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Physics 557 – Lecture 8 Quantum Numbers of the Standard Model

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Physics 557 – Lecture 8 Quantum Numbers of the Standard Model Physics 557 – Lecture 8 Quantum numbers of the Standard Model – (See Chapter 2 in Griffiths and Part B and Chapter 15 in Rolnick.) We want to organize the experimental results of the last 70 years produced by the accelerators and experiments introduced in Lecture 7 in terms of the “particle” content – the building blocks. To this end we will make extensive use of the idea of quantum numbers and symmetries. Recall that in quantum mechanics (and quantum field theory) we can characterize a complete set of basis states in terms of their eigenvalues with respect to a maximal set of commuting operators. Generally this set of commuting operators includes the full Hamiltonian, i.e., the states have definite (real) energy so that the states do not decay and the quantum numbers in question are conserved. For our purposes it is useful to expand these ideas a bit (to allow decay). In particular we imagine the Hamiltonian as having 3 distinct parts HHHH~strong EM weak . (8.1) For now we will ignore the role of both gravity and the Higgs sector. The former can be ignored on the grounds that it is a very weak interaction and, in any case, we do not have an adequate quantum description. The Higgs sector has yet to be established experimentally (although there have been hints) and we will return to it later. The separation into 3 sectors makes sense since the three interactions of the Standard -1 -7 model have distinctly different “strengths”, i.e., gstrong ~ 1, gEM ~ 10 , gweak ~ 10 , and preserve different quantum numbers (commute with different charges). Of course, the second two sectors are intimately related through electro-weak unification and symmetry-breaking but first we want to understand the organization of the observed particles. From this standpoint the masses of the quarks and massive vector bosons are determined by externally provided masses terms. The masses of the observed hadrons (strongly interaction particles) are “explained” by Hstrong and in that sense we think of Hstrong as the dominant member of this trio. The other two terms enter our discussion essentially as the explanation for particle decays. (Of course, HEM is also the source of atomic physics and thus of chemistry and life!). We will be interested in those generators of symmetry operations that commute with Hstrong, but not necessarily with HEM or Hweak. The idea of resonances will play an important in our understanding of the structure of the standard model. As discussed earlier, resonances correspond to bona fide states of the underlying theory, which, however, are produced and decay via the strong interaction. Thus their lifetimes (~ 10-23 s) are so short that these states are not Lecture 8 1 Physics 557 Autumn 2012 detected as observable tracks in detectors. Rather their existence is detected via enhanced interaction rates in specific channels, i.e., channels with specific quantum numbers. They play an essential role in our counting of the possible degrees of freedom and the confirmation of the Standard Model. In contrast, “particles” are characterized by living long enough to be “seen” as tracks (or at least displaced vertices) in detectors. However, it is still true that most of these particles eventually decay (often inside of today’s large detectors) and it is precisely by the analysis of these decays that we come to understand the identity of these particles and the structure of the Standard Model. Note that only the electron, the proton and the neutrinos are apparently free from decay and that this observation is presumably of limited validity, i.e., limited by the quality of our experiments. If the idea of Grand Unification is valid, there must be (extremely weak!) interactions (i.e., symmetry generators) that connect even the light quarks to the leptons and allow protons to decay. Likewise, since at least some of the neutrinos are massive, oscillations between different kinds of neutrinos are possible, and have been observed. The observed hierarchy of decay times is understood in terms of the conservation of certain quantum numbers by some of the interactions but not by the others. In particular, the strong interactions respect (commute with the charge operators of) essentially all readily observable quantum numbers. The underlying operators of the strong interaction operate on (i.e., do not commute with) the color charge, but for distances > 1 fm we observe only color singlets. Thus the strong interactions cannot contribute to processes that change any of the interesting quantum numbers (quark flavors, electric charge, parity, etc.). On