PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 28 Thursday, October 29, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Contents 1 Need for Color 1 2 Quark Model for Mesons 3 3 Valence and Sea Quarks 5 4 Weak Isospin 5 5 Color Symmetry 7 6 Gauge Bosons 7 1 Need for Color When we try to extend the quark model to all baryons we face a theoretical difficulty. Let us look at the ∆++ baryon. It is non-strange, carries two units of positive charge, and has spin angular momentum of 3=2. These properties immediately lead to the naive conclusion that the ∆++ can be described by three up quarks ∆++ = uuu: (1) This proposed substructure respects all the known quantum numbers. In the ground state of the ∆++, assuming that there are no contributions from relative orbital waves, the three up quarks can have parallel spins. 3 This can give a resultant value of J = 2 . 3 3 That is, l = 0, s = 2 giving J = 2 . PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 However, this leads to a paradox. The wave function for this state, representing three identical fermions, would therefore be sym- metric under the exchange of any two quarks. This is incompatible with the Pauli principle, which requires a wave function containing identical fermions to be totally antisymmetric. We can find a resolution if we assume that all quarks carry an additional internal quantum number, and that the state given in Eq. (1) is, in fact, antisymmetric in the space corresponding to this quantum number. This additional degree of freedom is referred to as color. Each of the quarks comes in three different colors. The quark multiplets take the form ! ! ! ua ca ta ; ; a = red; green; blue: (2) da sa ba Hadrons do not appear to carry any net color, and therefore correspond to bound states of quarks and antiquarks of zero total color quantum number. That is, hadrons are color-neutral bound states of quarks. This hypothesis leads to an excellent description of all known baryons as bound states of three quarks, and of mesons as bound states of quark-antiquark pairs. We have baryon = red + green + blue = colorless (3) meson = color + anticolor = colorless (4) In this language, the Ω− baryon, which has a S = −3 and s = 3=2, corresponds to the ground state of three strange quarks Ω− = sss: (5) The existence of color can be established as follows. Consider the annihilation of an electron and positron, leading to the creation of a µ+µ− pair or a quark-antiquark pair. The reaction can be thought of as proceeding through the production of an intermediate virtual photon, as shown in Fig. 1. The cross section for the production of hadrons in this process depends on the number of ways a photon can produce a quark-antiquark pair. This must therefore be proportional to the number of available quark colors. That is, the ratio of production cross sections σ(e−e+ ! hadrons) R = (6) σ(e−e+ ! µ−µ+) 2 / 8 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 e+ μ+(q¯) e− μ−(q) Figure 1: The annihilation of e+e− through a virtual photon into µ+µ− or qq¯ pair. is proportional to the number of quark colors. Experiments are consistent with exactly three colors. The production of hadrons through the mechanism in Fig. 1 depends, in addition, on the electric charges of the quarks. Such data also confirm the fractional nature of electric charge carried by quarks. 2 Quark Model for Mesons Let us apply the symmetry requirements of the strong interaction to qq¯ wave functions., and thereby deduce the quantum numbers that we would expect for the spectrum of charge-neutral meson states in a simple non-relativistic quark model. We will establish the restrictions on the spin (J), parity (P ) and charge conjugation (C) quantum numbers that apply to such systems. The qq¯ wave function is a product of separate wave functions Ψ = space spin charge; (7) where space denotes the space-time part of the qq¯ wave function, spin represents the intrinsic spin, and charge the charge conjugation properties. We have ignored the color part of the wave function because we know that this has even symmetry for mesons. The symmetry of space under the exchange of the q and q¯ is, as usual, determined by the spherical harmonics and the relative orbital angular momentum of the q and q¯. If we call the exchange operation X, then, schematically, we have l X space ∼ XYlm(θ; φ) = (−1) space: (8) Therefore, if Ψ is a state of definite parity, the spatial part of the wave function will be either symmetric or antisymmetric under exchange, depending on whether l is even or odd. The effect of the exchange operation on spin will depend on whether the two quark spins are in a singlet (s = 0) or triplet (s = 1) state. 