. The value of C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ clearly is not zero, thus demonstrating direct CP violation. (Historically, this was an important observation because it showed that CP violation could not be due to some kind of "superweak" model ala Wolfenstein 55.)

The non-zero value of C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ shows the presence of at least two amplitudes, presumably tree and penguin, in addition to the mixing amplitude, making extraction of α difficult. All is not lost, however. Gronau and London 56 showed that α could be extracted without theoretical error by doing a full isotopic spin analysis of the π π final state. The required measurements include: C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@, S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jr8tbaa@368F@, ℬ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@(B⁺ → π⁺π⁰), ℬ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ (B⁰ → π⁰π⁰), and ℬ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ (B¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ → π⁰π⁰). The last two items require a flavored tagged analysis that has not yet been done and whose prospects are bleak. Grossman and Quinn have showed, however, that an upper limit on Γ (B⁰ → π⁰π⁰)/Γ(B⁺ → π⁺π⁰) can be used to limit the penguin shift to α 575859. Unfortunately, current data show a relative large ℬ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@(B⁰ → π⁰π⁰) = (1.55 ± 0.19) × 10^-6 rate, compared with ℬ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@(B⁰ → π⁺ π^-) = (5.16 ± 0.22) × 10^-6, implying a large penguin contribution 1734, and the limit is very weak.

Use of the ρ⁺ ρ^- final state is in principle quite similar to π⁺ π^-, with some important caveats. First of all, it is a vector-vector final state and therefore could have both CP + and CP- components. It is however possible, doing a full angular analysis to measure the CP violating phase separately in each of these two amplitudes. The best method for this is in the "transversity" basis and will be discussed later 60 in Section 2.2.6. It is possible, however, for one polarization component to be dominant and then the angular analysis might not be necessary. In fact the longitudinal polarization is dominant in this case. BaBar measures the fraction as 0.992±0.024−0.013+0.026 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabiMda5iabiMda5iabikdaYiabgglaXkabicdaWiabc6caUiabicdaWiabikdaYiabisda0maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIZaWmaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqaI2aGnaaaaaa@419D@61, and Belle measures it as 0.941−0.040+0.034±0.030 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabiMda5iabisda0iabigdaXmaaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaI0aancqaIWaamaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIZaWmcqaI0aanaaGccqGHXcqScqaIWaamcqGGUaGlcqaIWaamcqaIZaWmcqaIWaamaaa@4193@62. Thus we can treat this final state without worrying about the angular analysis and just apply a correction for the amount of transverse amplitude.

In addition, ρ mesons are wide, so non-B backgrounds could be a problem and even if the proper B is reconstructed, there are non-resonant and possible a₁ π contributions.

Furthermore, it has been pointed out that the large ρ width could lead to the violation of the isospin constraints and this effect should be investigated 63. The relevant branching ratios are given in Table 1.

Nevertheless, the small branching ratio for ρ⁰ ρ⁰, if indeed it has been observed at all, shows that the effects of the Penguin diagram on the extracted value of sin(2α) are small, and this may indeed be a good way to extract a value of α. The time dependent decay rates separately for B⁰ → ρ⁺ ρ^- and B¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ → ρ⁺ ρ^- and their difference are shown in Figure 11 from BaBar. In is interesting to compare these results with those in Figure 8. We see that the measured asymmetry in the ρ⁺ ρ^- decay more or less cancels that from B⁰ - B¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ mixing inferring that sin(2α) is close to zero.

Results from the time dependent CP violation analysis from both Belle and BaBar are shown in Figure 12.

The data can be averaged and α determined by using the isospin analysis and the rates listed in Table 1. Unfortunately the precision of the data leads only to constraints. These have been determined by both the CKM fitter group 64 and the UT fit group 30. These groups disagree in some cases. The CKM fitter group use a frequentist statistical approach, while UT fit uses a Bayseian approach. The basic difference is the that the Bayesian approach the theoretical errors are taken as having a Gaussian distribution. Here we show in Figure 13 the results from CKM fitter.

The final state π⁺ π^- π⁰ can also be used to extract α. Snyder and Quinn proposed that a Dalitz plot analysis of B → ρ π → π⁺ π^- π⁰ can be used to unravel both α and the relative penguin-tree phases 65. The Dalitz plot for simulated events unaffected by detector acceptance is shown in Figure 14.

