CP Violation and Anti-matter

Contents

Matter Antimatter Asymmetry
Kaon Decay
Standard Model and CP Violation
B Meson Decay
Leptogenesis

Matter Antimatter Asymmetry

There are strong evidences that the universe is made of matter with very few antimatter. It is believed that particles and anti-particles were equally numerous in the early universe, but the former came to dominate as the universe cooled. Baryogenesis is the hypothetical physical process, that generated an asymmetry between baryons and anti-baryons in the very early universe. It is observed that only a small asymmetry is required in the early universe, as today we have only one leftover proton for 109 photons (assuming the photons were created by the annihilation of particles and antiparticles).

    There are three different theories trying to explain the matter-antimatter asymmetry:

  1. CP violation - The standard model of elementary particle suggests that when the universe was less than 10-12 sec old, the condition was ripe for the production of more matter than antimatter with CP violation to provide the mechanism for different reaction rate (to produce matter and antimatter). The explanation sounds reasonable as described in the subsequent sections. However, theoretical calculation as well as experimental measurement shows an excess far too small to account for the observed degree of asymmetry.
  2. Supersymmetry - Supersymmetry is one of the most promising extensions of the standard model of elementary particle. It demands many as-yet-unknown particles and perhaps new kind of interactions. It is suggested that new interaction outside the standard model might act differently on quarks and antiquarks, and produced the excess of quarks in our universe. Although there are inclusive claims, so far there is no hard evidence for supersymmetry from experiments.
  3. Leptogenesis - This explanation also requires new physics beyond the standard model. It assumes the existence of a new type of very heavy neutrino called singlet neutrino in the very early universe. According to the leptogenesis scenario, the singlet neutrinos would decay into either neutrinos or antineutrinos. Then the standard model predicts that certain reactions could occur in the very high-temperature conditions to convert antineutrinos into matter particles (and thus the lepton number is not conserved, e.g., from 0 to 2), eventually producing neutrons and protons leaving the universe devoid of antimatter (see neutrinoless double-beta decay). So far there is only one controversial claim in 2001 to have observed such reaction.

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Kaon Decay

CP Operation Elementary Particles As shown in Figure 01, charge conjugation (C) reverses the sign of quantum numbers such as electric charge, changing a particle to its antiparticle. Parity (P) reverses the arrow on all vectors associated with the object. The laws of classical mechanics and electromagnetism are invariant under either of these operations, as is the strong interaction of the Standard Model. These symmetries,

Figure 01 CP Violation
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Figure 02 Elementary Part-icles [view large image]

 however, are broken in the weak interaction. For many years, it appeared that the combined operation (CP) were invariant even for weak interactions until it was shown to be otherwise in 1964.
The neutral K meson Ko (a particle that contains a strange quark and a light quark as shown in Figure 02) manifests itself in two modes with respect to the weak interaction. In Figure 03 one mode is labelled Kshort with shorter half-life, a CP state of +1 and decays into two charged pions (also with CP=+1); the other is Klong with longer half-life, a CP state of -1 and
K Meson Decay decay into three neutral pions (also with CP=-1). The 1964 experiment by Cronin and Fitch looked for the decay products of Klong. They observed a few decays of the Klong turning into pairs of oppositely charged pions: about 1 out of a total of 500 decays. This kind of decay arrives at a final CP state that is different from the initial CP state and proves that CP symmetry was not preserved exactly by the weak interaction.

Figure 03 Ko Meson Decays
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Standard Model and CP Violation

The CKM matrix has its origin in the diagonalization of the mass term (the last term in Eq.(41c) of the Standard Model) and the interactions between the three quark families with the W bosons as prescribed by Eq.(41b) - more specifically when quarks undergo weak interactions and turn into quarks with different electric charge. Following is a recap of the mathematical derivation.

After the symmetry breaking of the Higgs field in the Standard Model, the mass terms for the d-type and u-type quarks become:
,
and
respectively, where 0 is the Higgs coupling constant, the Gijs are 3 X 3 complex matrices, d and u are the quark fields labeled as left-handed (L) and right-handed (R) components. These mass terms can be diagonalized by the unitary matrices DL and UL:

where the d'-type (including the d, s, b quarks) and u'-type (including the u, c, t quarks) are referred as "true" quark fields. The mass terms can be combined into:

The CKM matrix emerges when the "true" quark fields are substituted into part of the Lagrangian density for electroweak interaction via the W bosons:
.

There are three independent parameters, and one irreducible phase in the matrix. It can be shown that violation of the CP symmetry will occur if the matrix is not real. The CKM matrix and the quark fields (in the form of column matrices) are shown in Figure 04, where cij = cosij, sij = sinij, ij is the rotation angle in the internal space of the quark fields, and the phase angle turns some matrix elements into complex numbers, thereby violating CP invariance. The first form of the CKM matrix V in Figure 04 denotes a general unitary matrix, the second form is a more particular one advocated by the Particle Data Group, and the third form contains the magnitudes for each matrix element determined by experimental data (the phase angle responsible for CP violation is determined by another method below).

