On meson decay-modes in studying CP violation

In particle physics, CPT symmetry is an attribute of the universe that is held as fundamentally true by quantum field theory (QFT). It states that the laws of physics should not be changed and the opposite of all allowed motions be allowed (T symmetry) if a particle is replaced with its antiparticle (C symmetry) and then left and right are swapped (P symmetry).

What this implies is a uniformity of the particle’s properties across time, charge and orientation, effectively rendering them conjugate perspectives.

(T-symmetry, called so for an implied “time reversal”, defines that if a process moves one way in time, its opposite is signified by its moving the other way in time.)

The more ubiquitously studied version of CPT symmetry is CP symmetry with the assumption that T-symmetry is preserved. This is because CP-violation, when it was first observed by James Cronin and Val Fitch, shocked the world of physics, implying that something was off about the universe. Particles that ought to have remained “neutral” in terms of their properties were taking sides! (Note: CPT-symmetry is considered to be a “weaker symmetry” then CP-symmetry.)

In 1964, Oreste Piccioni, who had just migrated to the USA and was working at the Lawrence Berkeley National Laboratory (LBNL), observed that kaons, mesons each composed of a strange quark and an up/down antiquark, had a tendency to regenerate in one form when shot as a beam into matter.

The neutral kaon, denoted as K⁰, has two forms, the short-lived (K_S) and the long-lived (K_L). Piccioni found that kaons decay in flight, so a beam of kaons, over a period of time, becomes pure K_L because the K_S all decay away before them. When such a beam is shot into matter, the K⁰ is scattered by protons and neutrons whereas the K⁰* (i.e., antikaons) contribute to the formation of a class of particles called hyperons.

Because of this asymmetric interaction, (quantum) coherence between the two batches of particles is lost, resulting in the emergent beam being composed of K_S and K_L, where the K_S is regenerated by firing a K⁰-beam into matter.

When the results of Piccioni’s experiment were duplicated by Robert Adair in the same year, regeneration as a physical phenomenon became a new chapter in the study of particle physics. Later that year, that’s what Cronin and Fitch set out to do. However, during the decay process, they observed a strange phenomenon.

According to a theory formulated in the 1950s by Murray Gell-Mann and Kazuo Nishijima, and then by Gell-Mann and Abraham Pais in 1955-1957, the K_S meson was allowed to decay into two pions in order for certain quantum mechanical states to be conserved, and the K_L meson was allowed to decay into three pions.

For instance, the K_L (s*, u) decay happens thus:

1. s* → u* + W⁺ (weak interaction)
2. W+ → d* + u
3. u → g + d + d* (strong interaction)
4. u → u

In 1964, in their landmark experiment, Cronin and Fitch observed, however, that the K_L meson was decaying into two pions, albeit at a frequency of 1-in-500 decays. This implied an indirect instance of CP-symmetry violation, and subsequently won the pair the 1980 Nobel Prize for Physics.

An important aspect of the observation of CP-symmetry violation in kaons is that the weak force is involved in the decay process (even as observed above in the decay of the K_L meson). Even though the kaon is composed of a quark and an antiquark, i.e., held together by the strong force, its decay is mediated by the strong and the weak forces.

In all weak interactions, parity is not conserved. The interaction itself acts only on left-handed particles and right-handed anti-particles, and was parametrized in what is called the V-A Lagrangian for weak interactions, developed by Robert Marshak and George Sudarshan in 1957.

In fact, even in the case of the K_S and K_L kaons, their decay into pions can be depicted thus:

K_S → π⁺ + π⁰
K_L → π⁺ + π⁺ + π^–

Here, the “+” and “-” indicate a particle’s parity, or handedness. When a K_S decays into two pions, the result is one right-handed (“+”) and one neutral pion (“0”). When a K_L decays into three pions, however, the result is two right-handed pions and one left-handed (“-“) pion.

When kaons were first investigated via their decay modes, the different final parities indicated that there were two kaons that were decaying differently. Over time, however, as increasingly precise measurements indicated that only one kaon (now called K⁺) was behind both decays, physicists concluded that the weak interaction was responsible for resulting in one kind of decay some of the time and in another kind of decay the rest of the time.

To elucidate, in particle physics, the squares of the amplitudes of two transformations, B → f and B* → f*, are denoted thus.

Here,

B = Initial state (or particle); f = Final state (or particle)
B* = Initial antistate (or antiparticle); f* = Final antistate (or antiparticle)
P = Amplitude of transformation B → f; Q = Amplitude of transformation B* → f*
S = Corresponding strong part of amplitude; W = Corresponding weak part of amplitude; both treated as phases of the wave for which the amplitude is being evaluated

Subtracting (and applying some trigonometry):

The presence of the term sin(W_P – W_Q) is a sign that purely, or at least partly, weak interactions can occur in all transformations that can occur in at least two ways, and thus will violate CP-symmetry. (It’s like having the option of having two paths to reach a common destination: #1 is longer and fairly empty; #2 is shorter and congested. If their distances and congestedness are fairly comparable, then facing some congestion becomes inevitable.)

