CP violation

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

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

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

Overview

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

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

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

History

P-symmetry

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

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

CP-symmetry

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

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

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

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

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

Experimental status

Indirect CP violation

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

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

Direct CP violation

Kaon oscillation box diagram

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

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

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

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

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

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

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

CP violation in the Standard Model

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

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

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

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

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

Now the formula becomes:

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

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

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

Some further calculation gives:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Among these 36 potential CKM matrices, 4 of them

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

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

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

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

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

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

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

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

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

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

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

The best fit of the CKM elements are

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

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

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

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

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

Strong CP problem

Main article: Strong CP problem

Unsolved problem in physics:

Why is the strong nuclear interaction force CP-invariant?

(more unsolved problems in physics)

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

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

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

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

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

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

Matter–antimatter imbalance

Main articles: Baryon asymmetry and Baryogenesis

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

 

Unsolved problem in physics:

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

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

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

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

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

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

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

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

Spontaneous symmetry breaking

From Wikipedia, the free encyclopedia

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

Spontaneous symmetry breaking illustrated: At high energy levels (left), the ball settles in the center, and the result is symmetric. At lower energy levels (right), the overall "rules" remain symmetric, but the symmetric "sombrero" enforces an asymmetric outcome, since eventually the ball must rest at some random spot on the bottom, "spontaneously", and not all others.

Feynman diagram

Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry.

Overview

By definition, spontaneous symmetry breaking requires the existence of physical laws (e.g. quantum mechanics) which are invariant under a symmetry transformation (such as translation or rotation), so that any pair of outcomes differing only by that transformation have the same probability distribution. For example if measurements of an observable at any two different positions have the same probability distribution, the observable has translational symmetry.

Spontaneous symmetry breaking occurs when this relation breaks down, while the underlying physical laws remain symmetrical.

Conversely, in explicit symmetry breaking, if two outcomes are considered, the probability distributions of a pair of outcomes can be different. For example in an electric field, the forces on a charged particle are different in different directions, so the rotational symmetry is explicitly broken by the electric field which does not have this symmetry.

Phases of matter, such as crystals, magnets, and conventional superconductors, as well as simple phase transitions can be described by spontaneous symmetry breaking. Notable exceptions include topological phases of matter like the fractional quantum Hall effect.

Typically, when spontaneous symmetry breaking occurs, the observable properties of the system change in multiple ways. For example the density, compressibility, coefficient of thermal expansion, and specific heat will be expected to change when a liquid becomes a solid.

Examples

Sombrero potential

See also: Coleman–Weinberg potential

Consider a symmetric upward dome with a trough circling the bottom. If a ball is put at the very peak of the dome, the system is symmetric with respect to a rotation around the center axis. But the ball may spontaneously break this symmetry by rolling down the dome into the trough, a point of lowest energy. Afterward, the ball has come to a rest at some fixed point on the perimeter. The dome and the ball retain their individual symmetry, but the system does not.

Graph of Goldstone's "sombrero" potential function ${\displaystyle V(\varphi )}$.

In the simplest idealized relativistic model, the spontaneously broken symmetry is summarized through an illustrative scalar field theory. The relevant Lagrangian of a scalar field ${\displaystyle \varphi }$, which essentially dictates how a system behaves, can be split up into kinetic and potential terms,

${\displaystyle \mathcal{L}={\mathrm{\partial }}^{\mu }\varphi {\mathrm{\partial }}_{\mu }\varphi -V(\varphi ).}$

(1)

It is in this potential term ${\displaystyle V(\varphi )}$ that the symmetry breaking is triggered. An example of a potential, due to Jeffrey Goldstone is illustrated in the graph at the left.

${\displaystyle V(\varphi )=-5|\varphi {|}^2+|\varphi {|}^4}$.

(2)

This potential has an infinite number of possible minima (vacuum states) given by

${\displaystyle \varphi =\sqrt{5/2}{e}^{i\theta }}$.

