Movatterモバイル変換

[0]ホーム

Jump to content

CP violation

Edit links

From Wikipedia, the free encyclopedia

(Redirected fromCP-symmetry)

Violation of charge-parity symmetry in particle physics and cosmology

Beyond the Standard Model
SimulatedLarge Hadron Collider CMS particle detector data depicting aHiggs boson produced by colliding protons decaying into hadron jets and electrons
Standard Model
Evidence Hierarchy problem Dark matter Dark energy Quintessence Phantom energy Dark radiation Dark photon Cosmological constant problem Strong CP problem Neutrino oscillation
Theories Brans–Dicke theory Cosmic censorship hypothesis Fifth force F-theory Theory of everything Unified field theory Grand Unified Theory Technicolor Kaluza–Klein theory 6D (2,0) superconformal field theory Noncommutative quantum field theory Quantum cosmology Brane cosmology String theory Superstring theory M-theory Mathematical universe hypothesis Mirror matter Randall–Sundrum model N = 4 supersymmetric Yang–Mills theory Twistor string theory Dark fluid Doubly special relativity de Sitter invariant special relativity Causal fermion systems Black hole thermodynamics Unparticle physics Graviphoton Graviscalar Graviton Gravitino Massive gravity Gauge gravitation theory Gauge theory gravity CPT symmetry
Supersymmetry MSSM NMSSM Superstring theory M-theory Supergravity Supersymmetry breaking Extra dimensions Large extra dimensions
Quantum gravity False vacuum String theory Spin foam Quantum foam Quantum geometry Loop quantum gravity Quantum cosmology Loop quantum cosmology Causal dynamical triangulation Causal fermion systems Causal sets Canonical quantum gravity Semiclassical gravity Superfluid vacuum theory
Experiments ANNIE Gran Sasso INO LHC SNO Super-K Tevatron NOvA
v t e

Inparticle physics,CP violation is a violation ofCP-symmetry (orcharge conjugation parity symmetry): the combination ofC-symmetry (charge conjugation symmetry) andP-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 neutralkaons resulted in theNobel Prize in Physics in 1980 for its discoverersJames Cronin andVal Fitch.

It plays an important role both in the attempts ofcosmology to explain the dominance ofmatter overantimatter in the presentuniverse, and in the study ofweak interactions in particle physics.

Overview

[edit]

Until the 1950s, parity conservation was believed to be one of the fundamental geometricconservation laws (along withconservation of energy andconservation of momentum). After the discovery ofparity violation in 1956, CP-symmetry was proposed to restore order. However, while thestrong interaction andelectromagnetic interaction are experimentally found to be invariant under the combined CP transformation operation, further experiments showed that this symmetry is slightly violated during certain types ofweak decay.

Only a weaker version of the symmetry could be preserved by physical phenomena, which wasCPT symmetry. Besides C and P, there is a third operation, time reversalT, 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 ofquantum field theory, charge conjugation, parity, and time reversal are applied together. Direct observation of thetime reversal symmetry violation without any assumption of CPT theorem was done in 1998 by two groups,CPLEAR and KTeV collaborations, atCERN andFermilab, respectively.^[1] As early as 1970, Klaus Schubert observed T violation independent of assuming CPT symmetry by using the Bell–Steinberger unitarity relation.^[2]

History

[edit]

P-symmetry

[edit]

The idea behindparity 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 achemical reaction orradioactive decay) occurs at the same rate as the original reaction. However, in 1956 a careful critical review of the existing experimental data by theoretical physicistsTsung-Dao Lee andChen-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.^[3] They proposed several possible direct experimental tests.

The first test based onbeta decay ofcobalt-60 nuclei was carried out in 1956 by a group led byChien-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.^[4] However,parity symmetry still appears to be valid for all reactions involvingelectromagnetism andstrong interactions.

CP-symmetry

[edit]

Overall, the symmetry of aquantum mechanical system can be restored if another approximate symmetryS can be found such that the combined symmetryPS remains unbroken. This rather subtle point about the structure ofHilbert space was realized shortly after the discovery ofP violation, and it was proposed that charge conjugation,C, which transforms a particle into itsantiparticle, was the suitable symmetry to restore order.

