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{{Beyond the Standard Model|expanded=Evidence}}
In [[particle physics]], '''CP violation''' is a violation of '''CP-symmetry''' (or '''charge conjugation parity symmetry'''): the combination of [[C-symmetry]] ([[Charge (physics)|charge conjugation]] symmetry) and [[Parity (physics)|P-symmetry]] ([[Parity (physics)|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 [[kaon]]s 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 [[Physical cosmology|cosmology]] to explain the dominance of [[matter]] over [[antimatter]] in the present [[universe]], and in the study of [[weak interaction]]s in particle physics.
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|last1=Schwarzschild |first1=Bertram
|year=1999
|title=Two Experiments Observe Explicit Violation of
|journal=[[Physics Today]]
|volume=52 |issue=2 |pages=19–20
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Overall, the symmetry of a [[quantum mechanics|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,
|last1=Ioffe
|first1=B. L.
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===Indirect CP violation===
In 1964, [[James Cronin]], [[Val Fitch]] and coworkers provided clear evidence from [[kaon]] decay that CP-symmetry could be broken.
The kind of CP violation discovered in 1964 was linked to the fact that neutral [[kaon]]s can transform into their [[antiparticle]]s (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.
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==CP violation in the Standard Model==
"Direct" CP violation is allowed in the [[Standard Model]] if a [[Complex number|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 [[CKM matrix#Counting|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 ''[[Cabibbo–Kobayashi–Maskawa_matrix#The_unitarity_triangles| Jarlskog invariant]]'':
<math display="block">\ J = c_{12}\ c_{13}^2\ c_{23}\ s_{12}\ s_{13}\ s_{23}\ \sin \delta\ \approx\ 0.00003 \ ,</math>
for quarks, which is <math>\ 0.0003\ </math> times the maximum value of <math>\ J_{\max} = \tfrac{1}{6} \sqrt{ 3\ }\ \approx\ 0.1\ .</math> For leptons, only an upper limit exists: <math>\ |J| < 0.03\ .</math>
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Now the formula becomes:
<math display="block">\begin{align}
M &= |M|\ e^{i\theta}\ e^{+i\phi} \\
\bar{M} &= |M|\ e^{i\theta}\ e^{-i\phi}
\end{align}</math>
Physically measurable reaction rates are proportional to <math>\ |M|^{2}\ ,</math> thus so far nothing is different. However, consider that there are ''two different routes'': <math>\ a \overset{1}{\longrightarrow} b\ </math> and <math>\ a \overset{2}{\longrightarrow} b\ </math> or equivalently, two unrelated intermediate states: <math>\ a \rightarrow 1\rightarrow b\ </math> and <math>\ a \rightarrow 2\rightarrow b\ .</math> Now we have:
<math display="block">\begin{alignat}{3}
M &= |M_{1}|\ e^{i\theta_{1}}\ e^{i\phi_{1}} &&+ |M_{2}|\ e^{i\theta_{2}}\ e^{i\phi_{2}} \\
\bar{M} &= |M_{1}|\ e^{i\theta_{1}}\ e^{-i\phi_{1}} &&+ |M_{2}|\ e^{i\theta_{2}}\ e^{-i\phi_{2}}\ .
\end{alignat}</math>
Some further calculation gives:
<math display="block">|M|^{2} - |\bar{M}|^{2} = -4\ |M_{1}|\ |M_{2}|\ \sin(\theta_{1} - \theta_{2})\ \sin(\phi_{1} - \phi_{2})\ .</math>
Thus, we see that a complex phase gives rise to processes that proceed at different rates for particles and antiparticles, and CP is violated.
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Thus, there are two necessary conditions for getting a complex CKM matrix:
# At least one of
# If both of them are complex,
For a standard model with three fermion generations, the most general non-Hermitian pattern of its mass matrices can be given by
<math> M= \begin{bmatrix} 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{bmatrix}. </math>
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 <math> \mathbf{M^2} = M \cdot M^+ </math> can be given by
<math> \mathbf{M^2}=
\begin{bmatrix}
\mathbf{A_1} & \mathbf{B_1}+ i \mathbf{C_1} & \mathbf{B_2} + i \mathbf{C_2} \\ \mathbf{B_1}- i \mathbf{C_1} & \mathbf{A_2} & \mathbf{B_3}+ i \mathbf{C_3} \\ \mathbf{B_2}- i \mathbf{C_2}& \mathbf{B_3}- i \mathbf{C_3}& \mathbf{A_3}
\end{bmatrix},
</math>
and it has the same unitary transformation matrix U with M.