(3)

for any real θ between 0 and 2π. The system also has an unstable vacuum state corresponding to Φ = 0. This state has a U(1) symmetry. However, once the system falls into a specific stable vacuum state (amounting to a choice of θ), this symmetry will appear to be lost, or "spontaneously broken".

In fact, any other choice of θ would have exactly the same energy, and the defining equations respect the symmetry but the ground state (vacuum) of the theory breaks the symmetry, implying the existence of a massless Nambu–Goldstone boson, the mode running around the circle at the minimum of this potential, and indicating there is some memory of the original symmetry in the Lagrangian.

Other examples

• For ferromagnetic materials, the underlying laws are invariant under spatial rotations. Here, the order parameter is the magnetization, which measures the magnetic dipole density. Above the Curie temperature, the order parameter is zero, which is spatially invariant, and there is no symmetry breaking. Below the Curie temperature, however, the magnetization acquires a constant nonvanishing value, which points in a certain direction (in the idealized situation where we have full equilibrium; otherwise, translational symmetry gets broken as well). The residual rotational symmetries which leave the orientation of this vector invariant remain unbroken, unlike the other rotations which do not and are thus spontaneously broken.
• The laws describing a solid are invariant under the full Euclidean group, but the solid itself spontaneously breaks this group down to a space group. The displacement and the orientation are the order parameters.
• General relativity has a Lorentz symmetry, but in FRW cosmological models, the mean 4-velocity field defined by averaging over the velocities of the galaxies (the galaxies act like gas particles at cosmological scales) acts as an order parameter breaking this symmetry. Similar comments can be made about the cosmic microwave background.
• For the electroweak model, as explained earlier, a component of the Higgs field provides the order parameter breaking the electroweak gauge symmetry to the electromagnetic gauge symmetry. Like the ferromagnetic example, there is a phase transition at the electroweak temperature. The same comment about us not tending to notice broken symmetries suggests why it took so long for us to discover electroweak unification.
• In superconductors, there is a condensed-matter collective field ψ, which acts as the order parameter breaking the electromagnetic gauge symmetry.
• Take a thin cylindrical plastic rod and push both ends together. Before buckling, the system is symmetric under rotation, and so visibly cylindrically symmetric. But after buckling, it looks different, and asymmetric. Nevertheless, features of the cylindrical symmetry are still there: ignoring friction, it would take no force to freely spin the rod around, displacing the ground state in time, and amounting to an oscillation of vanishing frequency, unlike the radial oscillations in the direction of the buckle. This spinning mode is effectively the requisite Nambu–Goldstone boson.
• Consider a uniform layer of fluid over an infinite horizontal plane. This system has all the symmetries of the Euclidean plane. But now heat the bottom surface uniformly so that it becomes much hotter than the upper surface. When the temperature gradient becomes large enough, convection cells will form, breaking the Euclidean symmetry.
• Consider a bead on a circular hoop that is rotated about a vertical diameter. As the rotational velocity is increased gradually from rest, the bead will initially stay at its initial equilibrium point at the bottom of the hoop (intuitively stable, lowest gravitational potential). At a certain critical rotational velocity, this point will become unstable and the bead will jump to one of two other newly created equilibria, equidistant from the center. Initially, the system is symmetric with respect to the diameter, yet after passing the critical velocity, the bead ends up in one of the two new equilibrium points, thus breaking the symmetry.
• The two-balloon experiment is an example of spontaneous symmetry breaking when both balloons are initially inflated to the local maximum pressure. When some air flows from one balloon into the other, the pressure in both balloons will drop, making the system more stable in the asymmetric state.

In particle physics

In particle physics, the force carrier particles are normally specified by field equations with gauge symmetry; their equations predict that certain measurements will be the same at any point in the field. For instance, field equations might predict that the mass of two quarks is constant. Solving the equations to find the mass of each quark might give two solutions. In one solution, quark A is heavier than quark B. In the second solution, quark B is heavier than quark A by the same amount. The symmetry of the equations is not reflected by the individual solutions, but it is reflected by the range of solutions.