In 1956Reinhard 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.^[5]Charge violation was confirmed in theWu experiment and in experiments performed byValentine Telegdi andJerome Friedman andGarwin andLederman 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 byJohn Riley Holt at theUniversity of Liverpool.^[6]^[7]^[8]

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.^[5]^[9]

Lev Landau proposed in 1957CP-symmetry,^[10] often called justCP as the true symmetry between matter and antimatter.CP-symmetry is the product of twotransformations: C for charge conjugation and P for parity. In other words, a process in which all particles are exchanged with theirantiparticles 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 atDubna, on Okun's insistence, unsuccessfully searched for CP-violating kaon decay.^[11]

Experimental status

[edit]

Indirect CP violation

[edit]

In 1964,James Cronin,Val Fitch and coworkers provided clear evidence fromkaon decay that CP-symmetry could be broken.^[12] (cf. also Ref.^[13]). This work won them the 1980 Nobel Prize. This discovery showed that weak interactions violate not only the charge-conjugation symmetryC between particles and antiparticles and theP 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 (CPV) discovered in 1964 was linked to the fact that neutralkaons can transform into theirantiparticles (in which eachquark 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 calledindirect CP violation.

Direct CP violation

[edit]

The two box diagrams above are theFeynman 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 theNA31 experiment atCERN 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 atFermilab^[14] and theNA48 experiment atCERN.^[15]

Starting in 2001, a new generation of experiments, including theBaBar experiment at the Stanford Linear Accelerator Center (SLAC)^[16] and theBelle Experiment at the High Energy Accelerator Research Organisation (KEK)^[17] in Japan, observed direct CP violation in a different system, namely in decays of theB mesons.^[18] A large number of CP violation processes inB 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 didnot extend to the strong force, and furthermore, why this was not predicted by the unextendedStandard Model, despite the model's accuracy for "normal" phenomena.

In 2011, a hint of CP violation in decays of neutralD mesons was reported by theLHCb experiment atCERN using 0.6 fb⁻¹ of Run 1 data.^[19] However, the same measurement using the full 3.0 fb⁻¹ Run 1 sample was consistent with CP-symmetry.^[20]

In 2013 LHCb announced discovery of CP violation instrange B meson decays.^[21]

In March 2019, LHCb announced discovery of CP violation in charmed $D^{0}$ decays with a deviation from zero of 5.3 standard deviations.^[22]

In 2020, theT2K Collaboration reported some indications of CP violation in leptons for the first time.^[23]In this experiment, beams of muon neutrinos (
ν
_μ) and muon antineutrinos (
ν
_μ) were alternately produced by anaccelerator. By the time they got to the detector, a significantly higher proportion of electron neutrinos (
ν
_e) was observed from the
ν
_μ beams, than electron antineutrinos (
ν
_e) were from the
ν
_μ beams. Analysis of these observations was 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^[24] and is in slight tension with T2K.^[25]^[26]

CP violation in the Standard Model

[edit]

"Direct" CP violation is allowed in theStandard Model if acomplex phase appears in theCabibbo–Kobayashi–Maskawa matrix (CKM matrix) describingquark mixing, or thePontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix) describingneutrino 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 parametercan 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:

$\ J=c_{12}\ c_{13}^{2}\ c_{23}\ s_{12}\ s_{13}\ s_{23}\ \sin \delta \ \approx \ 0.00003\ ,$

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

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

Now the formula becomes:

${\begin{aligned}{\cal {M}}&=|{\cal {M}}|\ e^{i\theta }\ e^{+i\phi }\\{\bar {\cal {M}}}&=|{\cal {M}}|\ e^{i\theta }\ e^{-i\phi }\end{aligned}}$

Physically measurable reaction rates are proportional to $\ |{\cal {M}}|^{2}$ , thus so far nothing is different. However, consider that there aretwo different routes: $a{\overset {1}{\longrightarrow }}b$ and $a{\overset {2}{\longrightarrow }}b$ or equivalently, two unrelated intermediate states: $a\rightarrow 1\rightarrow b$ and $a\rightarrow 2\rightarrow b$ . This is exactly the case for the kaon where the decay is performed via different quark channels (see the Figure above). In this case we have:

${\begin{alignedat}{3}{\cal {M}}&=|{\cal {M}}_{1}|\ e^{i\theta _{1}}\ e^{i\phi _{1}}&&+|{\cal {M}}_{2}|\ e^{i\theta _{2}}\ e^{i\phi _{2}}\\{\bar {\cal {M}}}&=|{\cal {M}}_{1}|\ e^{i\theta _{1}}\ e^{-i\phi _{1}}&&+|{\cal {M}}_{2}|\ e^{i\theta _{2}}\ e^{-i\phi _{2}}\ .\end{alignedat}}$

