Questions and Experiments of Parity Violation in Weak Interactions and Further Developments of Theorem in Symmetries

Transcription

1 Revised version: Questions and Experiments of Parity Violation in Weak Interactions and Further Developments of Theorem in Symmetries Jin Namkung 1 1 Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea (Dated: 22 November 2017) Conservation laws are necessary for fully understanding a dynamical system especially in particle physics. Parity will be focused in this paper among various conserved properties. It is directly related to symmetry of spatial inversion and every elementary particle has a particular value of parity. The principle of parity conservation was first suggested by Wigner in 1927 and it was considered as a reasonable theory. However, an issue called θ-τ puzzle arose in the early 1950s. Moreover, evidences for parity conservation in weak interactions haven t been found for around 30 years. From these reasons, Tsung-Dao Lee and Chen-Ning Yang challenged to solve the problem by assuming parity is not a conserved quantity in weak interactions. Theoretical calculation and suggested experiments were demonstrated for β and meson decays, both of which are common examples for weak interactions. Experiments to notice parity violation all showed positive results, and therefore, Lee and Yang was awarded the Nobel prize in physics in This paper will basically follow Lee and Yang s approaches and introduce two experiments to explain their process of proving parity violation in weak interactions. Finally, some attempts to resolve parity violation and its relation with other symmetries, such as charge conjugation and time reversal, will also be discussed, consequently leading to CP violation and CPT theorem. I. INTRODUCTION General relationship between symmetry and conservation laws in physics can be explained from Noether s theorem. It can be stated as every symmetry of nature yields a conservation law, and conversely, every conservation law reflects an underlying symmetry [1]. For example, energy and momentum conservation are commonly used for analyzing dynamical reactions. It is possible since the conservation laws include symmetry of translations in time and space. In the world of quantum mechanics, Ehrenfest s theorem is widely used for confirming conservation for a given operator [2]. d dt A = 1 [A, H] + A i h t (1) The second term in RHS of the equation would be 0 for a time-independent operator A. Therefore, expectation value for A in LHS is conserved if operator A commutes with Hamiltonian H. Among various conserved properties, such as energy, momentum, parity, charge, and isospin, this paper will further introduce a basic concept of parity. Parity is a conserved property if there is a symmetry under space inversion. Geometrical meaning of space inversion or parity transformation is shown on Fig. 1 and it is possible to be described in coordinates. Letting parity operator P, it can be applied on Cartesian and spherical coordinates as following relations. P x y = x y, P r θ = r π θ (2) z z φ π + φ For example, the results for applying the operator to momentum and angular momentum would be P ( p) = FIG. 1. An example of spatial inversion with hands located in Cartesian coordinates [1]. d( r)/dt = p and P ( L) = ( r) ( p) = L. Since there is a correlation between angular momentum and spin momentum as an axial vector, spin is also invariant under parity transformation: P ( S) = S [3]. This idea will be used later for geometrically analyzing results from experiments. For finding a specific value of parity, assume there is a particle in a three-dimensional space. By applying parity operator twice, it is noticeable that the particle must come back to its original position. P 2 ψ( r) = P ψ( r) = ψ( r) (3) Therefore, eigenvalue for the parity operator is P = ±1. This value is called intrinsic parity, since above example didn t considered any systems or interactions between particles. Value for intrinsic parity is fixed as a convention. Parity of fermions is P = +1 and have an opposite parity for their antiparticles. For example, parity of electron and muon are fixed as P = +1 and also for proton and neutron, since they consist of three quarks. For another example, parity for π + = ud will be P π = 1 since it consists of quark and antiquark. Parity is a good example for discrete symmetry, and therefore, it is a multi-

2 2 plicative quantum number [4]. For example, let there be particle a and b for multiple particle system. The total parity in this case would be P = P a P b P s (4) where P a and P b indicate intrinsic parity of particles and P s indicates parity of the system which is obtainable from applying parity operator in wavefunction of the system. Importance of symmetry and conservation law with basic concept of parity have been shortly discussed in this section. From now on, this paper will follow Lee and Yang s approaches and calculations with their paper as a reference [5]. Experiments to show parity nonconservation in β and meson decays are also demonstrated and results are explained in detail. Finally, other symmetries, like charge conjugation and time reversal, are shortly explained and introduced to resolve parity violation. II. QUESTIONS OF PARITY VIOLATION IN WEAK INTERACTIONS The questions of parity problems usually start in the early 1950s with an enigma called θ τ puzzle. Two particles almost had same mass and lifetime with indistinguishable charge and spin. This information would be logically sufficient that the two particles must be identical. However, each particle had different decay patterns. One of them decayed into two pions, and the other decayed into three pions [1]: θ + π + + π 0, τ + π + + π + + π (5) By using the idea that parity of pions are P π = 1 and the parity of the system is P s = +1, total parity after decay of θ and τ are P θ = +1 and P τ = 1 each. There was a belief that parity is a conserved property for all interactions. Consequently, even though they were almost identical particles, it was unavoidable to label them as different particles. Actually, these two particles were later to be known as kaon(k + ) with different decay processes. This identical particle with two decay properties will be used for showing CP violation, which will be explained