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Quarks and leptons: introductory course in modern particle physics

Francis Halzen, Alan D. Martin

Chapter 2

Symmetries and Quarks - all with Video Answers

Educators

Chapter Questions

Problem 1

Justify the decomposition shown in (2.1) by either (1) considering the symmetry of the states under interchange of the nucleons or (2) using the angular momentum "step-down" operator.

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03:38

Problem 2

If the nucleons are in a state of relative orbital angular momentum $L=0$, use the Pauli exclusion principle to show that $S+I$ must be an odd integer.

There is much evidence to show that the nuclear force is invariant under isospin transformations [for example, that it is independent of the value of $I_3$ in the $I=1$ multiplet of (2.2)]. For instance, consider the three nuclei ${ }^6 \mathrm{He},{ }^6 \mathrm{Li}$, and ${ }^6 \mathrm{Be}$, which can be regarded respectively as an $\mathrm{nn}, \mathrm{np}$, and pp system attached to a ${ }^4 \mathrm{He}$ core of $I=0$. After correcting for the Coulomb repulsion between the protons and for the neutron-proton mass difference, the observed nuclear masses are as sketched in Fig. 2.2. Furthermore, isospin invariance requires that the same nuclear physics should be obtained for each of the three $I=1$ states $\left(I_3=\right.$ $-1,0,1)$, just as rotational invariance ensures that the $2 J+1$ substates of an isolated system of total angular momentum $J$ describe exactly equivalent physical systems.

Penny Riley
Penny Riley
Numerade Educator

Problem 3

Use isospin invariance to show that the reaction cross sections $\sigma$ must satisfy
$$
\frac{\sigma\left(\mathrm{pp} \rightarrow \pi^{+} \mathrm{d}\right)}{\sigma\left(\mathrm{np} \rightarrow \pi^0 \mathrm{~d}\right)}=2,
$$
given that the deuteron $\mathrm{d}$ has isospin $I=0$ and the $\pi$ has isospin $I=1$.

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Problem 4

Show that the four successive infinitesimal rotations ( $\varepsilon$ about the 1-axis, followed by $\eta$ about the 2 -axis, then $-\varepsilon$ about the 1 -axis, and finally $-\eta$ about the 2 -axis) are equivalent to the second-order rotation $\varepsilon \eta$ about the 3-axis. Hence, show that the generators satisfy
$$
\left[J_1, J_2\right]=i J_3 \text {. }
$$

Nonlinear functions of the generators which commute with all the generators are called invariants or Casimir operators. For the rotation group,
$$
J^2=J_1^2+J_2^2+J_3^2
$$
is the only Casimir operator,
$$
\left[J^2, J_i\right]=0 \quad \text { with } i=1,2,3 .
$$

It follows that we can construct simultaneous eigenstates $|j m\rangle$ of $J^2$ and one of the generators, say $J_3$. Using only (2.13), it is possible to show that
$$
\begin{aligned}
J^2|j m\rangle & =j(j+1)|j m\rangle \\
J_3|j m\rangle & =m|j m\rangle
\end{aligned}
$$
with $m=-j,-j+1, \ldots, j$, and where $j$ can take one of the values $0, \frac{1}{2}, 1, \frac{3}{2}, \ldots$

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Problem 5

Verify (2.16). To do this, it is useful to form the so-called "step-up" and "step-down" operators
$$
J_{ \pm}=J_1 \pm i J_2 .
$$

First, show that
$$
J_{ \pm}|j m\rangle=(C-m(m \pm 1))^{1 / 2}|j, m \pm 1\rangle,
$$
that is, $J_{ \pm}$step $m$ up and down by one unit, respectively. Show that $C=j(j+1)$.

A state $|j m\rangle$ is transformed under a rotation through an angle $\theta$ about the 2-axis into a linear combination of the $2 j+1$ states $\left|j m^{\prime}\right\rangle$, with $m^{\prime}=-j,-j+$ $1, \ldots, j$ :
$$
e^{-i \theta J_2}|j m\rangle=\sum_{m^{\prime}} d_{m^{\prime} m}^j(\theta)\left|j m^{\prime}\right\rangle,
$$
where the coefficients $d_{m^{\prime} m}^j$ are written in conventional notation and are frequently called rotation matrices. From (2.19), we see the states having the same $j$ but all possible $m$ values transform among themselves under rotations. In fact, all the $2 j+1$ states are mixed by rotations. They form the basis of a $(2 j+1)$ dimensional irreducible representation of the rotation group. The set of states is called a multiplet.

