[MUSIC] In this sixth module, we are discussing weak interactions. In this second video, we introduce the operations of charge conjugation, parity, and time reversal. After following this video, you will know discrete transformations of space time, which are parity and time reversal, the charge conjugation transformation, which reverses all charges, and the behavior of forces and matter under these transformations. The reversals of space and time directions are discrete transformations of space-time coordinates. For the space coordinates, the transformation is also called the parity, P. Both transformations can be applied as operators to a quantum state such as a particle. These operators are unitary, because a double application to the initial state reproduce the original state. Therefore the eigenvalues of the two operators are plus or minus 1 if they are applied to eigenstates of parity or time reversal. However, there are many wave functions which are not eigenstates of parity. For example, a function phi equal to cos x + sin x becomes under the application of parity cos x - sin x, which is neither +phi nor -phi. Eigenstates of P can be classified like in this table. Scalar wave functions describe the spin zero particles with positive parity. Pseudoscalar ones describe spin zero particles with parity minus. Vector and axial vector wave functions describe the spin one particles with parity negative and positive respectively. Parity is a multiplicative quantum number, so the total parity of a system in its ground state is the product of the individual parities of its components. If there is an angular momentum between the components, characterized by am l-type quantum number, it must be taken into account by a factor (-1)^l. Parity is conserved in interaction due to electromagnetic and strong forces, but not in weak interactions. Conservation means that if a system is in an eigenstate before the reaction, it will be in an eigenstate with the same eigenvalue after the reaction. For bound states, the total angular momentum, J=l + s, comes in, multiplying the intrinsic parity of the components. Consequently, the notation J^P is used to characterize particles. You find it in the PDG tables for each particle. Electromagnetic transitions in bound states, as the hydrogen atom for example, require that ∆l ± 1. So the parity of the photon must be -1. The photon is thus indeed a vector particle, as we noted in module 4. Hadrons are parity eigenstates, thus their parity eigenvalues can be used to characterize a particle, just like spin, electric charge or baryon number. The parity of fermions is opposite to that of antifermions. By convention, parities are arbitrarily fixed to +1 for leptons and quarks, and antiquarks and antileptions, therefore, have parity -1. For bosons, the parity is the same for particles and antiparticles. The photon parity, as we already said, is -1. Baryons which contain three quarks without a relative quantum angular momentum, such as the proton and neutron, have a parity (+1)^3, so, finally, parity +1. That is to say that J^P is (1/2)^+. The lightest mesons, such as pions as well as light kaons, contain quark and antiquark with anti parallel spins. They have parity (+1)(-1), so -1, and they are called pseudoscalar mesons. The states with the same quark contents but parallels spins, like the mesons rho, K^*, omega, and Phi are called vector mesons. They have J^P equal to 1^-. But parity is not just a theoretical concept, it has a measurable effect whenever it corresponds to a conserved observable. Let us look at the dominant decay of the π^0, which goes into photons. The study of this process has allowed to conclude that π^0 is a pseudoscalar boson with J^P = 0^-. Its decay products are two photons, vector boson of J^P = 1^-. We project spins on the only natural axis of the system, given by the direction of the two photons in the rest frame of the π^0. Conservation of angular momentum requires one of the two configurations sketched here, or a combination of both. The first configuration has two photons in a right circular polarization state. The second has the left polarized photons. Under parity, the momentum of the two photons change sign but their spin, being pseudovectors, do not change. So the parity operation changes the first configuration into the second one, and vice versa. The photon wave function is characterized by its polarization vector epsilon, which points in the direction of the electric field, and for real photons is normal to the direction of motion. As parity is conserved in decay process, this requires that the total wave function psi be an eigenstate of parity. Up to a normalization constant, the two possibilities correspond to the wave function psi_1 and psi_2, given here. The first is proportional to a scalar, so it has parity +1. The angle phi between the two polarization planes will be preferentially zero, with an intensity distribution proportional to the square of cosine phi. The second is proportional to the product between a pseudovector and a vector. So it has parity -1. The angle phi between the polarization planes of the two photons will be preferentially 90 degrees. Experimentally, it is found that they are indeed orthogonal. The party of the π^0 is therefore -1. The charge conjugation operator, indicated by C, is an example of a transformation of the field itself and not of coordinates. It transforms the wave function of a particle, phi_P, into the wave function of its antiparticle, phi_pbar. The operation therefore changes the sign of electric charge, color, baryon, and lepton number, in short, all the quantum numbers of the charge type. Thus the operator C applied to a fermion gives an antifermion, with all charges opposite, with the same mass, the same spin, and the same momentum. Again electromagnetic and strong interactions conserve charge conjugation. That is to say that this interactions have the same intensity for particles and anti-particles. Weak interactions on the contrary violate C, so they distinguish between the particles and the anti-particles. Particles which do not have any charge can be eigenstates of C. They are their own anti-particles. The photon and the Z boson are examples. The pseudoscalar meson π^0, which is also its own anti-particle, has C phi_pi = phi_pi, with eigenvalue C = +1. Since the photon is generated by moving charges, which change sign of the electric charge under C, C A_µ = -A_µ. So charge conjugation is another multiplicative quantum number. A system of n photons will have a C equal to (-1)^n. For example, π^0 decay into two photons respects the conservation of C, but π^0 decay into three photons is forbidden. Indeed, the branching ratio of this decay is measured to be less than 3.1 10^-8. Time reversal operation, indicated by T, transforms the coordinate for four-vector x, so (t,x,y,z) into x’ = (-t,x,y,z). When applied to a field, T also turns them into their complex conjugate, and this is necessary because of the transformation of the equation of motion. Except for real boson fields, there are no eigenstates of T alone. So no conserved quantum number associated. The importance of T is rather in combination with the other discrete transformations, parity and charge conservation. One can easily understand that all the local fields theories must be invariant under the joint action of CPT, transforming a process into itself. We will see what follows that weak interaction violated to a maximum extent the symmetry C and P, and even the combined operation CP, which transforms right handed fermions into a left hand antifermions, is not respected by weak interactions. This introduces a small difference between a reaction and its reverse. It gives an objective direction to time. We will introduce weak interactions in the next video. [MUSIC]