The task at hand is to do a time dependent analysis of the magnitude of the decay amplitudes and phases. The analyses of both collaborations allow for ρ(1450) and ρ(1700) contributions in addition to the ρ(770). There are a total of 26 free parameters in the fit. The statistics for such an analysis are not overwhelming: BaBar has about 2100 signal events 66, while Belle has about 1000 67.

The results for the confidence levels of α found by both collaborations are shown in Figure 15. Note that there is also a mirror solution at α + 180°. The Belle collaboration uses their measured decay rates for B⁺ → ρ⁺ π⁰ and B⁺ → ρ⁰ π⁺ coupled with isospin relations 6869 to help constrain α.

The LHCb experiment expects to be able to significantly improve on the determination of α using the ρ π mode. They expect 14,000 events in for an integrated luminosity of 2 fb^-1, with a signal to background ratio greater than 1 70. It is expected that this amount of data can be accumulated in one to two years of normal LHC operation.

Combining all the data, both the CKM fitter and UT fit groups derive a the confidence level plot for α shown in Figure 16. There is a clear disagreement between the two groups, UT fit preferring a solution in the vicinity of 160°, and CKM fitter a value closer to 90°. The CKM fitter group believes that this is due to the UT fit group's use of Bayesian statistics which they criticize 71.

The angle γ=arg⁡[−VudVub*VcdVcb*] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeo7aNjabg2da9iGbcggaHjabckhaYjabcEgaNnaadmaabaGaeyOeI0scfa4aaSaaaeaacqWGwbGvdaWgaaqaaiabdwha1jabdsgaKbqabaGaemOvay1aa0baaeaacqWG1bqDcqWGIbGyaeaacqGGQaGkaaaabaGaemOvay1aaSbaaeaacqWGJbWycqWGKbazaeqaaiabdAfawnaaDaaabaGaem4yamMaemOyaigabaGaeiOkaOcaaaaaaOGaay5waiaaw2faaaaa@470B@. It can be viewed as the phase of the |V_ub| matrix element, with respect to |V_cb|. Interference measurements are required to determine phases. We can use neutral or even charged B decays to measure γ, and there are several ways to do this without using any theoretical assumptions, as is the case for α and β. The first relies on the color suppressed tree diagrams shown in Figure 17, and the second on using the interference in B_s mixing. At first glance, the diagrams in Figure 17 don't appear to have any interfering components. However, if we insist that the final state is common to D⁰ and D¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdseaezaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D3B@, then we have two diagrams which differ in that one has a b → c amplitude and the other a b → u amplitude, the relative weak phase is then γ. Explicitly the amplitudes are defined as

A ( B − → D 0 K − ) ≡ A B A ( B − → D ¯ 0 K − ) ≡ A B r B e i ( δ B − γ ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@56F0@

where r_B reflects the amplitude suppression for the b → u mode and δ_B is the relative phase. We have not yet used identical final states for D⁰ and D¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdseaezaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D3B@ decays, yet we can see that there will be an ambiguity in our result between δ_B and γ that could be resolved by using different D⁰ decay modes.

There are several suggestions as to different D⁰ decay modes to use. In the original paper on this topic Gronau and Wyler 7273 propose using CP eigenstates for the D⁰ decay, such as K⁺ K^-, π⁺ π^- etc.., combining with charge specific decays and comparing B^- with B⁺ decays. In the latter, the sign of the strong phase is flipped with respect to the weak phase. In fact modes such as D*⁰ or K*^- can also be used. (When using D*⁰ there is a difference in δ_B of π between γ D⁰ and π⁰ D⁰ decay modes 74.)

It is convenient to define the follow variables:

A C P ± = [ Γ ( B − → D C P ± 0 ( * ) K − ( * ) ) − Γ ( B + → D C P ± 0 ( * ) K + ( * ) ) ] [ Γ ( B − → D C P ± 0 ( * ) K − ( * ) ) + Γ ( B + → D C P ± 0 ( * ) K + ( * ) ) ] , R C P ± = [ Γ ( B − → D C P ± 0 ( * ) K − ( * ) ) − Γ ( B + → D C P ± 0 ( * ) K + ( * ) ) ] [ Γ ( B − → D 0 ( * ) K − ( * ) ) + Γ ( B + → D ¯ 0 ( * ) K + ( * ) ) ] / 2 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakqaabeqaaiabdgeabnaaBaaaleaacqWGdbWqcqWGqbaucqGHXcqSaeqaaOGaeyypa0tcfa4aaSaaaeaadaWadaqaaiabfo5ahnaabmaabaGaemOqai0aaWbaaeqabaGaeyOeI0caaiabgkziUkabdseaenaaDaaabaGaem4qamKaemiuaaLaeyySaelabaGaeGimaaJaeiikaGIaeiOkaOIaeiykaKcaaiabdUealnaaCaaabeqaaiabgkHiTiabcIcaOiabcQcaQiabcMcaPaaaaiaawIcacaGLPaaacqGHsislcqqHtoWrdaqadaqaaiabdkeacnaaCaaabeqaaiabgUcaRaaacqGHsgIRcqWGebardaqhaaqaaiabdoeadjabdcfaqjabgglaXcqaaiabicdaWiabcIcaOiabcQcaQiabcMcaPaaacqWGlbWsdaahaaqabeaacqGHRaWkcqGGOaakcqGGQaGkcqGGPaqkaaaacaGLOaGaayzkaaaacaGLBbGaayzxaaaabaWaamWaaeaacqqHtoWrdaqadaqaaiabdkeacnaaCaaabeqaaiabgkHiTaaacqGHsgIRcqWGebardaqhaaqaaiabdoeadjabdcfaqjabgglaXcqaaiabicdaWiabcIcaOiabcQcaQiabcMcaPaaacqWGlbWsdaahaaqabeaacqGHsislcqGGOaakcqGGQaGkcqGGPaqkaaaacaGLOaGaayzkaaGaey4kaSIaeu4KdC0aaeWaaeaacqWGcbGqdaahaaqabeaacqGHRaWkaaGaeyOKH4Qaemiraq0aa0baaeaacqWGdbWqcqWGqbaucqGHXcqSaeaacqaIWaamcqGGOaakcqGGQaGkcqGGPaqkaaGaem4saS0aaWbaaeqabaGaey4kaSIaeiikaGIaeiOkaOIaeiykaKcaaaGaayjkaiaawMcaaaGaay5waiaaw2faaaaakiabcYcaSaqaaiabdkfasnaaBaaaleaacqWGdbWqcqWGqbaucqGHXcqSaeqaaOGaeyypa0tcfa4aaSaaaeaadaWadaqaaiabfo5ahnaabmaabaGaemOqai0aaWbaaeqabaGaeyOeI0caaiabgkziUkabdseaenaaDaaabaGaem4qamKaemiuaaLaeyySaelabaGaeGimaaJaeiikaGIaeiOkaOIaeiykaKcaaiabdUealnaaCaaabeqaaiabgkHiTiabcIcaOiabcQcaQiabcMcaPaaaaiaawIcacaGLPaaacqGHsislcqqHtoWrdaqadaqaaiabdkeacnaaCaaabeqaaiabgUcaRaaacqGHsgIRcqWGebardaqhaaqaaiabdoeadjabdcfaqjabgglaXcqaaiabicdaWiabcIcaOiabcQcaQiabcMcaPaaacqWGlbWsdaahaaqabeaacqGHRaWkcqGGOaakcqGGQaGkcqGGPaqkaaaacaGLOaGaayzkaaaacaGLBbGaayzxaaaabaWaamWaaeaacqqHtoWrdaqadaqaaiabdkeacnaaCaaabeqaaiabgkHiTaaacqGHsgIRcqWGebardaahaaqabeaacqaIWaamcqGGOaakcqGGQaGkcqGGPaqkaaGaem4saS0aaWbaaeqabaGaeyOeI0IaeiikaGIaeiOkaOIaeiykaKcaaaGaayjkaiaawMcaaiabgUcaRiabfo5ahnaabmaabaGaemOqai0aaWbaaeqabaGaey4kaScaaiabgkziUkqbdseaezaaraWaaWbaaeqabaGaeGimaaJaeiikaGIaeiOkaOIaeiykaKcaaiabdUealnaaCaaabeqaaiabgUcaRiabcIcaOiabcQcaQiabcMcaPaaaaiaawIcacaGLPaaaaiaawUfacaGLDbaacqGGVaWlcqaIYaGmaaGccqGGUaGlaaaa@E377@

These are related to the variables defined in Eq. 47 as

R C P ± = 1 + r B 2 ± 2 r cos ⁡ δ B cos ⁡ γ , A C P ± = ± 2 r sin ⁡ δ B sin ⁡ γ / R C P ± . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@675D@