The weak interaction is the only one in which a quark can change into another type (flavor) of quark or a lepton into another type of lepton. In this transformation, a quark is allowed only to change charge by a unit amount e (the charge of the electron). Because quarks and leptons can change flavor by weak interactions, only the lightest quarks and leptons are included in the stable matter of the world around us - all heavier ones decay to one or another of the lighter ones. If we look at all the ways in which one quark can turn into another quark with a charge change of e, e.g., all quarks with charge +2/3e (u, c, or t) turn into quarks with charge -1/3e (d, s, or b), it will produce nine possible pairings. Each of these pairings has its own weak charge
CKM Matrix associated with it, which is related to a physical constant which we call a "coupling constant" containing real and imaginary parts in general, i.e., it is a complex number. The set of coupling constants can be represented by the 3x3 CKM matrix (see Figure 04). In contrast with electric charge, which seems to come in a well-defined universal unit, each of these nine coupling constants is different. The triumph of the Standard Model is that it predicts a set of relationships between the nine elements of the CKM matrix and it predicts that they include properties that result in CP violation. The CP violation is related to the fact that the matrix elements include imaginary

Figure 04 CKM Matrix
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 numbers. If we look at enough decays that involve the different matrix elements, we can see whether the relationships are true (there are only four independent parameters in the matrix).

The CKM matrix has a property known as unitarity due to the physical properties it represents. Mathematically, this defines a set of equations that the matrix elements must satisfy. One of such equations is:

VudV*ub/|VcdV*cb| + 1 + VtdV*tb/|VcdV*cb|  =  |VudV*ub|/|VcdV*cb| ei + 1 + |VtdV*tb|/|VcdV*cb| ei  =  0

Complex numbers can be represented as points on a two-dimensional plane, or equivalently as vectors in a plane. Each term in the above equation is complex and can be drawn as a vector in the plane. Arranging the three vectors head-to-tail gives the sum. Since the sum is zero, the three vectors should form a closed polygon, that is, a triangle. Since one leg of the triangle is just 1.0, it lies on the x-axis and has a length of 1. The triangle is then defined by a single point on the plane at the apex. This is known as a unitarity triangle as shown in Figure 05. Currently, physicists are measuring the angle = arg(VtdVtb*/|VcdVcb*|), i.e., the polar angle of the complex number VtdVtb*/|VcdVcb*|, to determine the degree of CP-violation in Bo decay. The different modes of decay are also indicated in the diagram; D is the D meson containing a charm quark and a light quark.

Unitarity Triangle

Figure 05 Unitarity Triangle

We need to know all of the elements in the matrix, both their real and imaginary parts, to test the relationships predicted by the Standard Model. To measure the imaginary parts, we need to measure CP violation in many meson decays. For K meson decays the elements we can look at are mostly in the second column of the matrix. With B meson (a particle that contains a heavy bottom quark and a light quark) decays we can look in the third row and the third column. With the combination of both, we cover nearly all of the matrix and should be able to check all of the matrix relationships. Some decays are predicted to display CP violation, others are not. The fact that these coupling constants have to be determined by experimental measurements is considered to be a shortcoming of the Standard Model. A more fundamental theory is needed to explain the origin of these constants.

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B Meson Decay

Modern CP-violation experiments are designed to study decays of B mesons into those final states that have a definite CP number. If CP is violated, then the B and anti-B mesons would decay with different decay rates to the CP eigenstates. These heavier particles can spontaneously decay into matter and antimatter fragments in a greater number of ways than lighter particles, increasing the odds of finding something unexpected. The experimental challenge comes from the fact that B meson decays to CP eigenstates such as have very small branching ratios and in general low efficiencies for complete reconstruction of the final state. It is therefore necessary to produce a very large sample of B mesons to perform a CP measurement. At the High Energy Accelerator Research Organization (KEK) in Japan and the Stanford Linear Accelerator Center (SLAC) in California, accelerators have been designed to produce a plentiful supply of B mesons, through specially tuned electron - positron collisions. At each facility is a detector (Belle in Japan, and BaBar in California) to pick up and study
B Meson the decays of the many millions of B mesons created - hence these facilities are known as "B factories". Figure 06 shows the BaBar detector and the particle tracks from the B meson decay. The latest measurements yield a value of sine(2) = 0.78 0.08, which is exactly that needed to explain the magnitude of CP violation seen in the Cronin-Fitch experiment. However, these measurements also predict a value of one leftover proton to 1018 photons (resulting from the annihilation of particles and antiparticles). It is in disagreement with the observation of one in 109 by many order of magnitude.

Figure 06 B Meson Decay
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Leptogenesis

Leptogenesis The disagreement with observation is now supported by three years of B meson decay data. It seems that the Standard model alone is not able to explain the phenomenon of matter-antimatter asymmetry. A new theory called leptogenesis (Figure 07, also known as baryogenesis) suggests that an exceptionally heavy but unstable breed of Majorana neutrino existed in the very early universe. Their subsequent decay generated more anti-leptons than leptons. A mechanism then converted 1/3 of the excess anti-leptons into baryons leading to the imbalance between matter and antimatter at the dawn of time. Theoretical analysis shows that leptogenesis works best when the neutrino masses are in the range 0.1 ev - 1 Mev; and the reheating temperature must be larger than 109 Gev.

Figure 07 Leptogenesis
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NDB An experiment to look for neutrinoless double-beta decay (NDB, see Figure 08) is running since 1990 at the Gran Sasso laboratory near Rome. The claim for discovery is still in dispute. But the findings could confirm the neutrinos' changing behaviour (flip-floping between antineutrino and neutrino as shown in the lower diagram of Figure 08), and the existence of an extremely heavy form of neutrino at high temperatures (such as in the aftermath of the big bang).

Figure 08 NDB Decay
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