An actual measurement reflects only one solution, representing a breakdown in the symmetry of the underlying theory. "Hidden" is a better term than "broken", because the symmetry is always there in these equations. This phenomenon is called spontaneous symmetry breaking (SSB) because nothing (that we know of) breaks the symmetry in the equations. By the nature of spontaneous symmetry breaking, different portions of the early Universe would break symmetry in different directions, leading to topological defects, such as two-dimensional domain walls, one-dimensional cosmic strings, zero-dimensional monopoles, and/or textures, depending on the relevant homotopy group and the dynamics of the theory. For example, Higgs symmetry breaking may have created primordial cosmic strings as a byproduct. Hypothetical GUT symmetry-breaking generically produces monopoles, creating difficulties for GUT unless monopoles (along with any GUT domain walls) are expelled from our observable Universe through cosmic inflation.

Chiral symmetry

Main article: Chiral symmetry breaking

Chiral symmetry breaking is an example of spontaneous symmetry breaking affecting the chiral symmetry of the strong interactions in particle physics. It is a property of quantum chromodynamics, the quantum field theory describing these interactions, and is responsible for the bulk of the mass (over 99%) of the nucleons, and thus of all common matter, as it converts very light bound quarks into 100 times heavier constituents of baryons. The approximate Nambu–Goldstone bosons in this spontaneous symmetry breaking process are the pions, whose mass is an order of magnitude lighter than the mass of the nucleons. It served as the prototype and significant ingredient of the Higgs mechanism underlying the electroweak symmetry breaking.

Higgs mechanism

Main articles: Higgs mechanism and Yukawa interaction

The strong, weak, and electromagnetic forces can all be understood as arising from gauge symmetries, which is a redundancy in the description of the symmetry. The Higgs mechanism, the spontaneous symmetry breaking of gauge symmetries, is an important component in understanding the superconductivity of metals and the origin of particle masses in the standard model of particle physics. The term "spontaneous symmetry breaking" is a misnomer here as Elitzur's theorem states that local gauge symmetries can never be spontaneously broken. Rather, after gauge fixing, the global symmetry (or redundancy) can be broken in a manner formally resembling spontaneous symmetry breaking. One important consequence of the distinction between true symmetries and gauge symmetries, is that the massless Nambu–Goldstone resulting from spontaneous breaking of a gauge symmetry are absorbed in the description of the gauge vector field, providing massive vector field modes, like the plasma mode in a superconductor, or the Higgs mode observed in particle physics.

In the standard model of particle physics, spontaneous symmetry breaking of the SU(2) × U(1) gauge symmetry associated with the electro-weak force generates masses for several particles, and separates the electromagnetic and weak forces. The W and Z bosons are the elementary particles that mediate the weak interaction, while the photon mediates the electromagnetic interaction. At energies much greater than 100 GeV, all these particles behave in a similar manner. The Weinberg–Salam theory predicts that, at lower energies, this symmetry is broken so that the photon and the massive W and Z bosons emerge. In addition, fermions develop mass consistently.

Without spontaneous symmetry breaking, the Standard Model of elementary particle interactions requires the existence of a number of particles. However, some particles (the W and Z bosons) would then be predicted to be massless, when, in reality, they are observed to have mass. To overcome this, spontaneous symmetry breaking is augmented by the Higgs mechanism to give these particles mass. It also suggests the presence of a new particle, the Higgs boson, detected in 2012.

Superconductivity of metals is a condensed-matter analog of the Higgs phenomena, in which a condensate of Cooper pairs of electrons spontaneously breaks the U(1) gauge symmetry associated with light and electromagnetism.

Dynamical symmetry breaking

Dynamical symmetry breaking (DSB) is a special form of spontaneous symmetry breaking in which the ground state of the system has reduced symmetry properties compared to its theoretical description (i.e., Lagrangian).