later. Also, there were some theoretical evidences of parity conservation in strong and electromagnetic interactions [6, 7]. It is also trivial for gravitational interactions. However, there was no evidence of parity conservation for weak interactions. From these problems, Lee and Yang gave a question about parity conservation in weak interactions. They theoretically calculated the order of a factor of possessing the opposite parity, assuming it exists. They also suggested some experimental methods to show parity violation in the paper. This section will basically follow Lee and Yang s approach and refer to all complicated calculation they ve done [5]. Let fraction weight of states possessing the opposite parity F. Mixing parities is required for parity nonconservation and it will be of the order F 2. By introducing F, the violation of parity conservation will induce an electric dipole moment and its strength is roughly calculated to be ef (dimension of system). Upper limit of F 2 from experimental value should be F 2 < for showing parity conservation in the weak interactions, which seems complicated. There is a difficulty for directly finding out the nonconserving term for β decays. Let C and C be couplings of parity conserving and parity non-conserving interactions. The result of calculations includes terms of C 2 and C 2. However, without information of spin of neutrinos at that time, the couplings C and C were indistinguishable [8]. Therefore, CC terms were necessary to notice parity nonconserving interaction. According to the Appendix in Lee and Yang s paper, The angular distribution of β radiation from oriented nucleus has the following form if there is a possible interference term in β decay [5]: I(θ)dθ = (constant)(1 + αcosθ)sinθdθ (6) where α is related to the interference term CC. If there is an anisotropic distribution for β radiation, α 0 from equation (6) and this implies that interference term CC actually exists in β decay with parity non-conserving interactions. Some experiments will follow in next section to notice this distribution and it will properly verify parity violation in weak interactions. Another explanation of how asymmetry in distribution is geometrically related to parity violation will be discussed further using its original symmetric property: spatial inversion. III. EXPERIMENTS This section demonstrates experiments to show parity violation in weak interactions. The goal is to notice anisotropic distribution of β radiation to successfully show α 0 in equation (6). Chien-Shiung Wu noticed an anisotropy of β radiation from her famous experiment using Co 60 nuclei. Also, from an experiment held by Garwin, Lederman, and Weinrich, it was able to reconfirm the former result and show failure of parity conservation in meson decays. Therefore, all these experiments successfully showed parity nonconservation in weak interactions. A. Wu Experiment This experiment used β decay of Co 60 nucleus. There is a β γ correlation which means γ rays are emitted after β decay from metastable state of nucleus Co 60m 28 Ni + e + ν e 60 28Ni + e + ν e + 2γ (7)

3 3 FIG. 3. Results of counting rates for γ ray and β radiation with time and direction of nuclei spin [10]. FIG. 2. A simplified diagram of the lower part of the cryostat in Wu experiment [10]. Since the angular distribution of β radiation in equation (6) is based on the oriented nuclei with their spin, the challenge of this experiment would be effectively polarizing Co 60 nuclei. Magnetic moment of nucleus is usually small and just simply applying magnetic field won t properly polarize nuclei. Therefore, Rose-Gorter method, discovered in 1948, was used for polarizing Co 60 nuclei and they were able to successfully align Co 60 nuclei spins by using the coupling with electrons spin, known as hyperfine interaction [9]. Electrons are easily aligned in extremely low temperature. From a method of adiabatic demagnetization, temperature was decreased to the order of 10 3 K and polarized nuclei were attainable. For brief explanation of Fig. 2, two NaI scintillators located at equatorial and polar direction count γ rays. Since it was known that γ anisotropy was a result of polarized Co 60 nuclei from Rose and Gorter, the purpose was to notice γ ray anisotropy and check degree of polarization. Anthracene crystal and upper instruments count electrons and Co 60 nuclei can be properly polarized when they are grown on CeMg nitrate. Thermometer coils will apply magnetic field for aligning nuclei spins toward vertical direction of up or down. Results of the experiment are shown in Fig. 3 into two graphs. From counting rate of γ ray in two different counters, anisotropy is noticeable before 6 minutes and this indicates that Co 60 nuclei are properly polarized. Since temperature increases as time goes by, nuclei become disoriented after 6 minutes and isotropic distribution of γ ray is detected. For β radiation, a large asymmetry is noticed between Co 60 nuclei aligned into upper or lower direction. If Co 60 nuclei are oriented into lower direction, FIG. 4. (a) An analogical image for the result in the experiment. (b) The result after parity transformation [11]. more electrons were counted, which means more electrons were emitted in opposite direction of Co 60 spin. This result can be mathematically calculated by referring to counting rates. From equation (6), the sign of the asymmetry coefficient, α, must be negative and the value is approximately 0.4 [10]. This showed that α 0 and there is an interference term between couplings of parity conserving and nonconserving interactions present in β decay. Therefore, parity is not conserved in β decay and the result is more easy to understand by considering spatial inversion symmetry. An image of experimental result in an atomic scale is shown on Fig. 4(a). Notice that spin of Co 60 nucleus is pointing upwards and more electrons are emitted into downward direction. If the result is parity transformed as in Fig. 4(b), spin of nucleus is invariant, but momentum of emitted electron should point the opposite direction, which are explained in introduction of this paper. Since more electrons are emitted upwards in (b), physical laws are different and there is no symmetry between (a) and (b). Therefore, parity is not conserved in β decay.