Measurements have been made by the BaBar, Belle and CDF collaborations 7576777879. These data, however, are not statistically powerful enough by themselves to give a useful measurement of γ. Atwood, Dunietz and Soni suggested using double-Cabibbo suppressed decays as an alternative means of generating the interference between decay amplitudes 80. For example the final state B^- → D⁰ K^- can proceed via the tree level diagram in Figure 2(a) when the W^- → u¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdwha1zaaraaaaa@2C82@s. Then if we use the doubly-Cabibbo suppressed decay D⁰ → K⁺ π^-, we get a final state that interferes with the D¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdseaezaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D3B@K^- final state with the D¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdseaezaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D3B@ → K⁺ π^-, which is Cabibbo allowed. The advantage of this method, besides extending the number of useful final states, is that both amplitudes are closer to being equal than in the above method, which can generate larger interferences. Any doubly-Cabibbo suppressed decay can be used. Similar equations exist relating the measured parameters to γ and the strong phase shift.

Measurements have been made mostly using the K^± π^∓ final state 8182. Again, these attempts do not yet produce accurate results.

Thus far, the best method for measuring γ uses the three-body final state K_S π⁺ π^-. Since the final state is accessible by both D⁰ and D¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdseaezaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D3B@ decays interference results. By its very nature a three-body state is complicated, and consists of several resonant and non-resonant parts. It is typically analyzed by examining the Dalitz plot where two of the possible three sub-masses (squared) are plotted versus one another 83. In this way, the phase space is described as a uniform density and thus any resonant structure is clearly visible. For our particular case a practical method was suggested by Giri et al 84, where the decay is analyzed in separate regions of the Dalitz plot. Results from this analysis have been reported by BaBar 85 (they also include the mode D⁰ → K_S K⁺ K^-) and Belle 86. BaBar finds γ = (76 ± 22 ± 5 ± 5)°, and Belle finds γ=(76−13+12±4±5)∘ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeo7aNjabg2da9iabcIcaOiabiEda3iabiAda2maaDaaaleaacqGHsislcqaIXaqmcqaIZaWmaeaacqGHRaWkcqaIXaqmcqaIYaGmaaGccqGHXcqScqaI0aancqGHXcqScqaI1aqncqGGPaqkdaahaaWcbeqaaiablIHiVbaaaaa@3E49@. In both cases the first error is statistical, the second systematic, and the third refers to uncertainties caused by the Dalitz plot model. (There is also a mirror solution at ± 180°.)

It has been shown 8788 that measurement of the amplitude magnitudes and phases found in the decays of ψ (3770) → (CP ± Tag)(K_S π⁺ π^-)_D provide useful information that help the narrow model error. The CLEO collaboration is working on such an analysis, and preliminary results have been reported 89. (CLEO also is working on incorporating other D⁰ decay modes.)

Both the CKM fitter and UT fit groups have formed liklihoods for γ based on the measured results, as shown in Figure 18. In this case CKM fitter and UT fit agree on the general shape of the liklihood curve. The CKM fitter plot shows a small disagreement between the Daltiz method and a combination of the other two methods.

The angle χ shown in Figure 1 is the phase that appears in the box diagram for B_s mixing, similar to the diagram for B⁰ mixing shown in Figure 3, but with the d quark replaced by an s quark. The analogous mode to B⁰ → J/ψ K_s in the B_s system is B_s → J/ψ η. The Feynman diagrams are shown in Figure 19. This is very similar to measuring β so χ is often called β_s. We also have a relation between ϕ_s defined in Eq. 33, ϕ_s = -2χ.

Since there are usually two photons present in the η decay, experiments at hadron colliders, which can perform time-dependent studies of B_s mesons, preferentially use the J/ψ ϕ final state. This, unfortunately, introduces another complexity into the problem; as the B_s is spinless the total angular momentum of the final state particles must be zero. For a decay into a vector and scalar, such as J/ψ η, this forces the vector J/ψ to be fully longitudinally polarized with respect to the decay axis. For a vector-vector final state both angular momentum state vectors are either longitudinal (L), both are transverse with linear polarization vectors parallel (||) or they are perpendicular (⊥) to one another 90. Another way of viewing this is that a spin-0 B decay into two massive vector mesons can form CP even states with L = 0 or 2, and a CP odd state with L = 1. The relative populations in the two CP states are determined by strong interactions dynamics, but to study the weak phase here we are not particularly interested in the actual amount, unless of course one state dominated. We do not expect this to be the case however, since the SU(3) related decay B⁰ → J/ψ K*⁰, K*⁰ → K⁺ π^- has a substantial components of both CP states; the PDG quotes gives the longitudinal fraction as (80 ± 8 ± 5)% 17.

