PHY401 - Nuclear and Particle Physics

Home , Weak isospin

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 symmetric 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+)

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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 = ψspaceψspinψcharge, (7) where ψspace denotes the space-time part of the qq¯ wave function, ψspin represents the intrinsic spin, and ψcharge 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 ψspace 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 Xψspace ∼ XYlm(θ, φ) = (−1) ψspace. (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 ψspin will depend on whether the two quark spins are in a singlet (s = 0) or triplet (s = 1) state.

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We have

s = 0 : X (| ↑↓i − | ↓↑i) = − (| ↑↓i − | ↓↑i) s = 1 : X (| ↑↓i + | ↓↑i) = + (| ↑↓i + | ↓↑i) . (9)

Thus we deduce that s+1 Xψspin = (−1) ψspin. (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 ψcharge. 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 ψcharge. Thus, we require XΨ = −Ψ. (11)

Now, using the results of Eqs. (8), (10), and (11) we can write

XΨ = XψspaceXψspinXψcharge l s+1 = (−1) ψspace(−1) ψspinCψcharge = −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, ω, 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.

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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 interaction (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. Thus, for the (ν, e−) doublet we obtain

1 Y (ν) = 2 0 − = −1, (21) 2 1 Y (e−) = 2 −1 + = −1. (22) 2

Similarly, for the (u, d) quark doublet, we have

2 1 1 1 Y (u) = 2 − = 2 × = , (23) 3 2 6 3 1 1 1 1 Y (d) = 2 − + = 2 × = . (24) 3 2 6 3

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The Y quantum number for other quark and lepton doublets can be obtained in the same manner. In the Standard Model, only left-handed particles have a doublet structure. The right-handed quarks and the right-handed charged leptons are all singlets with I = 0. There are no right-handed neutrinos in the Standard Model (and also probably in our universe).

5 Color Symmetry

The color symmetry of quarks is also an internal symmetry. It can be shown that it is similar to isospin in that it involves rotations - however, the rotations are in an internal space of three dimensions - corresponding to the three distinct colors of the quarks. The relevant symmetry group is known as SU(3). The interactions of quarks are assumed to be invariant under such SU(3) rotations in color space, leading to an equivalence of quarks of diﬀerent color. (This is needed in order to have consistency with experimental observations.) Because the color quantum number is carried by quarks and not by leptons or photons, we expect this symmetry to be associated only with the strong interaction.

6 Gauge Bosons

The presence of a global symmetry can be used to classify particle states according to some quantum number (e.g., strong isospin). The presence of a local symmetry requires the introduction of forces. Since weak isospin and color symmetry are associated with rather distinct interactions, it is interesting to ask whether the corresponding physical forces - namely, the strong (color) and the weak forces - might arise purely from the requirement that these symmetries being local. It is the current understanding that the local symmetries underlying the electromagnetic, weak, and strong interactions have origin in the U(1)Y , SU(2)L, and SU(3)color symmetry groups, respec- tively.

The group corresponding to the weak hypercharge symmetry, U(1)Y , is a local Abelian symmetry group, while SU(2)L and SU(3)color are non-Abelian groups corresponding to the weak isospin and color symmetries. From the Gell-Mann-Nishijima formula we see that electric charge Q is related to weak hypercharge and weak isospin, from which it follows that the electromagnetic U(1)Q symmetry can be regarded as a particular combination of the weak isospin and weak hypercharge symmetries. Because the doublet structure of quarks and leptons involves only left-handed particles, the weak isospin symmetry group is also conventionally denoted by SU(2)L. This kind of structure is essential for incorporating the properties of neutrinos and of parity violation in weak interactions.

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Earlier we encountered the gauge principle, in which the local invariance necessarily leads to the introduction of gauge potentials, such as the vector potential in electromagnetic interactions. When these potentials are quantized, they provide the carriers of the force, otherwise known as gauge particles. The photon is the carrier of the electromagnetic interaction, or its gauge boson. All the gauge bosons have spin J = 1, and the number of gauge bosons associated with any symmetry reﬂects the nature of that symmetry group. There are three gauge bosons associated with the weak interactions, and they are known as the W +, W − and Z0 bosons. (These were discovered in 1983 at the antiproton-proton collider at the CERN.) For the strong interactions, there are eight gauge bosons, and all are referred to as gluons. (These are the same gluons we have been discussing in connection with the substructure of the nucleon.) The gluons, or the gauge bosons of color symmetry, are electrically neutral, but carry the color quantum number. This is in contrast to the photon, which is the carrier of the force between charged particles, but does not itself carry electric charge.

This diﬀerence can be attributed to the Abelian nature of the U(1)Q symmetry that describes the photon, and the non-Abelian nature of SU(3)color that describes gluons. In Fig. 2 we show the typical interactions among leptons and quarks and between gluons, mediated by diﬀerent fundamental gauge bosons.

e− νe μ− νμ c s e− e−

W− W− W+ Z0

− u d νe e d u νμ νμ

u u e− e− u u g g

γ γ g g

d d u u d d g g

Figure 2: Typical interactions among leptons and quarks and between gluons, mediated by diﬀerent fundamental gauge bosons.

References

[1] A. Das and T. Ferbel, Introduction To Nuclear And Particle Physics, World Scientiﬁc (2003).

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