Dynamical breaking of a global symmetry is a spontaneous symmetry breaking, which happens not at the (classical) tree level (i.e., at the level of the bare action), but due to quantum corrections (i.e., at the level of the effective action).

Dynamical breaking of a gauge symmetry is subtler. In conventional spontaneous gauge symmetry breaking, there exists an unstable Higgs particle in the theory, which drives the vacuum to a symmetry-broken phase (i.e, electroweak interactions.) In dynamical gauge symmetry breaking, however, no unstable Higgs particle operates in the theory, but the bound states of the system itself provide the unstable fields that render the phase transition. For example, Bardeen, Hill, and Lindner published a paper that attempts to replace the conventional Higgs mechanism in the standard model by a DSB that is driven by a bound state of top-antitop quarks. (Such models, in which a composite particle plays the role of the Higgs boson, are often referred to as "Composite Higgs models".) Dynamical breaking of gauge symmetries is often due to creation of a fermionic condensate — e.g., the quark condensate, which is connected to the dynamical breaking of chiral symmetry in quantum chromodynamics. Conventional superconductivity is the paradigmatic example from the condensed matter side, where phonon-mediated attractions lead electrons to become bound in pairs and then condense, thereby breaking the electromagnetic gauge symmetry.

In condensed matter physics

Most phases of matter can be understood through the lens of spontaneous symmetry breaking. For example, crystals are periodic arrays of atoms that are not invariant under all translations (only under a small subset of translations by a lattice vector). Magnets have north and south poles that are oriented in a specific direction, breaking rotational symmetry. In addition to these examples, there are a whole host of other symmetry-breaking phases of matter — including nematic phases of liquid crystals, charge- and spin-density waves, superfluids, and many others.

There are several known examples of matter that cannot be described by spontaneous symmetry breaking, including: topologically ordered phases of matter, such as fractional quantum Hall liquids, and spin-liquids. These states do not break any symmetry, but are distinct phases of matter. Unlike the case of spontaneous symmetry breaking, there is not a general framework for describing such states.

Continuous symmetry

The ferromagnet is the canonical system that spontaneously breaks the continuous symmetry of the spins below the Curie temperature and at h = 0, where h is the external magnetic field. Below the Curie temperature, the energy of the system is invariant under inversion of the magnetization m(x) such that m(x) = −m(−x). The symmetry is spontaneously broken as h → 0 when the Hamiltonian becomes invariant under the inversion transformation, but the expectation value is not invariant.

Spontaneously-symmetry-broken phases of matter are characterized by an order parameter that describes the quantity which breaks the symmetry under consideration. For example, in a magnet, the order parameter is the local magnetization.

Spontaneous breaking of a continuous symmetry is inevitably accompanied by gapless (meaning that these modes do not cost any energy to excite) Nambu–Goldstone modes associated with slow, long-wavelength fluctuations of the order parameter. For example, vibrational modes in a crystal, known as phonons, are associated with slow density fluctuations of the crystal's atoms. The associated Goldstone mode for magnets are oscillating waves of spin known as spin-waves. For symmetry-breaking states, whose order parameter is not a conserved quantity, Nambu–Goldstone modes are typically massless and propagate at a constant velocity.

An important theorem, due to Mermin and Wagner, states that, at finite temperature, thermally activated fluctuations of Nambu–Goldstone modes destroy the long-range order, and prevent spontaneous symmetry breaking in one- and two-dimensional systems. Similarly, quantum fluctuations of the order parameter prevent most types of continuous symmetry breaking in one-dimensional systems even at zero temperature. (An important exception is ferromagnets, whose order parameter, magnetization, is an exactly conserved quantity and does not have any quantum fluctuations.)

Other long-range interacting systems, such as cylindrical curved surfaces interacting via the Coulomb potential or Yukawa potential, have been shown to break translational and rotational symmetries. It was shown, in the presence of a symmetric Hamiltonian, and in the limit of infinite volume, the system spontaneously adopts a chiral configuration — i.e., breaks mirror plane symmetry.