4 4 B. Garwin, Lederman, and Weinrich Experiment Garwin, Lederman, and Weinrich also held an experiment for showing parity violation based on Lee and Yang s suggestions. Rather than β decay, they focused on pion and muon decays as the following reactions. π + µ + + ν µ, µ + e + + ν e + ν µ (8) In Wu experiment, they used polarized Co 60 and confirmed β asymmetry. If they haven t used polarized nuclei, the result should be symmetric since all nuclei possess randomly oriented direction of spin. Similarly, if the result shows an asymmetric positron distribution for muon decay in this experiment, parity conservation would be violated and automatically implies that µ + particles are polarized. For brief explanation of Fig. 5, π + beams are extracted from Nevis cyclotron and some of them decay in flight. Among mixtures of π + and µ + particles, π + particles are stopped in carbon absorber, letting µ + pass through it and stop in carbon target. To gather maximum number of µ + in the carbon target, 8- inches for size of carbon absorber is used based on the energy and mean range of π + [12]. If perpendicular magnetic field target is applied into the carbon target, muons will start to precess since magnetic field exerts a torque τ = µ B and direction of spin and magnetic field aren t parallel. From Larmor precession, muons precess at a rate shown below, where γ is a gyromagnetic ratio and s is spin of muon [2, 12]. FIG. 5. A simplified diagram of arrangement for Garwin experiment [12]. ω = γb = ( µ )B (9) s h Detecting counters count electrons in 0.75 and 2.0µsec for a given time considering lifetime of µ +. Strength of current was adjusted to change magnetic field, which will make muons to precess faster. Thus, distribution of larger range of angles can be detected for a given time. Since the counts have changed with respect to the strength of field current in Fig. 6, distribution of electrons from muon decay is asymmetric. After some calculation, value for α in equation (6) was found to be 1/3 in muon decay. Since α 0, parity is not conserved. Moreover, if muons in carbon target were not polarized, results for the experiment must have demonstrated symmetric distribution of electrons. Therefore, muons must be polarized after pion decay. By applying parity transformation, direction of pion beam would be the opposite, but polarization of muons would be the same, since spin is invariant under parity transformation. Therefore, parity is also violated in pion decay, which is an example of meson decay. Other important discoveries were found from the experiment. For example, the g-factor for µ + particle was found to be about ± From angular distribution of electrons and frequency of precession in equation (9), value of s = 1/2 was strongly suggested for muon spin. Value for α in angular distribution was approximately 1/20 for negative muon (µ ) decay, which is also quite interesting [12]. FIG. 6. Count rates of electrons with energy over 25MeV with precession field current [12]. IV. OTHER SYMMETRIES AND CP VIOLATION This section will introduce other important symmetries related to parity. First, charge conjugation operation (C) converts each particles into its antiparticles. C p = p (10) To be specific, it changes the sign of all internal quantum number, such as charge, baryon number, and lepton number while leaving mass, energy, momentum, and spin untouched [1]. When it is combined with parity operator (CP ), it actually solves problem of violation and gives a chance for CP invariance. For example, when CP operator is applied to the result of Wu experiment, positrons would head towards Co 60 nucleus with more positrons coming from lower direction. This is physically identical if the relation between electrons and positrons is considered. Also, weak interactions normally includes creation of neutrinos. They are always found out to be lefthanded, and conversely, antineutrinos are always right-

5 5 handed. If parity transformation is applied to a neutrino, its direction changes without influencing spin. Therefore, it becomes a right-handed neutrino, which haven t been noticed in nature. Thus, this gives a clue for some reason why parity is violated only in weak interactions. By introducing charge conjugation after parity transformation, above right-handed neutrino becomes right-handed antineutrino and problems are solved, which seems positive and leaves the system symmetric. This CP symmetry, or CP invariance, can be an alternative for parity and maybe applicable to all interactions. It was investigated further from neutral kaons, an idea suggested by Gell-Mann and Pais. K 0 can turn into its antiparticle K 0 through a second-order weak interactions [13]. Parity values for neutral kaons are P K 0 = 1 since they are pseudoscalars. Recalling the charge conjugation operator, applying CP to neutral kaons shows following results. CP K 0 = K 0, CP K 0 = K 0 (11) Therefore, eigenvalue should be CP = ±1 and corresponding normalized eigenstates of CP ( K 1 and K 2 ) are K 1,2 = ( 1 2 )( K 0 K 0 ) (12) Referring to the θ τ puzzle in equation (5), neutral kaons decay into 2π or 3π particles. If CP is assumed to be conserved in weak interactions, CP value for pions after decay must be the same. Since charge of pions should be neutral in total, C = +1 for both eigenstates, and parity for pion products will be P 1 = +1 and P 2 = 1 [1]. Therefore, states of K 1 and K 2 will each decay into two and three pions by recalling P π = 1. 