Some further calculation gives:

$|{\cal {M}}|^{2}-|{\bar {\cal {M}}}|^{2}=-4\ |{\cal {M}}_{1}|\ |{\cal {M}}_{2}|\ \sin(\theta _{1}-\theta _{2})\ \sin(\phi _{1}-\phi _{2}).$

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 $\ V_{\mathrm {CKM} }=U_{u}^{\dagger }U_{d}$ , where $U_{u}$ and $U_{d}$ are unitary transformation matrices which diagonalize the fermion mass matrices $M_{u}$ and $M_{d}$ , respectively.

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

At least one of $U_{u}$ and $U_{d}$ is complex, or the CKM matrix will be purely real.
If both of them are complex, $U_{u}$ and $U_{d}$ must be different, i.e., $U_{u}\neq U_{d}$ , 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

$M={\begin{bmatrix}A_{1}+iD_{1}&B_{1}+iC_{1}&B_{2}+iC_{2}\\B_{4}+iC_{4}&A_{2}+iD_{2}&B_{3}+iC_{3}\\B_{5}+iC_{5}&B_{6}+iC_{6}&A_{3}+iD_{3}\end{bmatrix}}.$

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 $\mathbf {M^{2}} =M\cdot M^{\dagger }$ can be given by

$\mathbf {M^{2}} ={\begin{bmatrix}\mathbf {A_{1}} &\mathbf {B_{1}} +i\mathbf {C_{1}} &\mathbf {B_{2}} +i\mathbf {C_{2}} \\\mathbf {B_{1}} -i\mathbf {C_{1}} &\mathbf {A_{2}} &\mathbf {B_{3}} +i\mathbf {C_{3}} \\\mathbf {B_{2}} -i\mathbf {C_{2}} &\mathbf {B_{3}} -i\mathbf {C_{3}} &\mathbf {A_{3}} \end{bmatrix}},$

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

${\begin{aligned}\mathbf {A_{1}} &=A_{1}^{2}+D_{1}^{2}+B_{1}^{2}+C_{1}^{2}+B_{2}^{2}+C_{2}^{2},\\\mathbf {A_{2}} &=A_{2}^{2}+D_{2}^{2}+B_{3}^{2}+C_{3}^{2}+B_{4}^{2}+C_{4}^{2},\\\mathbf {A_{3}} &=A_{3}^{2}+D_{3}^{2}+B_{5}^{2}+C_{5}^{2}+B_{6}^{2}+C_{6}^{2},\\\mathbf {B_{1}} &=A_{1}B_{4}+D_{1}C_{4}+B_{1}A_{2}+C_{1}D_{2}+B_{2}B_{3}+C_{2}C_{3},\\\mathbf {B_{2}} &=A_{1}B_{5}+D_{1}C_{5}+B_{1}B_{6}+C_{1}C_{6}+B_{2}A_{3}+C_{2}D_{3},\\\mathbf {B_{3}} &=B_{4}B_{5}+C_{4}C_{5}+B_{6}A_{2}+C_{6}D_{2}+A_{3}B_{3}+D_{3}C_{3},\\\mathbf {C_{1}} &=D_{1}B_{4}-A_{1}C_{4}+A_{2}C_{1}-B_{1}D_{2}+B_{3}C_{2}-B_{2}C_{3},\\\mathbf {C_{2}} &=D_{1}B_{5}-A_{1}C_{5}+B_{6}C_{1}-B_{1}C_{6}+A_{3}C_{2}-B_{2}D_{3},\\\mathbf {C_{3}} &=C_{4}B_{5}-B_{4}C_{5}+D_{2}B_{6}-A_{2}C_{6}+A_{3}C_{3}-B_{3}D_{3}.\end{aligned}}$

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

The $M {\displaystyle M}$ and $\mathbf {M^{2}}$ 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 $\mathbf {M^{2}}$ matrix has only 9 parameters, it is still too complicated to be diagonalized directly. Thus, an assumption