Generalisation and technical usage

For spontaneous symmetry breaking to occur, there must be a system in which there are several equally likely outcomes. The system as a whole is therefore symmetric with respect to these outcomes. However, if the system is sampled (i.e. if the system is actually used or interacted with in any way), a specific outcome must occur. Though the system as a whole is symmetric, it is never encountered with this symmetry, but only in one specific asymmetric state. Hence, the symmetry is said to be spontaneously broken in that theory. Nevertheless, the fact that each outcome is equally likely is a reflection of the underlying symmetry, which is thus often dubbed "hidden symmetry", and has crucial formal consequences. (See the article on the Goldstone boson.)

When a theory is symmetric with respect to a symmetry group, but requires that one element of the group be distinct, then spontaneous symmetry breaking has occurred. The theory must not dictate which member is distinct, only that one is. From this point on, the theory can be treated as if this element actually is distinct, with the proviso that any results found in this way must be resymmetrized, by taking the average of each of the elements of the group being the distinct one.

The crucial concept in physics theories is the order parameter. If there is a field (often a background field) which acquires an expectation value (not necessarily a vacuum expectation value) which is not invariant under the symmetry in question, we say that the system is in the ordered phase, and the symmetry is spontaneously broken. This is because other subsystems interact with the order parameter, which specifies a "frame of reference" to be measured against. In that case, the vacuum state does not obey the initial symmetry (which would keep it invariant, in the linearly realized Wigner mode in which it would be a singlet), and, instead changes under the (hidden) symmetry, now implemented in the (nonlinear) Nambu–Goldstone mode. Normally, in the absence of the Higgs mechanism, massless Goldstone bosons arise.

The symmetry group can be discrete, such as the space group of a crystal, or continuous (e.g., a Lie group), such as the rotational symmetry of space. However, if the system contains only a single spatial dimension, then only discrete symmetries may be broken in a vacuum state of the full quantum theory, although a classical solution may break a continuous symmetry.

Nobel Prize

On October 7, 2008, the Royal Swedish Academy of Sciences awarded the 2008 Nobel Prize in Physics to three scientists for their work in subatomic physics symmetry breaking. Yoichiro Nambu, of the University of Chicago, won half of the prize for the discovery of the mechanism of spontaneous broken symmetry in the context of the strong interactions, specifically chiral symmetry breaking. Physicists Makoto Kobayashi and Toshihide Maskawa, of Kyoto University, shared the other half of the prize for discovering the origin of the explicit breaking of CP symmetry in the weak interactions. This origin is ultimately reliant on the Higgs mechanism, but, so far understood as a "just so" feature of Higgs couplings, not a spontaneously broken symmetry phenomenon.

Chain reaction

From Wikipedia, the free encyclopedia

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

A chain reaction is a sequence of reactions where a reactive product or by-product causes additional reactions to take place. In a chain reaction, positive feedback leads to a self-amplifying chain of events.

Chain reactions are one way that systems which are not in thermodynamic equilibrium can release energy or increase entropy in order to reach a state of higher entropy. For example, a system may not be able to reach a lower energy state by releasing energy into the environment, because it is hindered or prevented in some way from taking the path that will result in the energy release. If a reaction results in a small energy release making way for more energy releases in an expanding chain, then the system will typically collapse explosively until much or all of the stored energy has been released.

A macroscopic metaphor for chain reactions is thus a snowball causing a larger snowball until finally an avalanche results ("snowball effect"). This is a result of stored gravitational potential energy seeking a path of release over friction. Chemically, the equivalent to a snow avalanche is a spark causing a forest fire. In nuclear physics, a single stray neutron can result in a prompt critical event, which may finally be energetic enough for a nuclear reactor meltdown or (in a bomb) a nuclear explosion.

Another metaphor for a chain reaction is the domino effect, named after the act of domino toppling, where the simple action of toppling one domino leads to all dominoes eventually toppling, even if they are significantly larger.

