Time Symmetries in Particle Physics

Space –time symmetries in particle physics

• Read Chapter 5 in M&S, 4th ed. • Some of the conservation laws discussed in this chapter – those for linear and angular momentum – are universal laws of nature, valid for all interactions. Others we shall meet, like parity, are only conserved in the approximation that weak interactions are neglected. Their violation will be discussed in Chapter 11. Here we shall neglect weak interactions and concentrate on the strong and electromagnetic interactions with which we will be primarily concerned in the next two chapters. • The conservation laws we discuss have their origin in the symmetries Translational and rotational invariance

• Translational invariance expresses the fact that all positions in space are indistinguishable for a closed system, the Hamiltonian is invariant under this translation It leads to the conservation law for linear momentum (derivation in M&S):

Similarly, the rotational invariance leads to the conservation of angular momentum Adding spin to angular momentum

• Note, that if the particle has spin, then the total angular momentum J is the sum of the orbital and spin angular momenta, J = L + S Iy leads to a conservation of the total angular momentum: a sum of L+S Note, that it does not lead to a separate conservation of L and S

because of the existence of spin dependent forces Classification of particles

• we use the word particle to describe both elementary particles, which have no internal structure, and composite particles, which are bound states of smaller constituents • a composite particle may have ‘good’ quantum numbers, related to its internal structure, which take on deﬁnite values and therefore characterize the particle. The associated operators with the “good” quantum numbers commute with the Hamiltonian. Classification of the particles (2) • In the rest frame of the particle, the total angular momentum J of the constituents that make up the particle is conserved, but the total orbital angular momentum L and the total spin angular momentum S are not separately conserved. However, despite this, it is often a very good approximation to assume that L2 and S2, are conserved separately (it is an approximation!) Classification of particles: spectroscopic notation

• This approximation corresponds to assuming that the forces can, for example, ﬂip the direction of the spin, thereby changing its components, but not its magnitude. In this approximation, the orbital angular momentum and spin quantum numbers L and S are also good quantum numbers, so that the particle is characterized by SP = J, L and S, while a fourth Jz depends on the orientation of its spin. This is the basis of the so-called ‘spectroscopic notation’ where, instead of the numerical value of L, usually the historical S, P, D, F, ... 3 for L =0, 1, 2, 3,.... are used. Thus a composite particle denoted by S1 has L =0, 2 S =1 and SP = J =1, while particle D3/2 has L =2, S =1/2, and SP = J =3/2. In all of this it is of course crucial to distinguish between the spin

SP = J of the particle and the spin S of its constituents. An example (deuteron) consisting of proton and neutron from nuclear physics:

3 • the deuteron is not a pure S1 lowest state with L=0 of proton and neutron as one would naturally assume, but has, as experiment 3 shows, a few percent admixture of the D1 state with L =2. Thus the assumption that L is a good quantum is a good approximation, but not exact. The same holds in particle physics. Angular momentum in the quark model

• Mesons are qq¯ bound states, with the rest frame of the meson corresponding to the centror-of-mass frame of the qq¯ system. In this frame there is a single orbital angular momentum L, but two con- stituent spins, so in an obvious notation:

Since both q and q¯ have spin 1/2, the only possibilities are S =0and S =1.ForL =0,J = S, and in the spectroscopic notation (5.27) the possible states are Angular momentum in the quark model (2) Angular momentum in the quark model (3)

• if (as we expect) the lightest meson states have L = 0, then they can have spin 0 or 1, so we would expect two low-mass states with spin 0 and spin 1. This is exactly what is found experimentally, and the lower lying of the two states always has spin-0. Examples are the π, K and D 1 mesons discussed, which are all spin-0 mesons corresponding to S0 states of the appropriate qq¯ systems. Angular momentum in the quark model (4)

• Baryons are qqq bound states, so in the rest frame of the baryon, corresponding to the centre-of-mass frame of the qqq system, there are two orbital angular momenta associated with the relative motion of the three quarks. The ﬁrst is conveniently taken to be the orbital angular momentum L12 of a chosen pair of quarks in their mutual center-of-mass frame. The second is then the orbital angular momentum L3 of the third quark about the center-of-mass of the pair in the overall center-of-mass frame. This is illustrated in Figure. The total orbital angular momentum is given by

L = L12 + L3, Angular momentum in the quark model (5)

the spin is the sum of the spins of the three quarks

S = S1 + S2 + S3, so that S =1/2or S =3/2. It is now straightforward to show that the possible baryon states in the spectroscopic notation are: Angular momentum in the quark model (6)

• The baryon spectrum is clearly very complicated! However, if we restrict ourselves to the lowest-lying states with zero orbital angular momentum L12 = L3 = 0, then the baryon spin SB = J can be ½ or 3/2. We therefore expect the lightest baryons to have spin-1/2 or spin-3/2. Again this is in accord with experiment. 2 The p, n, Λ, Λc and Λb, we now identify with the S1/2 states PARITY

• The parity transformation is parity is not an exact symmetry, but is violated by the weak interaction. This will be discussed in Chapter 11. Parity is conserved in strong and em interactions, discussed here

Parity operator deﬁned by

Two successive parity transformations leave the system unchanged: Parity of a particle • For a particle with arbitrary momentum

Then for a particle at rest, with p = 0, eigenvalue Pa is called the intrinsic parity of particle a. The words ‘at rest’ usually left implicit For many-particle systems

In addition to a particle at rest, a particle with a deﬁnite orbital angular momentum is also an eigenstate of parity. The wavefunction for such a particle has the form Parity for a particle with an orbitaI momentum l

It can be shown (M&S) that for a particle with definite orbital momentum Intrinsic Parity of fermion and antifermion • From Dirac equation the product of parities of electron and positron (or for pointlike fermion-antifermion) have opposite values, individual parities of particles and antiparticles have no meaning .They are produced only in pairs in strong and em interactions. From Dirac equation For spin ½ fermions

It is convenient to define for fermions (hence opposite true for antifermions) Parity of quarks and hadrons

• Parity for quarks and antiquarks:

Hence the intrinsic parity of a meson M (bound quark a and antiquark b):

For a baryon:

For antibaryon: Parity of hadrons (2) Parity of hadrons (3) Parity od the charged pion and of photon

• Parity of the charged pion has been measured experimetally, and it is in agreement with predictions of parity of quark-aniquark system, assuming L=0

• Parity of photon can be deduced theoretically using the principle of correspondence with classical e.m. theory (M&S): Charge conjugation C

• C : changes particles to antiparticles (in the same state) • Charge and magnetic moment of every particle reversed • Is symmetry of strong and e.m interactions, but violated in weak (Ch11, M&S) Here we assume in strong and e.m. interactions: Charge conjugation (2) Charge conjugation (3)

• For multiparticle states we have:

• Interesting that (see also for details M&S) Charge conjugation (4) C-parity of the photon and neutral pion decays • C-parity of the photon (under charge conjugation operation C) Can be also deduced using correspondance princile (M&S):

Hence

Cannot happen (forbidden) Experimentally it is confirmed.