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

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

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

$\mathbf {M^{2}} ={\begin{bmatrix}\mathbf {A} +\mathbf {B} (xy-{x \over y})&y\mathbf {B} &x\mathbf {B} \\y\mathbf {B} &\mathbf {A} +\mathbf {B} ({y \over x}-{x \over y})&\mathbf {B} \\x\mathbf {B} &\mathbf {B} &\mathbf {A} \end{bmatrix}}+i{\begin{bmatrix}0&{\mathbf {C} \over y}&-{\mathbf {C} \over x}\\-{\mathbf {C} \over y}&0&\mathbf {C} \\{\mathbf {C} \over x}&-\mathbf {C} &0\end{bmatrix}}\equiv \mathbf {M^{2}} _{R}+i\mathbf {M^{2}} _{I},$

where $\mathbf {A} \equiv \mathbf {A_{3}} ,\mathbf {B} \equiv \mathbf {B_{3}} ,\mathbf {C} \equiv \mathbf {C_{3}} ,x\equiv \mathbf {B_{2}/B_{3}} ,$ and $y\equiv \mathbf {B_{1}/B_{3}}$ .

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

$\mathbf {m_{1}} ^{2}=\mathbf {A} -\mathbf {B} {x \over y}-\mathbf {C} {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}} \over xy},$

$\mathbf {m_{2}} ^{2}=\mathbf {A} -\mathbf {B} {x \over y}+\mathbf {C} {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}} \over xy},$

$\mathbf {m_{3}} ^{2}=\mathbf {A} +\mathbf {B} {(x^{2}+1)y \over x},$

and the $U {\displaystyle U}$ matrix for up-type quarks can then be given by

However, the order of the eigenvalues and correspondingly the order of the columns of $U_{u}$ does not necessarily have to be $(\mathbf {m_{1}} ^{2},\mathbf {m_{2}} ^{2},\mathbf {m_{3}} ^{2})$ but can be any permutation of those.

After obtaining a general $U {\displaystyle U}$ matrix pattern, the same procedure can be applied to down-type quarks by introducing primed parameters. To construct the CKM matrix, the conjugate transpose of the $U {\displaystyle U}$ matrix for up-type quarks, denoted as $U_{u}^{\dagger }$ , has to be multiplied with the $U {\displaystyle U}$ matrix for down-type quarks, denoted as $U_{d}$ . As mentioned earlier, there are no inherent constraints that dictate the assignment of eigenvalues to specific quark flavors. All $3!\times 3!=36$ potential permutations of eigenvalues are listed elsewhere.^[27]^[28]

Among these 36 potential CKM matrices, 4 of them

$V[52]=V{\begin{bmatrix}1\\3\\2\end{bmatrix}}{\begin{bmatrix}2&3&1\end{bmatrix}}=V[25]^{*}=V^{*}{\begin{bmatrix}2\\3\\1\end{bmatrix}}{\begin{bmatrix}1&3&2\end{bmatrix}}={\begin{bmatrix}s&p&r\\p^{\prime }&q&p^{\prime *}\\r^{*}&p^{*}&s^{*}\end{bmatrix}}$ and

$V[22]=V{\begin{bmatrix}2\\3\\1\end{bmatrix}}{\begin{bmatrix}2&3&1\end{bmatrix}}=V[55]^{*}=V^{*}{\begin{bmatrix}1\\3\\2\end{bmatrix}}{\begin{bmatrix}1&3&2\end{bmatrix}}={\begin{bmatrix}r^{*}&p^{*}&s^{*}\\p^{\prime *}&q&p^{\prime }\\s&p&r\end{bmatrix}},$

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

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

$r={{(x^{2}+y^{2})(x'^{2}+y'^{2})+(xx'+yy')(xyx'y'+{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}}$

 $+i{{(xy'-x'y)(x'y'{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}+xy{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},$

$s={{(x^{2}+y^{2})(x'^{2}+y'^{2})+(xx'+yy')(xyx'y'-{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}}$

 $+i{{(xy'-x'y)(x'y'{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}-xy{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},$

$p={{[y'y^{2}(x-x')+x'x^{2}(y-y')]+i(xy'-x'y){\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}} \over {{\sqrt {2}}{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},$

$p^{\prime }={{[yy'^{2}(x'-x)+xx'^{2}(y'-y)]+i(xy'-x'y){\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}} \over {{\sqrt {2}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},$

$q={{xx'+yy'+xyx'y'} \over {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},$

The best fit of the CKM elements are

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

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

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

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

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

[edit]

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 inquantum chromodynamics. As there is no known reason for it to be conserved in QCD specifically, this is a "fine tuning" problem known as thestrong CP problem.