Numerous chain reactions can be represented by a mathematical model based on Markov chains.

Chemical chain reactions

History

In 1913, the German chemist Max Bodenstein first put forth the idea of chemical chain reactions. If two molecules react, not only molecules of the final reaction products are formed, but also some unstable molecules which can further react with the parent molecules with a far larger probability than the initial reactants. (In the new reaction, further unstable molecules are formed besides the stable products, and so on.)

In 1918, Walther Nernst proposed that the photochemical reaction between hydrogen and chlorine is a chain reaction in order to explain what is known as the quantum yield phenomena. This means that one photon of light is responsible for the formation of as many as 10⁶ molecules of the product HCl. Nernst suggested that the photon dissociates a Cl₂ molecule into two Cl atoms which each initiate a long chain of reaction steps forming HCl.

In 1923, Danish and Dutch scientists J. A. Christiansen and Hendrik Anthony Kramers, in an analysis of the formation of polymers, pointed out that such a chain reaction need not start with a molecule excited by light, but could also start with two molecules colliding violently due to thermal energy as previously proposed for initiation of chemical reactions by van' t Hoff.

Christiansen and Kramers also noted that if, in one link of the reaction chain, two or more unstable molecules are produced, the reaction chain would branch and grow. The result is in fact an exponential growth, thus giving rise to explosive increases in reaction rates, and indeed to chemical explosions themselves. This was the first proposal for the mechanism of chemical explosions.

A quantitative chain chemical reaction theory was created later on by Soviet physicist Nikolay Semyonov in 1934. Semyonov shared the Nobel Prize in 1956 with Sir Cyril Norman Hinshelwood, who independently developed many of the same quantitative concepts.

Typical steps

The main types of steps in chain reaction are of the following types.

• Initiation (formation of active particles or chain carriers, often free radicals, in either a thermal or a photochemical step)
• Propagation (may comprise several elementary steps in a cycle, where the active particle through reaction forms another active particle which continues the reaction chain by entering the next elementary step). In effect the active particle serves as a catalyst for the overall reaction of the propagation cycle. Particular cases are:
  • chain branching (a propagation step where one active particle enters the step and two or more are formed);
  • chain transfer (a propagation step in which the active particle is a growing polymer chain which reacts to form an inactive polymer whose growth is terminated and an active small particle (such as a radical), which may then react to form a new polymer chain).
• Termination (elementary step in which the active particle loses its activity; e. g. by recombination of two free radicals).

The chain length is defined as the average number of times the propagation cycle is repeated, and equals the overall reaction rate divided by the initiation rate.

Some chain reactions have complex rate equations with fractional order or mixed order kinetics.

Detailed example: the hydrogen-bromine reaction

The reaction H₂ + Br₂ → 2 HBr proceeds by the following mechanism:

• Initiation

Br₂ → 2 Br• (thermal) or Br₂ + hν → 2 Br• (photochemical)

each Br atom is a free radical, indicated by the symbol "•" representing an unpaired electron.

• Propagation (here a cycle of two steps)

Br• + H₂ → HBr + H•

H• + Br₂ → HBr + Br•

the sum of these two steps corresponds to the overall reaction H₂ + Br₂ → 2 HBr, with catalysis by Br• which participates in the first step and is regenerated in the second step.

• Retardation (inhibition)

H• + HBr → H₂ + Br•

this step is specific to this example, and corresponds to the first propagation step in reverse.

• Termination 2 Br• → Br₂

recombination of two radicals, corresponding in this example to initiation in reverse.

As can be explained using the steady-state approximation, the thermal reaction has an initial rate of fractional order (3/2), and a complete rate equation with a two-term denominator (mixed-order kinetics).