The even and odd CP components can be disentangled by measuring the appropriate angular quantities of each event. Following Dighe et al 91, we can decompose the decay amplitude for a B_s as

A ( B s → J / ψ ϕ ) = A 0 ( m ϕ ) / E ϕ e J / ψ * L − A | | e J / ψ * T / 2 − i A ⊥ e ϕ * ⋅ p ^ / 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGbbqqcqGGOaakcqWGcbGqdaWgaaWcbaGaem4CamhabeaakiabgkziUkabdQeakjabc+caViabeI8a5jabew9aMjabcMcaPiabg2da9iabdgeabnaaBaaaleaacqaIWaamaeqaaOGaeiikaGIaemyBa02aaSbaaSqaaiabew9aMbqabaGccqGGPaqkcqGGVaWlcqWGfbqrdaWgaaWcbaGaeqy1dygabeaaruGqLXgBGC0B0HwAJbYu0rgicXwyJTgaiqGakiab=vgaLnaaDaaaleaacqWGkbGscqGGVaWlcqaHipqEaeaacqGGQaGkcqWGmbataaGccqGHsislcqWGbbqqdaWgaaWcbaGaeiiFaWNaeiiFaWhabeaakiab=vgaLnaaDaaaleaacqWGkbGscqGGVaWlcqaHipqEaeaacqGGQaGkcqWGubavaaGccqGGVaWldaGcaaqaaiabikdaYaWcbeaakiabgkHiTiabdMgaPjabdgeabnaaBaaaleaacqGHLkIxaeqaaOGae8xzau2aa0baaSqaaiabew9aMbqaaiabcQcaQaaakiabgwSixlqbhchaWzaajaGaei4la8YaaOaaaeaacqaIYaGmaSqabaGccqGGSaalaaa@70DE@

where ϵ_J/ψ and ϵ_ϕ are polarization 3-vectors in the J/ψ rest frame, p^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbhchaWzaajaaaaa@2C74@ is a unit vector giving the direction of the ϕ momentum in the J/ψ rest frame, and E_ϕ is the energy of the ϕ in the J/ψ rest frame. We note that the corresponding amplitude for the B¯s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaSbaaSqaaiabdohaZbqabaaaaa@2DB7@ decay are A¯0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdgeabzaaraWaaSbaaSqaaiabicdaWaqabaaaaa@2D34@ = A₀, A¯|| MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdgeabzaaraWaaSbaaSqaaiabcYha8jabcYha8bqabaaaaa@2F46@ = A_||, and A¯⊥ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdgeabzaaraWaaSbaaSqaaiabgwQiEbqabaaaaa@2DF7@ = -A_⊥. The amplitudes are normalized so that

dΓ(B_s → J/ψ ϕ)/dt = |A₀|² + |A_|||² + |A_⊥|².

The ϕ meson direction in the J/ψ rest frame defines the x^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdIha4zaajaaaaa@2C80@ direction. The z^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdQha6zaajaaaaa@2C84@ direction is perpendicular to the decay plane of the K⁺ K^- system, where p_y (K⁺) ≥ 0. The decay direction of the ℓ⁺ in the J/ψ rest frame is described by the angles (θ, ϕ). The angle ψ is that formed by the K⁺ direction with the x^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdIha4zaajaaaaa@2C80@-axis in the ϕ rest frame. Figure 20 shows the angles.