2π decay of K 1 (τ s) is much faster compared to 3π decay of K 2 (τ s), and therefore, it is possible to distinguish K 1 and K 2 using time difference of decays. By starting with a beam of K 0 = (1/ 2)( K 1 + K 2 ), K 1 would decay after a few centimeters, and K 2 would decay after some meters. However, this expectation came out to be untrue from an experiment by Cronin and Fitch. About 45 counts of 2π decays were noticed out of decays at the end of a beam of 57 feet ( 17.4m) long [14]. Therefore, for a K 0 long, it is not composed of the state only with K 2, but also contains some fraction of K 1, which is somewhat similar to Lee and Yang s approach of including mixing parity terms. K L = ɛ 2 ( K 2 + ɛ K 1 ) (13) The value was found out to be ɛ and this result became a strong evidence of CP violation [14]. Therefore, Cronin and Fitch was awarded the Noble prize in physics in 1980 for showing CP violation experimentally. Another conservation law for weak interactions showed to be a failure. Finally, by combining time reversal symmetry shown below to CP, the struggling of achieving fundamental symmetry can be finished with CPT symmetry. T : t t (14) CPT theorem implies that any quantum field theory based on a Hermitian, local, and Lorentz invariance guarantees a corresponding symmetry of operating C, P, and T together in any order [15]. Since an example of CP violation have been demonstrated in this section, corresponding violation of time reversal must be present. Since no experiments have shown CPT violation, the theorem is currently acknowledged as a fundamental symmetry. If CPT or Lorentz violation ever happens, some explanations are required beyond standard model, which are still being researched in theoretical physics. Historically, violation of parity influenced a lot especially in theoretical, nuclear, and particle physics. Pauli asserted that each C, P, and T was an independently conserved property before parity violation was noticed [16]. Combining them were necessary to show fundamental symmetry, which was explained briefly in the former paragraph. Results of the direction and amount of asymmetry for weak interactions showed that weak interaction is left-handed and shows maximal parity violation [17]. This is directly related to helicity of neutrinos and gave more information of various elementary particles. Finally, based on a group theory with knowledge of particles, it was possible to develop standard model and unify weak and electromangetic interaction into electroweak interaction. To fully understand CP violation, concept of flavor changing was necessary, and consequently, CKM matrix properly included information of strength for flavor changing in weak decays [18]. V. CONCLUSION The paper began by emphasizing the relation between conservation law and symmetry. Properties of parity were elucidated in detail. Starting with θ τ puzzle, arising questions of parity conservation in weak interactions were discussed from Lee and Yang. Parity violation was demonstrated in weak interactions from experiments using β and meson decays. Furthermore, charge conjugation and time reversal symmetries were introduced to resolve violation of parity. However, CP violation was experimentally noticed from neutral kaon decay, and consequently, it developed into CPT theorem and included some necessities of CPT symmetry based on quantum field theory. After a long journey, symmetry in laws of physics developed a lot and maybe reached the end, known as the goal of fundamental symmetry. Nevertheless, physics is still asking question, such as why only one value of helicity is possible for neutrinos or why particularly C, P, and T are related together among various symmetries. Fundamental symmetry may have been achieved, but still some explanation are required.

6 6 [1] D. Griffiths, Introduction to elementary particles (John Wiley & Sons, 2008). [2] D. J. Griffiths, Introduction to quantum mechanics (Cambridge University Press, 2016). [3] L. Wolfenstein and J. P. Silva, Exploring fundamental particles (CRC Press, 2010). [4] R. Mann, An introduction to particle physics and the standard model (CRC press, 2011). [5] T.-D. Lee and C.-N. Yang, Physical Review 104, 254 (1956). [6] R. Haas, L. Leipuner, and R. Adair, Physical Review 116, 1221 (1959). [7] L. Landau, Nuclear Physics 3, 127 (1957). [8] C.-N. Yang and J. Tiomno, Physical Review 79, 495 (1950). [9] M. W. Zemansky, Temperatures very low and very high (Courier Corporation, 1981). [10] C.-S. Wu, E. Ambler, R. Hayward, D. Hoppes, and R. P. Hudson, Physical review 105, 1413 (1957). [11] J. Dunietz, The experiment that taught us what left really means,. [12] R. L. Garwin, L. M. Lederman, and M. Weinrich, Physical Review 105, 1415 (1957). [13] M. Gell-Mann and A. Pais, in Murray Gell-Mann: Selected Papers (World Scientific, 2010) pp [14] J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, Physical Review Letters 13, 138 (1964). [15] M. Sozzi, Discrete symmetries and CP violation: From experiment to theory (Oxford University Press, 2008). [16] W. Pauli, Niels Bohr and the development of physics, 30 (1955). [17] G. T. Garvey and S. J. Seestrom, Los Alamos Science, 156 (1993). [18] A. Höcker and Z. Ligeti, Annu. Rev. Nucl. Part. Sci. 56, 501 (2006).