QCD does not violate the CP-symmetry as easily as theelectroweak theory; unlike the electroweak theory in which the gauge fields couple tochiral currents constructed from thefermionic 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 theelectric dipole moment of theneutron 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 QCDLagrangian that are able to break the CP-symmetry.

${\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {n_{f}g^{2}\theta }{32\pi ^{2}}}F_{\mu \nu }{\tilde {F}}^{\mu \nu }+{\bar {\psi }}\left(i\gamma ^{\mu }D_{\mu }-me^{i\theta '\gamma _{5}}\right)\psi$

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 $\scriptstyle {\tilde {\theta }}$ 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 afine-tuning problem in physics, and is typically solved byphysics beyond the Standard Model.

There are several proposed solutions to solve the strong CP problem. The most well-known isPeccei–Quinn theory, involving newscalar particles calledaxions. A newer, more radical approach not requiring the axion is a theory involvingtwo time dimensions first proposed in 1998 by Bars, Deliduman, and Andreev.^[29]

Matter–antimatter imbalance

[edit]

Main articles:Baryon asymmetry andBaryogenesis

Unsolved problem in physics:

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

(more unsolved problems in physics)

This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.(November 2020) (Learn how and when to remove this message)

The observable universe is made chiefly ofmatter, rather than consisting of equal parts of matter andantimatter as might be expected^{[additional citation(s) needed]}. It can be demonstrated that, to create an imbalance in matter and antimatter from an initial condition of balance, theSakharov conditions must be satisfied, one of which is the existence of CP violation during the extreme conditions of the first seconds after theBig 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.^[30]

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 withantiprotons,electrons withpositrons,neutrons withantineutrons, 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.^[30]

The Standard Model contains at least three sources of CP violation. The first of these, involving theCabibbo–Kobayashi–Maskawa matrix in thequark 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 theelectric 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 thePontecorvo–Maki–Nakagawa–Sakata matrix in thelepton sector. The current long-baseline neutrino oscillation experiments,T2K andNOνA, 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 andDUNE, 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, aneutrino factory could be sensitive to nearly all possible values of the CP violating Dirac phase. If neutrinos areMajorana fermions, thePMNS 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 ofneutrinoless double-beta decay. The best limits come from theGERDA experiment. CP violation in the lepton sector generates a matter-antimatter asymmetry through a process calledleptogenesis. 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.^[31]

If CP violation in the lepton sector is experimentally determined to be too small to account for matter-antimatter asymmetry, some newphysics 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.^[30]

Sakharov proposed a way to restore CP-symmetry using T-symmetry, extending spacetimebefore the Big Bang. He described completeCPT reflections of events on each side of what he called the "initial singularity". Because of this, phenomena with an oppositearrow of time att < 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 att < 0 from contracting matter having an excess of antiquarks, that they pass "one through the other" at the instantt = 0 when the density is infinite, and decay with an excess of quarks whent > 0, realizing total CPT symmetry of the universe. All the phenomena att < 0 are assumed in this hypothesis to be CPT reflections of the phenomena att > 0.
— Andrei Sakharov, inCollected Scientific Works (1982).^[32]

References

[edit]