the other hand, the weak interaction respects very few quantum numbers. Once we have discussed the general properties of quantum numbers, we will proceed to discuss the various particles observed, essentially in historical order, and describe how the various relevant quantum numbers were introduced. We can think of the quantum numbers as being the eigenvalues of some operator acting on the appropriate “single” (or multi-) particle state (with the “” ’s to remind us that we will include resonances in the counting of 1-particle states). Generally, for the case of an eigenstate of an operator, we can write Op Op XX X . (8.2) Lecture 8 2 Physics 557 Autumn 2012 The quantum numbers of interest will generally fall into two classes – multiplicative and additive. This distinction arises from the way in which the quantum numbers are combined when we consider 2 (or more) particle states. Thus for multiplicative quantum numbers we simply multiply the quantum numbers of the two particles together to find the corresponding quantum number for the 2-particle state (ignoring for now the impact of any interactions and the possible influence of the composite state), 1 2 3:Op3 3 Op 1 Op 2 1 2 . Such quantum numbers are typically associated with discrete symmetry groups (see Chapter 4 in Griffiths and Chapter 12 of Rolnick) and the typical values of are 1. Examples are parity P, charge conjugation C, the combination CP and time reversal T. The symmetry group for all of these operations is isomorphic to the two-element group we discussed earlier. P,1 G : P-1 = P, P2 = 1. Additive quantum numbers, on the other hand, are, as the name implies, additive, 1 2 3:Op3 3 Op 1 Op 2 1 2 . Examples are electric charge, baryon number, lepton number, etc. In these cases the quantum numbers of composite states are the scalar sums of the quantum numbers of the parts. More generally, the addition of the quantum numbers may be of a vector nature or be even more complex, as with SU(3) quantum numbers. Examples include momentum, angular momentum, isospin, etc. The symmetry group associated with additive quantum numbers is generally a continuous group, which may or may not be a local symmetry. Multiplicative Quantum Numbers (and discrete symmetries) Parity, P: the operation of parity inverts all vectors through the origin. We can think of the corresponding passive operation as reflecting each of the unit vectors through the origin. Thus in the transformed frame the directions of all (real) vectors are opposite, i.e., the signs of all components have changed by a factor of -1: rP r r, (8.3) x,,,,,,. y z P x y z x y z Thus a usual 3-vector is odd under P, i.e., has eigenvalue = –1. The usual scalar, e.g., the scalar product of two vectors Lecture 8 3 Physics 557 Autumn 2012 s r1 r 2 P (8.4) s r1 r 2 r 1 r 2 r 1 r 2 s, is unchanged by P and thus has eigenvalue = +1. As we know there are quantities, with which we are already familiar, that behave in a different manner – pseudovectors (or axial vectors) and pseudoscalars. A good example of the former is angular momentum, which involves two vectors: L r p P (8.5) L r p r p r p L. Unlike a vector, a pseudovector has P eigenvalue +1. With 3 independent (ordinary) vectors available we can construct a pseudoscalar using the familiar scalar triple product, which transforms as ps r1 r 2 r 3 P (8.6) ps r1 r 2 r 3 r 1 r 2 r 3 ps. Unlike a true scalar, a pseudoscalar has P eigenvalue –1. (Note that this pseudoscalar product is constructed with our old friend jkl.) When we combine two (or more) particles, the parity of the resulting composite state will depend on the product of the intrinsic parities of the two particles, times any collective parity of the two particle state. This latter factor is easy to determine for states of definite angular momentum, l. Such states are described by the spherical harmonic functions Yl,m(,). In the transformed frame = -, = + and it is easy to verify that Yl, m , P l (8.7) YYYl,,, m, l m , 1 l m , . Thus the parity eigenvalue of the composite state of two particles with relative P P angular momentum l and intrinsic parities 1 and 2 is the product PPPl composite12 1 1 2 . (8.8) Lecture 8 4 Physics 557 Autumn 2012 An extremely useful application of this form is the parity eigenvalue for a state composed of a spin ½ fermion and its anti-particle. It is straightforward to verify that the structure of the Dirac equation requires that a fermion and its antiparticle have opposite intrinsic parity (see page 141 in Griffiths and Eq. 12.8 and problem 12.1 in Rolnick). In particular, as we will see in more detail later, if t, r is a solution of the Dirac equation in the original frame, then 0 t, r is a solution in the frame reached after a parity transformation.
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