7 First version: Questions and Experiments of Parity Violation in Weak Interactions and Further Developments of Theorem in Symmetries Jin Namkung 1 1 Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea (Dated: 22 November 2017) Conservation laws are necessary for fully understanding a dynamical system especially in particle physics. Parity will be focused in this paper among various conserved properties. It is directly related to symmetry of spatial inversion and every elementary particle has a particular value of parity. The principle of parity conservation was first suggested by Wigner in 1927 and it was considered as a reasonable theory. However, an issue called θ-τ puzzle arose in the early 1950s. Moreover, evidences for parity conservation in weak interactions haven t been found for 30 years. From these reasons, Tsung-Dao Lee and Chen-Ning Yang challenged to solve the problem by assuming parity is not a conserved quantity in weak interactions. Theoretical calculation and suggested experiments were demonstrated for β and meson decays, both of which are common examples for weak interactions. Experiments to notice parity violation all showed positive results, and therefore, Lee and Yang was awarded the Nobel prize in physics in This paper will basically follow Lee and Yang s approaches and introduce two experiments to explain their process of proving parity violation in weak interactions. Finally, some attempts to resolve parity violation and its relation with other symmetries, such as charge conjugation or time reversal, will also be discussed, consequently leading to CP violation and CPT theorem. I. INTRODUCTION General relationship between symmetry and conservation laws in physics can be explained from Noether s theorem. It can be stated as every symmetry of nature yields a conservation law, and conversely, every conservation law reflects an underlying symmetry [1]. For example, energy and momentum conservation are commonly used for analyzing dynamical reactions. It is possible since the conservation laws include symmetry of translations in time and space. In the world of quantum mechanics, Ehrenfest s theorem is widely used for confirming conservation for a given operator [2]. d dt A = 1 [A, H] + A i h t (1) The second term on the RHS of the equation would be 0 for a time-independent operator A. Therefore, expectation value for A in LHS is conserved if operator A commutes with Hamiltonian H. Among various conserved properties, such as energy, momentum, parity, charge, and isospin, this paper will further introduce a basic concept of parity. Parity is a conserved property if there is a symmetry under space inversion. Geometrical meaning of space inversion or parity transformation is shown on Fig. 1 and it is possible to be described in coordinates. Letting parity operator P, it can be applied on Cartesian and spherical coordinates as following relations. P x y = x y, P r θ = r π θ (2) z z φ π + φ For example, the results for applying the operator to momentum and angular momentum would be P ( p) = FIG. 1. An example of spatial inversion with hands located in Cartesian coordinates [1]. d( r)/dt = p and P ( L) = ( r) ( p) = L. Since there is a correlation between angular momentum and spin momentum as an axial vector, spin is also invariant under parity transformation: P ( S) = S [3]. This idea will be used later for geometrically analyzing the results from experiments. For finding a specific value of parity, assume there is a particle in a three-dimensional space. By applying parity operator twice, it is noticeable that the particle must come back to its original position. P 2 ψ( r) = P ψ( r) = ψ( r) (3) Therefore, eigenvalue for the parity operator is P = ±1. This value is called intrinsic parity, since above example didn t considered any systems or interactions between particles. Value for intrinsic parity is fixed as a convention. Parity of fermions is P = +1 and have an opposite parity for their antiparticles. For example, parity of electron and muon are fixed as P = +1 and also for proton and neutron, since they consist of three quarks. For another example, parity for π + = ud will be P π = 1 since it consists of quark and antiquark. Parity is a good example for discrete symmetry, and therefore, it is a multi-

8 2 plicative quantum number [4]. For example, let there be particle a and b for multiple particle system. The total parity in this case would be P = P a P b P s (4) where P a and P b indicate intrinsic parity of particles and P s indicates parity of the system which is obtainable from applying parity operator in wavefunction of the system. Importance of symmetry and conservation law with basic concepts of parity have been shortly discussed in this section. From now on, this paper will follow Lee and Yang s approaches and calculations with their paper as a reference [5]. Experiments to show parity nonconservation in β and meson decays are also demonstrated and results are explained in detail. Finally, other symmetries, like charge conjugation and time reversal, are shortly explained and introduced to resolve parity violation. II. QUESTIONS OF PARITY VIOLATION IN WEAK INTERACTIONS The questions of parity problems usually start in the early 1950s with an enigma called θ τ puzzle. Two particles almost had same mass and lifetime with indistinguishable charge and spin. This information would be logically sufficient that the two particles must be identical. However, each particle had different decay patterns. One of them decayed into two pions, and the other decayed into three pions [1]: θ + π + + π 0, τ + π + + π + + π (5) By using the idea that parity of pions are P π = 1 and the parity of the system is P s = +1, total parity after decay of θ and τ are P θ = +1 and P τ = 1 each. There was a belief that parity is a conserved property for all interactions. Consequently, even though they were almost identical particles, it was unavoidable to label them as different particles. Actually, these two particles were later to be known as kaon(k + ) with different decay processes. This identical particle with two decay properties will be used for showing CP violation, which will be explained later. Also, there were some theoretical evidences of parity