Besides, parameters in <math> \mathbf{M^2} </math> are correlated to those in M directly in the ways shown below
<math> \mathbf{ A_1} = A_1^2 + D_1^2 + B_1^2 + C_1^2 + B_2^2 + C_2^2, </math>
<math>\mathbf{ A_2} = A_2^2 + D_2^2 + B_3^2 + C_3^2 + B_4^2 + C_4^2, </math>
<math> \mathbf{ A_3} = A_3^2 + D_3^2 + B_5^2 + C_5^2 + B_6^2 + C_6^2, </math>
<math> \mathbf{ B_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, </math>
<math> \mathbf{ B_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, </math>
<math> \mathbf{ B_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, </math>
<math> \mathbf{ C_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, </math>
<math>\mathbf{ C_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, </math>
<math> \mathbf{ C_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.
</math>
That means if we diagonalize an <math> \mathbf{M^2} </math> matrix with 9 parameters, it has the same effect as diagonalizing an M matrix with 18 parameters. Therefore, diagonalizing the <math> \mathbf{M^2} </math> matrix is certainly the most reasonable choice.
The M and <math> \mathbf{M^2} </math> 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 <math> \mathbf{M^2} </math> matrix has only 9 parameters, it is still too complicated to be diagonalized directly. Thus, an assumption
<math> \mathbf{M_R^2}} \cdot \mathbf{M_I^{2+}} - \mathbf{M_I^2} \cdot \mathbf{M_R^{2+} =0 </math>
was employed to simplify the pattern, where <math> \mathbf{ M^2_R} </math> is the real part of <math> \mathbf{M^2} </math> and <math> \mathbf{M^2_I} </math> is the imaginary part.
Such an assumption could further reduce the parameter number from 9 to 5 and the reduced <math> \mathbf{M^2} </math> matrix can be given by
<math>
\mathbf {M^2} = \begin{bmatrix} \mathbf{ A} + \mathbf{B (x y- {x \over y})} & \mathbf{ y B} &\mathbf{ x B} \\\mathbf{y B} & \mathbf{A +B ({y \over x}-{x \over y})} & \mathbf{ B} \\ \mathbf{ x B} & \mathbf{ B} & \mathbf{ A} \end{bmatrix}
</math>
<math>
+ i \begin{bmatrix} 0 & \mathbf{ C \over y} & - \mathbf{C \over x} \\ - \mathbf{ C \over y} & 0 & \mathbf{ C} \\ i \mathbf{C \over x} & - \mathbf{C} & 0 \end{bmatrix}
\equiv \mathbf{ M^2_R}+ i \mathbf{ M^2_I},
</math>
where <math> \mathbf{A \equiv A_3}, \mathbf{ B \equiv B_3}, \mathbf{ C \equiv C_3}, \mathbf{x \equiv B_2 / B_3},</math> and <math> \mathbf{ y \equiv B_1 / B_3} </math>.