^Schwarzschild, Bertram (1999). "Two Experiments Observe Explicit Violation of Time-Reversal Symmetry".Physics Today.52 (2):19–20.Bibcode:1999PhT....52b..19S.doi:10.1063/1.882519.
^Schubert, K.R. (2015). "T violation and CPT tests in neutral-meson systems".Progress in Particle and Nuclear Physics.81:1–38.arXiv:1409.5998.Bibcode:2015PrPNP..81....1S.doi:10.1016/j.ppnp.2014.12.001.S2CID 117740717.
^Lee, T. D.; Yang, C. N. (1956)."Question of Parity Conservation in Weak Interactions".Physical Review.104 (1):254–258.Bibcode:1956PhRv..104..254L.doi:10.1103/PhysRev.104.254.
^Wu, C. S.; Ambler, E.; Hayward, R. W.; Hoppes, D. D.; Hudson, R. P. (1957)."Experimental Test of Parity Conservation in Beta Decay".Physical Review.105 (4):1413–1415.Bibcode:1957PhRv..105.1413W.doi:10.1103/PhysRev.105.1413.
^^a ^bIoffe, B. L.; Okun, L. B.; Rudik, A. P. (1957)."The Problem of Parity Non-conservation in Weak Interactions"(PDF).Journal of Experimental and Theoretical Physics.32:328–330.^{[permanent dead link‍]}
^Friedman, J. I.; Telegdi, V. L. (1957). "Nuclear Emulsion Evidence for Parity Nonconservation in the Decay Chain π⁺→μ⁺→e⁺".Physical Review.106 (6):1290–1293.Bibcode:1957PhRv..106.1290F.doi:10.1103/PhysRev.106.1290.
^Garwin, R. L.; Lederman, L. M.; Weinrich, M. (1957)."Observations of the Failure of Conservation of Parity and Charge Conjugation in Meson Decays: The Magnetic Moment of the Free Muon".Physical Review.105 (4):1415–1417.Bibcode:1957PhRv..105.1415G.doi:10.1103/PhysRev.105.1415.
^Culligan, G.; Frank, S. G. F.; Holt, J. R. (1959). "Longitudinal polarization of the electrons from the decay of unpolarized Positive and Negative Muons".Proceedings of the Physical Society.73 (2): 169.Bibcode:1959PPS....73..169C.doi:10.1088/0370-1328/73/2/303.
^Lee, T. D.; Oehme, R.; Yang, C. N. (1957)."Remarks on Possible Noninvariance under Time Reversal and Charge Conjugation".Physical Review.106 (2):340–345.Bibcode:1957PhRv..106..340L.doi:10.1103/PhysRev.106.340. Archived fromthe original on 5 August 2012.
^Landau, L. (1957). "On the conservation laws for weak interactions".Nuclear Physics.3 (1):127–131.Bibcode:1957NucPh...3..127L.doi:10.1016/0029-5582(57)90061-5.
^Anikina, M. Kh.; Neagu, D. V.; Okonov, E. O.; Petrov, N. I.; Rozanova, A. M.; Rusakov, V. A."An experimental investigation of some consequences of CP-invariance in K⁰
₂ meson decays"(PDF).Soviet Physics JETP.15 (1):93–96. Archived fromthe original(PDF) on 27 January 2021. Retrieved3 April 2021.
^Christenson, J. H.; Cronin, J. W.; Fitch, V. L.; Turlay, R. (1964)."Evidence for the 2π Decay of the K⁰
₂ Meson System".Physical Review Letters.13 (4): 138.Bibcode:1964PhRvL..13..138C.doi:10.1103/PhysRevLett.13.138.
^The Fitch-Cronin Experiment
^Alavi-Harati, A.; et al. (KTeV Collaboration) (1999). "Observation of Direct CP Violation in K_S,L→ππ Decays".Physical Review Letters.83 (1):22–27.arXiv:hep-ex/9905060.Bibcode:1999PhRvL..83...22A.doi:10.1103/PhysRevLett.83.22.S2CID 119333352.
^Fanti, V.; et al. (NA48 Collaboration) (1999). "A new measurement of direct CP violation in two pion decays of the neutral kaon".Physics Letters B.465 (1–4):335–348.arXiv:hep-ex/9909022.Bibcode:1999PhLB..465..335F.doi:10.1016/S0370-2693(99)01030-8.S2CID 15277360.
^Aubert, B; et al. (2001). "Measurement of CP-Violating Asymmetries in B⁰ Decays to CP Eigenstates".Physical Review Letters.86 (12):2515–22.arXiv:hep-ex/0102030.Bibcode:2001PhRvL..86.2515A.doi:10.1103/PhysRevLett.86.2515.PMID 11289970.S2CID 24606837.
^Abe K; et al. (2001). "Observation of Large CP Violation in the Neutral B Meson System".Physical Review Letters.87 (9): 091802.arXiv:hep-ex/0107061.Bibcode:2001PhRvL..87i1802A.doi:10.1103/PhysRevLett.87.091802.PMID 11531561.S2CID 3197654.