conservation in strong and electromagnetic interactions [6, 7]. It is also trivial for gravitational interactions. However, there was no evidence of parity conservation for weak interaction. From these problems, Lee and Yang gave a question about parity conservation in weak interactions. They theoretically calculated the order of a factor of possessing the opposite parity, assuming it exists. They also suggested some experimental methods to show parity violation in the paper. This section will basically follow Lee and Yang s approach and refer to all complicated calculation they ve done [5]. Let fraction weight of states possessing the opposite parity F. Mixing parities is required for parity nonconservation and it will be of the order F 2. By introducing F, the violation of parity conservation will induce an electric dipole moment and its strength is roughly calculated to be ef (dimension of system). Upper limit of F 2 from experimental value should be F 2 < for showing parity conservation in the weak interactions, which seems complicated. There is a difficulty for directly finding out the nonconserving term for β decays. Let C and C be couplings of parity conserving and parity non-conserving interactions. The result of calculations includes terms of C 2 and C 2. However, without information of spin of neutrinos at that time, the couplings C and C were indistinguishable [8]. Therefore, CC terms were necessary to notice parity nonconserving interaction. According to the Appendix in Lee and Yang s paper, The angular distribution of β radiation from oriented nucleus has the following form if there is a possible interference term in β decay [5]: I(θ)dθ = (constant)(1 + αcosθ)sinθdθ (6) where α is related to the interference term CC. If there is an anisotropic distribution for β radiation, α 0 from equation (6) and this implies that interference term CC actually exists in β decay with parity non-conserving interactions. Some experiments will follow in next section to notice this distribution and it will properly verify parity violation in weak interactions. Another explanation of how asymmetry in distribution is geometrically related to parity violation will be discussed further using its original symmetric property: spatial inversion. III. EXPERIMENTS This section demonstrates introduce experiments to show parity violation in weak interactions. The goal is to notice anisotropic distribution of β radiation to successfully show α 0 in equation (6). Chien-Shiung Wu noticed an anisotropy of β radiation from her famous experiment using Co 60 nuclei. Also, from an experiment by Garwin, Lederman, and Weinrich, it was able to reconfirm the former result and show failure of parity conservation in meson decays. Therefore, all these experiments successfully showed parity nonconservation in weak interactions. A. Wu Experiment This experiment used β decay of Co 60 nucleus. There is a β γ correlation which means γ rays are emitted after β decay from metastable state of nucleus Co 60m 28 Ni + e + ν e 60 28Ni + e + ν e + 2γ (7)

9 3 FIG. 3. Results of counting rates for γ ray and β radiation with time and direction of nuclei spin [10]. FIG. 2. A simplified diagram of the lower part of the cryostat in Wu experiment [10]. Since the angular distribution of β radiation in equation (6) is based on the oriented nuclei with their spin, the challenge of this experiment would be effectively polarizing Co 60 nuclei. Magnetic moment of nucleus is usually small and just simply applying magnetic field won t properly polarize nuclei. Therefore, Rose-Gorter method, discovered in 1948, was used for polarizing Co 60 nuclei and they were able to successfully align Co 60 nuclei spins by using the coupling with the electrons spin known as hyperfine interaction [9]. Electrons are easily aligned in extremely low temperature. From a method of adiabatic demagnetization, temperature decreased to the order of 10 3 K polarized nuclei were attainable. For brief explanation of Fig. 2, two NaI scintillators located at equatorial and polar direction count γ rays. Since it was known that γ anisotropy was a result of polarized Co 60 nuclei from Rose and Gorter, the purpose was to notice γ ray anisotropy and check degree of polarization. Anthracene crystal and upper instruments count β ray and Co 60 nuclei can be properly polarized when they are grown on CeMg nitrate. Thermometer coils will apply magnetic field for aligning polarized nuclei spins toward vertical direction of up or down. Results of the experiment are shown in Fig. 3 into two graphs. From counting rate of γ ray in two different counters, anisotropy is noticeable before 6 minutes and this indicates that Co 60 nuclei are properly polarized. Since temperature increases as time goes by, nuclei become disoriented after 6 minutes and isotropic distribution of γ ray is detected. For β radiation, a large asymmetry is noticed between Co 60 nuclei aligned into upper or lower direction. If Co 60 nuclei are oriented into lower direction, FIG. 4. (a) An analogical image for the result in the experiment. (b) The result after parity transformation [11]. more electrons were counted, which means more electrons were emitted in opposite direction of Co 60 spin. This result can be mathematically calculated by referring to counting rates. From equation (6), the sign of the asymmetry coefficient, α, must be negative and the value is approximately 0.4 [10]. This showed that α 0 and there is an interference term between couplings of parity conserving and nonconserving interactions present in β decay. Therefore, parity is not conserved in β decay and the result is more easy to understand by considering spatial inversion symmetry. An image of experimental result in an atomic scale is shown on Fig. 4(a). Notice that spin of Co 60 nucleus is pointing upwards and more electrons are emitted into downward direction. If the result is parity transformed as in Fig. 4(b), spin of nucleus is invariant, but momentum of emitted electron should point the opposite direction, which are explained in introduction of this paper. Since more electrons are emitted upwards in (b), physical laws are different and there is no symmetry between (a) and (b). Therefore, parity is not conserved in β decay.