Further chemical examples

• The reaction 2 H₂ + O₂ → 2 H₂O provides an example of chain branching. The propagation is a sequence of two steps whose net effect is to replace an H atom by another H atom plus two OH radicals. This leads to an explosion under certain conditions of temperature and pressure.
  • H• + O₂ → •OH + •O•
  • •O• + H₂ → •OH + H•
• In chain-growth polymerization, the propagation step corresponds to the elongation of the growing polymer chain. Chain transfer corresponds to transfer of the activity from this growing chain, whose growth is terminated, to another molecule which may be a second growing polymer chain. For polymerization, the kinetic chain length defined above may differ from the degree of polymerization of the product macromolecule.
• Polymerase chain reaction, a technique used in molecular biology to amplify (make many copies of) a piece of DNA by in vitro enzymatic replication using a DNA polymerase.

Acetaldehyde pyrolysis and rate equation

The pyrolysis (thermal decomposition) of acetaldehyde, CH₃CHO (g) → CH₄ (g) + CO (g), proceeds via the Rice-Herzfeld mechanism:

• Initiation (formation of free radicals):

CH₃CHO (g) → •CH₃ (g) + •CHO (g) k₁

The methyl and CHO groups are free radicals.

• Propagation (two steps):

•CH₃ (g) + CH₃CHO (g) → CH₄ (g) + •CH₃CO (g) k₂

This reaction step provides methane, which is one of the two main products.

•CH₃CO (g) → CO (g) + •CH₃ (g) k₃

The product •CH₃CO (g) of the previous step gives rise to carbon monoxide (CO), which is the second main product.

The sum of the two propagation steps corresponds to the overall reaction CH₃CHO (g) → CH₄ (g) + CO (g), catalyzed by a methyl radical •CH₃.

• Termination:

•CH₃ (g) + •CH₃ (g) → C₂H₆ (g) k₄

This reaction is the only source of ethane (minor product) and it is concluded to be the main chain ending step.

Although this mechanism explains the principal products, there are others that are formed in a minor degree, such as acetone (CH₃COCH₃) and propanal (CH₃CH₂CHO).

Applying the Steady State Approximation for the intermediate species CH₃(g) and CH₃CO(g), the rate law for the formation of methane and the order of reaction are found:

The rate of formation of the product methane is

${\displaystyle (1)...\frac{d[{\text{CH}}_4^{}]}{dt}=k_2[{\text{CH}}_3^{}][{\text{CH}}_3^{}\text{CHO}]}$

For the intermediates

${\displaystyle (2)...\frac{d[{\text{CH}}_3^{}]}{dt}=k_1[{\text{CH}}_3^{}\text{CHO}]-k_2[{\text{CH}}_3^{}][{\text{CH}}_3^{}\text{CHO}]+k_3[{\text{CH}}_3^{}\text{CO}]-2k_4{[{\text{CH}}_3^{}]}^2=0}$ and

${\displaystyle (3)...\frac{d[{\text{CH}}_3^{}\text{CO}]}{dt}=k_2[{\text{CH}}_3^{}][{\text{CH}}_3^{}\text{CHO}]-k_3[{\text{CH}}_3^{}\text{CO}]=0}$

Adding (2) and (3), we obtain ${\displaystyle k_1[{\text{CH}}_3^{}\text{CHO}]-2k_4{[{\text{CH}}_3^{}]}^2=0}$

so that ${\displaystyle (4)...[{\text{CH}}_3^{}]=\frac{k_1}{2k_4}{[{\text{CH}}_3^{}\text{CHO}]}^{1/2}}$

Using (4) in (1) gives the rate law ${\displaystyle (5)\frac{d[{\text{CH}}_4^{}]}{dt}=\frac{k_1}{2k_4}{k}_2{[{\text{CH}}_3^{}\text{CHO}]}^{3/2}}$, which is order 3/2 in the reactant CH₃CHO.