The decay width can be written as

d 4 Γ [ B s → ( ℓ + ℓ − ) J / ψ ( K + K − ) ϕ ] d cos ⁡ θ   d ϕ   d cos ⁡ ψ   d t = 9 32 π [ 2 | A 0 | 2 cos ⁡ 2 ψ ( 1 − sin ⁡ 2 θ cos ⁡ 2 ϕ ) + sin ⁡ 2 ψ { | A | | | 2 ( 1 − sin ⁡ 2 θ sin ⁡ 2 ϕ ) + | A ⊥ | 2 sin ⁡ 2 θ − Im ⁡ ( A | | * A ⊥ ) sin ⁡ 2 θ sin ⁡ ϕ } + 1 2 sin ⁡ 2 ψ { Re ⁡ ( A 0 * A | | ) sin ⁡ 2 θ sin ⁡ 2 ϕ + Im ⁡ ( A 0 * A ⊥ ) sin ⁡ 2 θ cos ⁡ ϕ } ] . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGceaqabeaajuaGdaWcaaqaaiaadsgadaahaaqabeaacaaI0aaaaiabfo5ahjaacUfacaWGcbWaaSbaaeaacaWGZbaabeaacqGHsgIRcaGGOaGaeS4eHW2aaWbaaeqabaGaey4kaScaaiabloriSnaaCaaabeqaaiabgkHiTaaacaGGPaWaaSbaaeaacaWGkbGaai4laiabeI8a5bqabaGaaiikaiaadUeadaahaaqabeaacqGHRaWkaaGaam4samaaCaaabeqaaiabgkHiTaaacaGGPaWaaSbaaeaacqaHvpGzaeqaaiaac2faaeaacaWGKbGaci4yaiaac+gacaGGZbGaeqiUdeNaaeiiaiaadsgacqaHvpGzcaqGGaGaamizaiGacogacaGGVbGaai4CaiabeI8a5jaabccacaWGKbGaamiDaaaakiabg2da9KqbaoaalaaabaGaaGyoaaqaaiaaiodacaaIYaGaeqiWdahaaOGaai4waiaaikdacaGG8bGaamyqamaaBaaaleaacaaIWaaabeaakiaacYhadaahaaWcbeqaaiaaikdaaaGcciGGJbGaai4BaiaacohadaahaaWcbeqaaiaaikdaaaGccqaHipqEcaGGOaGaaGymaiabgkHiTiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabeI7aXjGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabew9aMjaacMcaaeaacqGHRaWkciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaHipqEcaGG7bGaaiiFaiaadgeadaWgaaWcbaGaaiiFaiaacYhaaeqaaOGaaiiFamaaCaaaleqabaGaaGOmaaaakiaacIcacaaIXaGaeyOeI0Iaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeqiUdeNaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeqy1dyMaaiykaiabgUcaRiaacYhacaWGbbWaaSbaaSqaaiabgwQiEbqabaGccaGG8bWaaWbaaSqabeaacaaIYaaaaOGaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeqiUdeNaeyOeI0Iaciysaiaac2gacaGGOaGaamyqamaaDaaaleaacaGG8bGaaiiFaaqaaiaacQcaaaGccaWGbbWaaSbaaSqaaiabgwQiEbqabaGccaGGPaGaci4CaiaacMgacaGGUbGaaGOmaiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMjaac2haaeaacqGHRaWkjuaGdaWcaaqaaiaaigdaaeaadaGcaaqaaiaaikdaaeqaaaaakiGacohacaGGPbGaaiOBaiaaikdacqaHipqEcaGG7bGaciOuaiaacwgacaGGOaGaamyqamaaDaaaleaacaaIWaaabaGaaiOkaaaakiaadgeadaWgaaWcbaGaaiiFaiaacYhaaeqaaOGaaiykaiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabeI7aXjGacohacaGGPbGaaiOBaiaaikdacqaHvpGzcqGHRaWkciGGjbGaaiyBaiaacIcacaWGbbWaa0baaSqaaiaaicdaaeaacaGGQaaaaOGaamyqamaaBaaaleaacqGHLkIxaeqaaOGaaiykaiGacohacaGGPbGaaiOBaiaaikdacqaH4oqCciGGJbGaai4BaiaacohacqaHvpGzcaGG9bGaaiyxaiaac6caaaaa@E9A9@

The decay rate for B¯s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaSbaaSqaaiabdohaZbqabaaaaa@2DB7@ can be found by replacing A_⊥ in the above expression with -A_⊥.