10 4 B. Garwin, Lederman, and Weinrich Experiment Garwin, Lederman, and Weinrich also held an experiment for showing parity violation based on Lee and Yang s suggestions. Rather than β decay, they focused on pion and muon decays as the following reactions. π + µ + + ν µ, µ + e + + ν e + ν µ (8) In Wu experiment, they used polarized Co 60 and confirmed β asymmetry. If they haven t used polarized nuclei, the result should be symmetric since all nuclei possess randomly oriented direction of spin. Similarly, if the result shows an asymmetric positron distribution for muon decay in this experiment, parity conservation would be violated and automatically implies that µ + particles are polarized. For brief explanation of Fig. 5, π + beams are extracted from Nevis cyclotron and some of them decay in flight. Among mixtures of π + and µ + particles, π + particles are stopped in carbon absorber, letting µ + pass through it and stop in carbon target. To gather maximum number of µ + in the carbon target, 8- inches for size of carbon absorber is used based on the energy and mean range of π + [12]. If perpendicular magnetic field target is applied into the carbon target, muons will start to precess since magnetic field exerts a torque τ = µ B and direction of spin and magnetic field aren t parallel. From Larmor precession, muons precess at a rate shown below, where γ is a gyromagnetic ratio and s is spin of muon [2, 12]. FIG. 5. A simplified diagram of arrangement for Garwin experiment [12]. ω = γb = ( µ )B (9) s h Detecting counters count electrons in 0.75 and 2.0µsec for a given time considering lifetime of µ +. Strength of current was adjusted to change magnetic field, which will make muons to precess faster. Thus, distribution of larger range of angles can be detected for a given time. Since the counts have changed with respect to the strength of field current in Fig. 6, distribution of electrons from muon decay is asymmetric. After some calculation, value for α in equation (6) was found to be 1/3 in muon decay. Since α 0, parity is not conserved. Moreover, if muons in carbon target were not polarized, results for the experiment must have demonstrated symmetric distribution of electrons. Therefore, muons must be polarized after pion decay. By applying parity transformation, direction of pion beam would be the opposite, but polarization of muons would be the same, since spin is invariant under parity transformation. Therefore, parity is also violated in pion decay, which is an example of meson decay. Other important discoveries were found from the experiment. For example, the g-factor for µ + particle was found to be about ± From angular distribution of electrons and frequency of precession in equation (9), value of s = 1/2 was strongly suggested for muon spin. Value for α in angular distribution was approximately 1/20 for negative muon (µ ) decay, which is also quite interesting [12]. FIG. 6. Count rates of electrons with energy over 25MeV with precession field current [12]. IV. OTHER SYMMETRIES AND CP VIOLATION This section will introduce other important symmetries related to parity. First, charge conjugation operation (C) converts each particles into its antiparticles. C p = p (10) To be specific, it changes the sign of all internal quantum number, such as charge, baryon number, and lepton number while leaving mass, energy, momentum, and spin untouched [1]. When it is combined with parity operator (CP ), it actually solves problem of violation and gives a chance for CP invariance. For example, when CP operator is applied to the result of Wu experiment, positrons would head towards Co 60 nucleus with more positrons coming from lower direction. This is physically identical if the relation between electrons and positrons is considered. Also, weak interactions normally includes creation of neutrinos. They are always found out to be lefthanded, and conversely, antineutrinos are always right-

11 5 handed. If parity transformation is applied to a neutrino, its direction changes without influencing spin. Therefore, it becomes a right-handed neutrino, which haven t been noticed in nature. Thus, this gives a clue for some reason why parity is violated only in weak interactions. By introducing charge conjugation after parity transformation, above right-handed neutrino becomes right-handed antineutrino and problems are solved, which seems positive and leaves the system symmetric. This CP symmetry, or CP invariance, can be an alternative for parity and amybe applicable to all interactions. It was investigated further from neutral kaons, an idea suggested by Gell-Mann and Pais. K 0 can turn into its antiparticle K 0 through a second-order weak interactions [13]. Parity values for neutral kaons are P K 0 = 1 since they are pseudoscalars. Recalling the charge conjugation operator, applying CP to neutral kaons shows following results. CP K 0 = K 0, CP K 0 = K 0 (11) Therefore, eigenvalue should be CP = ±1 and corresponding normalized eigenstates of CP ( K 1 and K 2 ) are K 1,2 = ( 1 2 )( K 0 K 0 ) (12) Referring to the θ τ puzzle in equation (5), neutral kaons decay into 2π or 3π particles. If CP is assumed to be conserved in weak interactions, CP value for pions after decay must be the same. Since charge of pions should be neutral in total, C = +1 for both eigenstates, and parity for pion products will be P 1 = +1 and P 2 = 1 [1]. Therefore, states of K 1 and K 2 will each decay into two and three pions by recalling P π = 1. 