^Rodgers, Peter (August 2001)."Where did all the antimatter go?".Physics World. p. 11.
^Carbone, A. (2012). "A search for time-integrated CP violation in D⁰→h⁻h⁺ decays".arXiv:1210.8257 [hep-ex].
^LHCb Collaboration (2014). "Measurement of CP asymmetry in D⁰→K⁺K⁻ and D⁰→π⁺π⁻ decays".Journal of High Energy Physics.2014 (7): 41.arXiv:1405.2797.Bibcode:2014JHEP...07..041A.doi:10.1007/JHEP07(2014)041.S2CID 118510475.
^Aaij, R.; et al. (LHCb Collaboration) (30 May 2013). "First Observation of CP Violation in the Decays of B⁰_s Mesons".Physical Review Letters.110 (22): 221601.arXiv:1304.6173.Bibcode:2013PhRvL.110v1601A.doi:10.1103/PhysRevLett.110.221601.PMID 23767711.S2CID 20486226.
^R. Aaij; et al. (LHCb Collaboration) (2019)."Observation of CP Violation in Charm Decays"(PDF).Physical Review Letters.122 (21): 211803.arXiv:1903.08726.Bibcode:2019PhRvL.122u1803A.doi:10.1103/PhysRevLett.122.211803.PMID 31283320.S2CID 84842008.
^Abe, K.; Akutsu, R.; et al. (T2K Collaboration) (16 April 2020). "Constraint on the matter-antimatter symmetry-violating phase in neutrino oscillations".Nature.580 (7803):339–344.arXiv:1910.03887.Bibcode:2020Natur.580..339T.doi:10.1038/s41586-020-2177-0.PMID 32296192.S2CID 203951445.
^Himmel, Alex; et al. (NOvA Collaboration) (2 July 2020)."New Oscillation Results from the NOvA Experiment".Neutrino2020.doi:10.5281/zenodo.3959581.
^Kelly, Kevin J.; Machado, Pedro A.N.; Parke, Stephen J.; Perez-Gonzalez, Yuber F.; Funchal, Renata Zukanovich (2021). "Neutrino mass ordering in light of recent data".Physical Review D.103 (1): 013004.arXiv:2007.08526.Bibcode:2021PhRvD.103a3004K.doi:10.1103/PhysRevD.103.013004.S2CID 220633488.
^Denton, Peter B.; Gehrlein, Julia; Pestes, Rebekah (2021). "CP-Violating Neutrino Non-Standard Interactions in Long-Baseline-Accelerator Data".Physical Review Letters.126 (5): 051801.arXiv:2008.01110.Bibcode:2021PhRvL.126e1801D.doi:10.1103/PhysRevLett.126.051801.PMID 33605742.S2CID 220961778.
^Lin, C.L. (2021). "Exploring the Origin of CP Violation in the Standard Model".Letters in High Energy Physics.221: 1.arXiv:2010.08245.Bibcode:2021LHEP....4..221L.doi:10.31526/LHEP.2021.221.S2CID 245641205.
^Lin, C.L. (2023)."BAU Production in the SN-Breaking Standard Model".Symmetry.15 (5): 1051.arXiv:2209.12490.Bibcode:2023Symm...15.1051L.doi:10.3390/sym15051051.
^I. Bars; C. Deliduman; O. Andreev (1998). "Gauged Duality, Conformal Symmetry, and Spacetime with Two Times".Physical Review D.58 (6): 066004.arXiv:hep-th/9803188.Bibcode:1998PhRvD..58f6004B.doi:10.1103/PhysRevD.58.066004.S2CID 8314164.
^^a ^b ^cSakharov, Andrei (January 1967). "Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe".Andrei Sakharov.5 (24):24–27.doi:10.1070/PU1991v034n05ABEH002497 – via American Institute of Physics (AIP).
^Mauger, Christopher (17 July 2023)."CP violation searches in neutrino oscillations"(PDF).Nature.580 (7805):339–344 – via PubMed.
^Sakharov, A. D. (7 December 1982).Collected Scientific Works.Marcel Dekker.ISBN 978-0824717148.

External links

[edit]

Cern Courier article

v t e C, P, and T symmetries
C-symmetry P-symmetry T-symmetry
CP CP symmetry CPT symmetry
Chirality Pin group Symmetry (physics)

Standard Model

Background

Constituents

Beyond the
Standard Model

Evidence	Hierarchy problem Dark matter Cosmological constant problem Strong CP problem Neutrino oscillation
Theories	Technicolor Kaluza–Klein theory Grand Unified Theory Theory of everything
Supersymmetry	MSSM NMSSM Split supersymmetry Supergravity
Quantum gravity	String theory Superstring theory Loop quantum gravity Causal dynamical triangulation Canonical quantum gravity Superfluid vacuum theory Twistor theory