Another complexity arises from the expectation that the width difference ΔΓ_s/Γ_s ≈ 15%. This complicates the time dependent rate equations. For convenience, setting ρ→ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbeg8aYzaalaaaaa@2CC9@ ≡ (cos θ, ϕ, cosψ), we have for the decay width for B_s:

d 4 P ( t , ρ → ) d t d ρ → ∝ | A 0 | 2 T + f 1 ( ρ → ) + | A | | | 2 T + f 2 ( ρ → ) + | A ⊥ | 2 T − f 3 ( ρ → ) + | A | | | | A ⊥ | U + f 4 ( ρ → ) + | A 0 | | A | | | cos ⁡ ( δ | | ) T + f 5 ( ρ → ) + | A 0 | | A ⊥ | V + f 6 ( ρ → ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqaaeabdaaaaKqbagaadaWcaaqaaiabdsgaKnaaCaaabeqaaiabisda0aaacqWGqbaucqGGOaakcqWG0baDcqGGSaalcuaHbpGCgaWcaiabcMcaPaqaaiabdsgaKjabdsha0jabdsgaKjqbeg8aYzaalaaaaaGcbaGaeyyhIulabaGaeiiFaWNaemyqae0aaSbaaSqaaiabicdaWaqabaGccqGG8baFdaahaaWcbeqaaiabikdaYaaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaakiab=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rr8vnaaBaaaleaacqGHRaWkaeqaaOGaemOzay2aaSbaaSqaaiabisda0aqabaGccqGGOaakcuaHbpGCgaWcaiabcMcaPaqaaaqaaiabgUcaRaqaaiabcYha8jabdgeabnaaBaaaleaacqaIWaamaeqaaOGaeiiFaWNaeiiFaWNaemyqae0aaSbaaSqaaiabcYha8jabcYha8bqabaGccqGG8baFcyGGJbWycqGGVbWBcqGGZbWCcqGGOaakcqaH0oazdaWgaaWcbaGaeiiFaWNaeiiFaWhabeaakiabcMcaPiab=nr8unaaBaaaleaacqGHRaWkaeqaaOGaemOzay2aaSbaaSqaaiabiwda1aqabaGccqGGOaakcuaHbpGCgaWcaiabcMcaPaqaaaqaaiabgUcaRaqaaiabcYha8jabdgeabnaaBaaaleaacqaIWaamaeqaaOGaeiiFaWNaeiiFaWNaemyqae0aaSbaaSqaaiabgwQiEbqabaGccqGG8baFcqWFveVvdaWgaaWcbaGaey4kaScabeaakiabdAgaMnaaBaaaleaacqaI2aGnaeqaaOGaeiikaGIafqyWdiNbaSaacqGGPaqkcqGGSaalaaaaaa@C624@

where

T ± = [ ( 1 ± cos ⁡ ( 2 χ ) ) e − Γ L t + ( 1 ∓ cos ⁡ ( 2 χ ) ) e − Γ H t ] / 2 , U ± = ± e − Γ t × [ sin ⁡ ( δ ⊥ − δ | | ) cos ⁡ ( Δ m s t ) − cos ⁡ ( δ ⊥ − δ | | ) cos ⁡ ( 2 χ ) sin ⁡ ( Δ m s t ) ± cos ⁡ ( δ ⊥ − δ | | ) sin ⁡ ( 2 χ ) sinh ⁡ ( Δ Γ t / 2 ) ] , V ± = ± e − Γ t × [ sin ⁡ ( δ ⊥ ) cos ⁡ ( Δ m s t ) − cos ⁡ ( δ ⊥ ) cos ⁡ ( 2 χ ) sin ⁡ ( Δ m s t ) ± cos ⁡ ( δ ⊥ ) sin ⁡ ( 2 χ ) sinh ⁡ ( Δ Γ t / 2 ) ] . f 1 ( ρ → ) = 2 cos ⁡ 2 ψ ( 1 − sin ⁡ 2 θ cos ⁡ 2 ϕ ) , f 2 ( ρ → ) = sin ⁡ 2 ψ ( 1 − sin ⁡ 2 θ sin ⁡ 2 ϕ ) , f 3 ( ρ → ) = sin ⁡ 2 ψ sin ⁡ 2 θ , f 4 ( ρ → ) = − sin ⁡ 2 ψ sin ⁡ ( 2 θ ) sin ⁡ ϕ , f 5 ( ρ → ) = sin ⁡ ( 2 ψ ) sin ⁡ 2 θ sin ⁡ ( 2 ϕ ) / 2 , f 6 ( ρ → ) = sin ⁡ ( 2 ψ ) sin ⁡ ( 2 θ ) cos ⁡ ( 2 ϕ ) / 2 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@011F@

The quantities δ_⊥ and δ_|| are the strong phases of A_⊥ and A_|| relative to A₀, respectively 92. The expression for B¯s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaSbaaSqaaiabdohaZbqabaaaaa@2DB7@ mesons can be found by substituting U+ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=rr8vnaaBaaaleaacqGHRaWkaeqaaaaa@37A1@ → U− MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=rr8vnaaBaaaleaacqGHsislaeqaaaaa@37AC@ and V+ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=vr8wnaaBaaaleaacqGHRaWkaeqaaaaa@37A3@ → V− MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=vr8wnaaBaaaleaacqGHsislaeqaaaaa@37AE@.