Diagonalizing <math> \mathbf{M^2} </math> analytically, the eigenvalues are given by
<math> \mathbf{m^2_1} = \mathbf{ A-B {x \over y} -C {\sqrt {\mathbf{ x^2 +y^2 +x^2 y^2} \over {x y}}}},</math>
<math> \mathbf{ m^2_2} = \mathbf{A-B{x \over y} + C{\sqrt \mathbf{ x^2 +y^2 +x^2 y^2} \over {x y}}}, </math>
<math> \mathbf{ m^2_3} = \mathbf{ A+B{{(x^2+1) y} \over x}},</math>
and the U matrix for up-type quarks can then be given by
<math>
\mathbf{U^u} =
\begin{bmatrix}
{-\sqrt{\mathbf{x^2+y^2}} \over \sqrt{\mathbf{2(x^2+y^2+x^2 y^2)}}}
& {\mathbf{{x(y^2-i \sqrt{\mathbf{x^2+y^2+x^2 y^2}})}} \over {\sqrt{2} \sqrt{\mathbf{x^2+y^2}} \sqrt{\mathbf{x^2+y^2+x^2 y^2}}} }
& {\mathbf{{y(x^2+i \sqrt{\mathbf{x^2+y^2+x^2 y^2}})}} \over {\sqrt{2} \sqrt{\mathbf{x^2+y^2}} \sqrt{\mathbf{x^2+y^2+x^2 y^2}}}}
\\ {-\sqrt{\mathbf{x^2+y^2}} \over \sqrt{ \mathbf{2(x^2+y^2+x^2 y^2)}}}
& {\mathbf{x(y^2+i \sqrt{\mathbf{x^2+y^2+x^2 y^2}})} \over {\sqrt{ 2} \sqrt{ \mathbf{ x^2+y^2} \sqrt{ \mathbf{ x^2+y^2+x^2 y^2}}}}}
& {\mathbf{y(x^2-i \sqrt{\mathbf{x^2+y^2+x^2 y^2})} \over {\sqrt{\bf 2} \sqrt{\bf x^2+y^2} \sqrt{\bf x^2+y^2+x^2 y^2}}}}
\\ {\mathbf{x y} \over \sqrt{\mathbf{x^2+y^2+x^2 y^2}}}
& {\mathbf{y \over \sqrt{\mathbf{x^2+y^2+x^2 y^2}}} }
& {\mathbf{x \over \sqrt{\mathbf{x^2+y^2+x^2 y^2}}}}
\end{bmatrix}.
</math>
However, the eigenvalues' order does not necessarily have to be <math> (\mathbf{ m^2_1} , \mathbf{ m^2_2} , \mathbf{ m^2_3} ) </math>; 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 <math> U^u </math>, can be multiplied with the conjugate transpose of the U matrix for down-type quarks, denoted as <math> U^{d+} </math>. 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 <ref> .
{{cite journal
|last1=Lin |first1=C.L.
|year=2021
|title=Exploring the Origin of CP Violation in the Standard Model
|journal=[[Letters in High Energy Physics]]
|volume=221 |pages=1
|arxiv=2010.08245
|doi=10.31526/LHEP.2021.221
|bibcode=2021LHEP....4..221L
|s2cid=245641205
}}</ref>
<ref>
{{cite journal
|last1=Lin |first1=C.L.
|year=2023
|title=BAU Production in the SN-Breaking Standard Model
|journal=[[Symmetry]]
|volume=15 |issue=5
|pages=1051
|arxiv=2209.12490
|doi= 10.3390/sym15051051
|bibcode=2023Symm...15.1051L
|doi-access=free
}}</ref>
Among these 36 potential CKM matrices, 4 of them
<math> V[52]= V \begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix} \begin{bmatrix} 2 & 3 & 1 \end{bmatrix} = V[52]^*=V^* \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \end{bmatrix} = \begin{bmatrix} s & p^* & r^* \\ p^{\prime *} & q & p^{\prime} \\ r & p & s^* \end{bmatrix}
</math> and
<math>
V[22]= V \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} \begin{bmatrix} 2 & 3 & 1 \end{bmatrix} = V^*[55]=V^*\begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \end{bmatrix}
= \begin{bmatrix} r & p & s^* \\ p^{\prime *} & q & p^{\prime} \\ s & p^* & r^* \end{bmatrix},
</math>
fit experimental data to the order of <math> \lambda^{1/2} </math> or better, at tree level, where <math> \lambda </math> is one of the Wolfenstein parameters.