2π decay of K 1 (τ s) is much faster compared to 3π decay of K 2 (τ s), and therefore, it is possible to distinguish K 1 and K 2 using time difference of decays. By starting with a beam of K 0 = (1/ 2)( K 1 + K 2 ), K 1 would decay after a few centimeters, and K 2 would decay after some meters. However, this expectation came out to be untrue from an experiment by Cronin and Fitch. About 45 counts of 2π decays were noticed out of decays at the end of a beam of 57 feet ( 17.4m) long [14]. Therefore, for a K 0 long, it is not composed of the state only with K 2, but also contains some fraction of K 1, which is somewhat similar to Lee and Yang s approach of including mixing parity terms. K L = ɛ 2 ( K 2 + ɛ K 1 ) (13) The value was found out to be ɛ and this result became a strong evidence of CP violation [14]. Therefore, Cronin and Fitch was awarded the Noble prize in physics in 1980 for showing CP violation experimentally. Another conservation law for weak interactions showed to be a failure. Finally, by combining time reversal symmetry shown below, the struggling of achieving fundamental symmetry can be finished with CPT symmetry. T : t t (14) CPT theorem implies that any quantum field theory based on a Hermitian, local, and Lorentz invariance guarantees a corresponding symmetry of operating C, P, and T together in any order [15]. Since an example of CP violation have been demonstrated in this section, corresponding violation of time reversal must be present. Since no experiments have showed CPT violation, the theorem is currently acknowledged as a fundamental symmetry. If CPT or Lorentz violation ever happens, some explanations are required beyond standard model, which are still being researched in theoretical physics. V. CONCLUSION The paper began by emphasizing the relation between conservation law and symmetry. Properties of parity were elucidated in detail. Starting with θ τ puzzle, arising questions of parity conservation in weak interactions were discussed from Lee and Yang. Parity violation was demonstrated in weak interactions from experiments using β and meson decays. Furthermore, charge conjugation and time reversal symmetries were introduced to resolve violation of parity. However, CP violation was experimentally noticed from neutral kaon decay. Consequently, it developed into CPT theorem and included some necessities of CPT symmetry based on quantum field theory. After a long journey, symmetry in laws of physics developed a lot and maybe reached the end, known as the goal of fundamental symmetry. Nevertheless, physics is still asking question, such as why only one value of helicity is possible for neutrinos or why particularly C, P, and T are related together among various symmetries. Fundamental symmetry may have been achieved, but still some explanation are required. [1] D. Griffiths, Introduction to elementary particles (John Wiley & Sons, 2008). [2] D. J. Griffiths, Introduction to quantum mechanics (Cambridge University Press, 2016).

12 6 [3] L. Wolfenstein and J. P. Silva, Exploring fundamental particles (CRC Press, 2010). [4] R. Mann, An introduction to particle physics and the standard model (CRC press, 2011). [5] T.-D. Lee and C.-N. Yang, Physical Review 104, 254 (1956). [6] R. Haas, L. Leipuner, and R. Adair, Physical Review 116, 1221 (1959). [7] L. Landau, Nuclear Physics 3, 127 (1957). [8] C.-N. Yang and J. Tiomno, Physical Review 79, 495 (1950). [9] M. W. Zemansky, Temperatures very low and very high (Courier Corporation, 1981). [10] C.-S. Wu, E. Ambler, R. Hayward, D. Hoppes, and R. P. Hudson, Physical review 105, 1413 (1957). [11] J. Dunietz, The experiment that taught us what left really means,. [12] R. L. Garwin, L. M. Lederman, and M. Weinrich, Physical Review 105, 1415 (1957). [13] M. Gell-Mann and A. Pais, in Murray Gell-Mann: Selected Papers (World Scientific, 2010) pp [14] J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, Physical Review Letters 13, 138 (1964). [15] M. Sozzi, Discrete symmetries and CP violation: From experiment to theory (Oxford University Press, 2008).