Electromagnetism, strong interactions, and gravitation do not display any features that could give rise to the distinction between right and left, however. This disparity is also called the ‘strong CP problem’ and is one of the unsolved problems of physics. It is especially puzzling because the QCD Lagrangian, which is a function describing the dynamics of the strong interaction, includes terms that could break the CP-symmetry.

(The best known resolution – one that doesn’t resort to spacetime with two time-dimensions – is the Peccei-Quinn theory put forth by Roberto Peccei and Helen Quinn in 1977. It suggests that the QCD-Lagrangian be extended with a CP-violating parameter whose value is 0 or close to 0.

This way, CP-symmetry is conserved during the strong interactions while CP-symmetry “breakers” in the QCD-Lagrangian have their terms cancelled by an emergent, dynamic field whose flux is encapsulated by massless Goldstone bosons called axions.)

Now, kaons are a class of mesons whose composition includes a strange quark (or antiquark). Another class of mesons, called B-mesons, are identified by their composition including a bottom antiquark, and are also notable for the role they play in studies of CP-symmetry violations in nature. (Note: A B-meson composed of a bottom antiquark and a bottom quark is not called a meson but a bottomonium.)

According to the Standard Model (SM) of particle physics, there are some particles – such as quarks and leptons – that carry a property called flavor. Mesons, which are composed of quarks and antiquarks, have an overall flavor inherited from their composition as a result. The presence of non-zero flavor is significant because SM permits quarks and leptons of one flavor to transmute into the corresponding quarks and leptons of another flavor, a process called oscillating.

And the B-meson is no exception. Herein lies the rub: during oscillations, the B-meson is favored over its antiparticle counterpart. Given the CPT theorem’s assurance of particles and antiparticles being differentiable only by charge and handedness, not mass, etc., the preference of B*-meson for becoming the B-meson more than the B-meson’s preference for becoming the B*-meson indicates a matter-asymmetry. Put another way, the B-meson decays at a slower rate than the B*-meson. Put yet another way, matter made of the B-meson is more stable than antimatter made of the B*-meson.

Further, if the early universe started off as a perfect symmetry (in every way), then the asymmetric formation of B-mesons would have paved the way for matter to take precedence over anti-matter. This is one of the first instances of the weak interaction possibly interfering with the composition of the universe. How? By promising never to preserve parity, and by participating in flavor-changing oscillations (in the form of the W/Z boson).

The prevalence of matter over antimatter in our universe is credited to a hypothetical process called baryogenesis. In 1967, Andrei Sakharov, a Soviet nuclear physicist, proposed three conditions for asymmetric baryogenesis to have occurred.

1. Baryon-number violation
2. Departure from thermal equilibrium
3. C- and CP-symmetry violation

The baryon-number of a particle is defined as one-third of the difference between the number of quarks and number of antiquarks that make up the particle. For a B-meson composed of a bottom antiquark and a quark, the value’s 0; of a bottom antiquark and another antiquark, the value’s 1. Baryon-number violation, while theoretically possible, isn’t considered in isolation of what is called “B – L” conservation (“L” is the lepton number, and is equal to the number of leptons minus the number of antileptons).

Now, say a proton decays into a pion and a position. A proton’s baryon-number is 1, L-number is 0; a pion has both baryon- and L-numbers as 0; a positron has baryon-number 0 and L-number -1. Thus, neither the baryon-number nor the lepton-number are conserved, but their difference (1) definitely is. If this hypothetical process were ever to be observed, then baryogenesis would make the transition from hypothesis to reality (and the question of matter-asymmetry become conclusively answered).

Therefore, in recognition of the role of B-mesons (in being able to present direct evidence of CP-symmetry violation through asymmetric B-B* oscillations involving the mediation of the weak-force) and their ability to confirm or deny an “SM-approved” baryogenesis in the early universe, what are called the B-factories were built: a collider-based machine whose only purpose is to spew out B-mesons so they can be studied in detail by high-precision detectors.

The earliest, and possibly most well-known, B-factories were constructed in the 1990s and shut down in the 2000s: the BaBar experiment at SLAC (2008), Stanford, and the Belle experiment at the KEKB collider (2010) in Japan. In fact, a Belle II plant is under construction and upon completion will boast the world’s highest-luminosity experiment.

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Equations generated thanks to the Daum equations editor.