The full expressions of parameters <math> p, q, r, s, </math> and <math> p^{\prime} </math> are given by
<math>
r = {{(x^2+y^2)(x'^2+y'^2)+(x x' +y y')(x y x' y' +\sqrt{x^2+y^2+x^2 y^2} \sqrt{x'^2+ y'^2 +x'^2 y'^2})}
\over {2 \sqrt{ x^2+y^2} \sqrt{ x'^2+y'^2}\sqrt{x^2+y^2+x^2 y^2} \sqrt{x'^2+y'^2+x'^2 y'^2} }} </math>
<math> + i {{ (x y' -x' y)(x' y' \sqrt{x^2+y^2+x^2 y^2} +x y \sqrt{x'^2 +y'^2 +x'^2 y'^2})}
\over {2 \sqrt{ x^2+y^2} \sqrt{ x'^2+y'^2}\sqrt{x^2+y^2+x^2 y^2}\sqrt{x'^2+y'^2+x'^2 y'^2}}}, </math>
<math> s= {{(x^2+y^2)(x'^2+y'^2)+(x x' +y y')(x y x' y' -\sqrt{x^2+y^2+x^2 y^2} \sqrt{x'^2+ y'^2 +x'^2 y'^2})}
\over {2 \sqrt{ x^2+y^2} \sqrt{ x'^2+y'^2}\sqrt{x^2+y^2+x^2 y^2} \sqrt{x'^2+y'^2+x'^2 y'^2} }} </math>
<math> + i {{ (x y' -x' y)(x' y' \sqrt{x^2+y^2+x^2 y^2} -x y \sqrt{x'^2 +y'^2 +x'^2 y'^2})}
\over {2 \sqrt{ x^2+y^2} \sqrt{ x'^2+y'^2}\sqrt{x^2+y^2+x^2 y^2}\sqrt{x'^2+y'^2+x'^2 y'^2}}}, </math>
<math> p = {{[y' y^2(x-x')+ x' x^2 (y-y')]+ i (x y' -x' y) \sqrt{x^2+y^2+x^2 y^2}}
\over {\sqrt{2} \sqrt{ x^2+y^2} \sqrt{x^2+y^2+x^2 y^2}\sqrt{x'^2+y'^2+x'^2 y'^2}}}, </math>
<math> p^{\prime} ={{[y y'^2 (x'-x)+ x x'^2 (y'-y)]+ i (x y' -x' y) \sqrt{x'^2+y'^2+x'^2 y'^2}}
\over {\sqrt{2} \sqrt{ x^2+y^2+x^2 y^2} \sqrt{ x'^2+y'^2}\sqrt{x'^2+y'^2+x'^2 y'^2}} }, </math>
<math> q = {{x x'+y y' +x y x' y'}\over {\sqrt{ x^2+y^2+x^2 y^2} \sqrt{x'^2+y'^2+x'^2 y'^2}}},
</math>
The best fit of the CKM elements are
<math> |V_{ud} |=|V_{tb} | \sim 0.9925, </math>
<math> |V_{ub} |=|V_{td} | \sim 0.0075, </math>
<math> |V_{us} |=|V_{ts} | =|V_{cd} |=|V_{cb} | \sim 0.122023, </math>
and
<math> |V_{cs} |\sim 0.9845. </math>
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==
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==Matter–antimatter imbalance==
{{main|Baryon asymmetry|Baryogenesis}}
{{See also|T-symmetry|Arrow of time|Lorentz transformation}}
{{unsolved|physics|Why does the universe have so much more matter than antimatter?}}
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*[[Neutral particle oscillation]]
*[[Electron electric dipole moment]]
== In popular culture ==
* The video game [[Half-Life 2]] has a song in its soundtrack titled [https://www.youtube.com/watch?v=dqlG3E769Tc CP Violation].
==References==
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*{{cite journal
|author1=Mark Trodden |title=Electroweak Baryogenesis |year=1999 |journal=[[Reviews of Modern Physics]] |volume=71 |issue=5 |pages=1463–1500 |arxiv=hep-ph/9803479 |bibcode = 1999RvMP...71.1463T |doi=10.1103/RevModPhys.71.1463|s2cid=17275359 }}
* {{cite web | author = Davide Castelvecchi | title = What is direct CP-violation? | url = http://www2.slac.stanford.edu/tip/special/cp.htm | publisher = [[SLAC]] | access-date = 2009-07-01 | archive-url = https://web.archive.org/web/20140503090147/http://www2.slac.stanford.edu/tip/special/cp.htm
* An elementary discussion of parity violation and CP violation is given in chapter 15 of this student level textbook [https://www.routledge.com/Fundamentals-of-Molecular-Symmetry/Bunker-Jensen/p/book/9780750309417]
{{Refend}}
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