Nuclear chain reactions

Main article: Nuclear chain reaction

A nuclear chain reaction was proposed by Leo Szilard in 1933, shortly after the neutron was discovered, yet more than five years before nuclear fission was first discovered. Szilárd knew of chemical chain reactions, and he had been reading about an energy-producing nuclear reaction involving high-energy protons bombarding lithium, demonstrated by John Cockcroft and Ernest Walton, in 1932. Now, Szilárd proposed to use neutrons theoretically produced from certain nuclear reactions in lighter isotopes, to induce further reactions in light isotopes that produced more neutrons. This would in theory produce a chain reaction at the level of the nucleus. He did not envision fission as one of these neutron-producing reactions, since this reaction was not known at the time. Experiments he proposed using beryllium and indium failed.

Later, after fission was discovered in 1938, Szilárd immediately realized the possibility of using neutron-induced fission as the particular nuclear reaction necessary to create a chain-reaction, so long as fission also produced neutrons. In 1939, with Enrico Fermi, Szilárd proved this neutron-multiplying reaction in uranium. In this reaction, a neutron plus a fissionable atom causes a fission resulting in a larger number of neutrons than the single one that was consumed in the initial reaction. Thus was born the practical nuclear chain reaction by the mechanism of neutron-induced nuclear fission.

Specifically, if one or more of the produced neutrons themselves interact with other fissionable nuclei, and these also undergo fission, then there is a possibility that the macroscopic overall fission reaction will not stop, but continue throughout the reaction material. This is then a self-propagating and thus self-sustaining chain reaction. This is the principle for nuclear reactors and atomic bombs.

Demonstration of a self-sustaining nuclear chain reaction was accomplished by Enrico Fermi and others, in the successful operation of Chicago Pile-1, the first artificial nuclear reactor, in late 1942.

Electron avalanche in gases

An electron avalanche happens between two unconnected electrodes in a gas when an electric field exceeds a certain threshold. Random thermal collisions of gas atoms may result in a few free electrons and positively charged gas ions, in a process called impact ionization. Acceleration of these free electrons in a strong electric field causes them to gain energy, and when they impact other atoms, the energy causes release of new free electrons and ions (ionization), which fuels the same process. If this process happens faster than it is naturally quenched by ions recombining, the new ions multiply in successive cycles until the gas breaks down into a plasma and current flows freely in a discharge.

Electron avalanches are essential to the dielectric breakdown process within gases. The process can culminate in corona discharges, streamers, leaders, or in a spark or continuous electric arc that completely bridges the gap. The process may extend huge sparks — streamers in lightning discharges propagate by formation of electron avalanches created in the high potential gradient ahead of the streamers' advancing tips. Once begun, avalanches are often intensified by the creation of photoelectrons as a result of ultraviolet radiation emitted by the excited medium's atoms in the aft-tip region. The extremely high temperature of the resulting plasma cracks the surrounding gas molecules and the free ions recombine to create new chemical compounds.

The process can also be used to detect radiation that initiates the process, as the passage of a single particles can be amplified to large discharges. This is the mechanism of a Geiger counter and also the visualization possible with a spark chamber and other wire chambers.

Avalanche breakdown in semiconductors

An avalanche breakdown process can happen in semiconductors, which in some ways conduct electricity analogously to a mildly ionized gas. Semiconductors rely on free electrons knocked out of the crystal by thermal vibration for conduction. Thus, unlike metals, semiconductors become better conductors the higher the temperature. This sets up conditions for the same type of positive feedback—heat from current flow causes temperature to rise, which increases charge carriers, lowering resistance, and causing more current to flow. This can continue to the point of complete breakdown of normal resistance at a semiconductor junction, and failure of the device (this may be temporary or permanent depending on whether there is physical damage to the crystal). Certain devices, such as avalanche diodes, deliberately make use of the effect.

Living organisms

Main article: Chain reactions in living organisms

Examples of chain reactions in living organisms include excitation of neurons in epilepsy and lipid peroxidation. In peroxidation, a lipid radical reacts with oxygen to form a peroxyl radical (L• + O₂ → LOO•). The peroxyl radical then oxidises another lipid, thus forming another lipid radical (LOO• + L–H → LOOH + L•). A chain reaction in glutamatergic synapses is the cause of synchronous discharge in some epileptic seizures.