The most interesting quantities to be extracted from the data are χ and ΔΓ. There are many experimental challenges: the angular and lifetime distributions must be corrected for experimental acceptances; flavor tagging efficiencies and dilutions must be evaluated; backgrounds must be measured. Both the CDF 93 and D0 94 experiments have done this complicated analysis. Updated results as of this writing are summarized by the CKM fitter derived limits shown in Figure 21.

The Standard Model allowed region is a very thin vertical band centered near zero at ϕ_s of -0.036 ± 0.002 (shown in red). (Recall ϕ_s ≡ -2χ ≡ -2β_s.) The region labeled "all" (green) shows the allowed region at 68% confidence level. Although the fit uses several input components besides the CP asymmetry measurements in B_s → J/ψ ϕ, including use of measured total widths introduced via the constraint equation ΔΓs=cos⁡ϕsΔΓsSM MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabfs5aejabfo5ahnaaBaaaleaacqWGZbWCaeqaaOGaeyypa0Jagi4yamMaei4Ba8Maei4CamNaeqy1dy2aaSbaaSqaaiabdohaZbqabaGccqqHuoarcqqHtoWrdaqhaaWcbaGaem4CamhabaGaem4uamLaemyta0eaaaaa@3EBD@, it is the measurement of ϕ_s that dominates the fit result. We see that there is a discrepancy that may be as large as 2.7 standard deviations 9596. While this is as not yet significant, it is very tantalizing. The LHCb experiment plans to vastly improve this measurement 97.

It has been pointed out, however, that there is likely an S-wave K⁺ K^- contribution in the region of the ϕ that contributes 5–10% of the event rate as estimated using Ds+ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdseaenaaDaaaleaacqWGZbWCaeaacqGHRaWkaaaaaa@2E86@ decays 98. For example, analysis of the analogous channel B⁰ → J/ψ K*⁰ reveals about 8% S-wave in the K_π system under the K*⁰, and BaBar has used this to extract a value for cos 2β thus removing an ambiguity in β 99. The S-wave amplitude and phase needs to be added to Eq. 53. Note that the errors will increase due to the addition of another amplitude and phase. The S-wave can manifest itself as a π⁺ π^-, so it is suggested that the decay B_s → J/ψ f₀ (980) be used to measure χ; here the f₀ → π⁺ π^- 98; the estimate is that the useful f₀ rate would be about 20% of the ϕ rate, but the J/ψ f₀ is a CP eigenstate, so an an angular analysis is unnecessary, and these events may provide a determination of χ with an error comparable to that using J/ψ ϕ.

Time integrated asymmetries of B mesons produced at the ϒ(4S) resonance can only be due to direct CP asymmetry, as the mixing generated asymmetry must integrate to zero due to the fact that the initial state has J^PC = 1^--. The first evidence for such direct CP violation at the greater than four standard deviation level in the K^∓ π^± final state was given by BaBar 100. The latest BaBar result is 101

A K − π + = Γ ( B ¯ 0 → K − π + ) − Γ ( B 0 → K + π − ) Γ ( B ¯ 0 → K − π + ) + Γ ( B 0 → K + π − ) = − 0.107 ± 0.016 − 0.004 + 0.006 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@87DD@

showing a large statistical significance.

This result was confirmed by the Belle collaboration, but Belle also measured the isospin conjugate mode. Consider the two-body decays of B mesons into a kaon and a pion, shown in Figure 22. For netural and charged decays, it can proceed via a tree level diagram (a) or a Penguin diagram (b). There are two additional decay diagrams allowed for the B^-, the color-suppressed tree level diagram (c) and the elusive "Electroweak" Penguin diagram in (d). (So named because of the intermediate γ or Z boson.) Since it is expected that diagrams (c) and (d) are small, the direct CP violating asymmetries in both charged and neutral modes should be the same. Yet Belle observed 102

A K ∓ π 0 = Γ ( B − → K − π 0 ) − Γ ( B + → K + π 0 ) Γ ( B − → K − π 0 ) + Γ ( B + → K + π 0 ) = 0.07