3 / 8 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 We have s = 0 : X (j "#i − j #"i) = − (j "#i − j #"i) s = 1 : X (j "#i + j #"i) = + (j "#i + j #"i) : (9) Thus we deduce that s+1 X spin = (−1) spin: (10) Under the action of the exchange operator, q and q¯ become interchanged, and consequently, we can think of this as the operation of charge conjugation in the space of charge. To determine the charge conjugation properties of such a state, let us impose the Pauli principle on our two-fermion system, namely, let us require that the overall wave function change sign under an interchange of q and q¯. Note that we are using a generalized form of the Pauli principle, which treats q and q¯ as identical fermions corresponding to spin up and spin down states in the space of charge. Thus, we require XΨ = −Ψ: (11) Now, using the results of Eqs. (8), (10), and (11) we can write XΨ = X spaceX spinX charge l s+1 = (−1) space(−1) spinC charge = −Ψ: (12) Consequently, for Eq. (12) to hold, we conclude that the meson state must be an eigenstate of charge conjugation with charge parity l+s ηC = (−1) : (13) Thus, for meson states that are eigenstates of charge conjugation, Eq. (13) establishes a rela- tionship between the orbital wave, the intrinsic spin value, and the C quantum number of the qq¯ system. The only relevant quantum number that is still missing in our discussion is the parity of the allowed states. The overall parity of Ψ is determined by the effect of spatial inversion and the intrinsic parities of the quarks. A detailed analysis of relativistic quantum theories reveals that bosons have the same intrinsic parities as their antiparticles, whereas the relative intrinsic parity of fermions and their antiparticles is odd (opposite). Thus the total parity Ψ is P Ψ = −(−1)lΨ = (−1)l+1Ψ; (14) 4 / 8 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 or the total parity quantum number is l+1 ηP = (−1) : (15) Since the spins of mesons are obtained from the addition of the orbital and intrinsic angular momenta of the qq¯ pair J~ = L~ + S;~ (16) we now have all the ingredients for forming an allowed spectrum of mesons. Table 1 lists the possible lowest-lying states, all of which correspond to known mesons. l s j ηP ηC Meson 0 0 0 − + π0; η 0 1 1 − 1 ρ0; !; φ 0 1 0 1 + − b1(1235) 1 1 0 + + a0(1980); f0(975) 0 1 1 1 + + a1(1260); f1(1285) 0 1 1 2 + + a2(1320); f2(1270) Table 1: Lowest-lying meson states expected in the quark model. 3 Valence and Sea Quarks The quarks that characterize the quantum numbers of hadrons are called valance quarks. In addition to this, hadrons contain a sea of quark-antiquark pairs known as sea quarks, as well as gluons. There can be other kind of hadrons such as tetra-quarks (qqq¯ q¯), penta-quarks (qqqqq¯), hybrid mesons (qqg¯ ), and glueballs (gg). 4 Weak Isospin Leptons and quarks come as doublets, or in pairs. Quarks also carry a color quantum number. The existence of such groupings, and the color degrees of freedom, suggest the presence of new underlying symmetries for this overall structure. We can associate a doublet structure with a non-commuting (non-Abelian) symmetry group SU(2) for leptons. We will continue to refer to this underlying symmetry group as isospin, since it is an internal symmetry. 5 / 8 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 Unlike strong isospin, which is used only to classify hadrons, the isospin in the present case also classifies leptons. Leptons, on the other hand, interact weakly and, therefore, this symmetry must be related to the weak interaction. Correspondingly, the isospin symmetry associated with the weak interactions of quarks and leptons is referred to as weak isospin. As with strong isospin, where the symmetry is discernible only when the electromagnetic inter- action (electric charge) can be ignored, so is the essential character of the weak isospin symmetry also apparent only when the electromagnetic force is “turned off”. Under such circumstances, the up and down states ! ! ! ν ν ν e ; µ ; τ (17) e− µ− τ − and ! ! ! u c t ; ; (18) d s b become equivalent and cannot be distinguished. For the case of weak-isospin, we can define a weak hypercharge for each quark and lepton, based on a general form of the Gell-Mann-Nishijima relation Y Q = I + (19) 3 2 or Y = 2(Q − I3); (20) where Q is the charge of the particle, Y is the weak hypercharge, and I3 the projection of its weak isospin quantum number.