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03:24

Problem 6

Show that the rotation matrices
$$
d_{m^{\prime} m}^j(\theta)=\left\langle j m^{\prime}\left|e^{-i \theta J_2}\right| j m\right\rangle
$$
for $j=\frac{1}{2}$ and $j=1$ are
$$
j=\frac{1}{2}\left\{\begin{array}{l}
d_{++}=d_{--}=\cos \frac{1}{2} \theta \\
d_{-+}=-d_{+-}=\sin \frac{1}{2} \theta
\end{array}\right.
$$

where \pm denote $m= \pm \frac{1}{2}$, respectively, and
$$
j=1\left\{\begin{array}{l}
d_{01}=-d_{10}=-d_{0-1}=d_{-10}=\sqrt{\frac{1}{2}} \sin \theta \\
d_{11}=d_{-1-1}=\frac{1}{2}(1+\cos \theta) \\
d_{-11}=d_{1-1}=\frac{1}{2}(1-\cos \theta) \\
d_{00}=\cos \theta .
\end{array}\right.
$$

Thane Stiles
Thane Stiles
Numerade Educator
01:05

Problem 7

Show that the rotation of a spin- $\frac{1}{2}$ system through a finite angle $\theta$ about the 2 -axis corresponds to the unitary transformation
$$
e^{-i \theta \sigma_2 / 2}=\cos \frac{\theta}{2}-i \sigma_2 \sin \frac{\theta}{2} .
$$

AG
Ankit Gupta
Numerade Educator
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Problem 8

Obtain the matrix representations of the $\lambda_i$ of Fig. 2.3. Show that
$$
\left[\frac{\lambda_i}{2}, \frac{\lambda_j}{2}\right]=i \sum_k f_{i j k} \frac{\lambda_k}{2}
$$
where the $S U(3)$ structure constants $f_{i j k}$ are fully antisymmetric under interchange of any pair of indices (see exercise 14.9) and the nonvanishing values are permutations of
$$
\begin{gathered}
f_{123}=1, \quad f_{458}=f_{678}=\sqrt{3} / 2, \\
f_{147}=f_{165}=f_{246}=f_{257}=f_{345}=f_{376}=\frac{1}{2} .
\end{gathered}
$$

Victor Salazar
Victor Salazar
Numerade Educator
01:34

Problem 9

Make use of (2.54) and (2.53) to predict the decay modes and branching ratios of the $\phi$-meson (mass $1020 \mathrm{MeV}$ ). Comment on the width of the resonance.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
00:54

Problem 10

Explain why a particularly good way of identifying mesons coupled to the $\pi \pi$-channel is to study the reaction $\pi \mathrm{N} \rightarrow(\pi \pi) \mathrm{N}$ at high energies. Show that $I+J$ must be an even integer for these mesons.

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
03:28

Problem 11

Write down the quark composition of the three "dds" states.

Carlos Henrique De Lima
Carlos Henrique De Lima
Numerade Educator

Problem 12

Determine the structure of the six "uds" states. In particular, show that the $S U(3)$ singlet state is the completely antisymmetric combination
$$
\left.(\mathrm{qqq})_{\text {singlet }}=\sqrt{\frac{1}{6}} \text { (uds }- \text { usd }+ \text { sud }- \text { sdu }+ \text { dsu }- \text { dus }\right) .
$$

In the ground state, the baryon spin is found simply by the addition of three spin- $\frac{1}{2}$ angular momenta. Writing the decomposition in terms of the multiplicities of the spin states, we found [see (2.32)]
$$
2 \otimes 2 \otimes 2=(3 \oplus 1) \otimes 2=4 \oplus 2 \oplus 2,
$$
or, in other words, baryon spin multiplets with $S=\frac{3}{2}, \frac{1}{2}, \frac{1}{2}$. The subscripts on the mixed symmetry doublets $\left(M_S, M_A\right)$ indicate that the spin states are symmetric or antisymmetric under interchange of the first two quarks. The four $S=\frac{3}{2}$ spin states are totally symmetric.

Note that in deriving eqs. (2.60)-(2.62), we were working in the $S U(2)$ isospin sector of $S U(3)$. We can therefore apply these results directly to $S U(2)$ spin if we make the replacements $\mathbf{u} \rightarrow \uparrow$ and $\mathrm{d} \rightarrow \downarrow$. Using this analogy, we immediately obtain the composition of the spin "up" state belonging to each of the three spin multiplets
$$
\begin{aligned}
\chi(S) & =\sqrt{\frac{1}{3}}(\uparrow \uparrow \downarrow+\uparrow \downarrow \uparrow+\downarrow \uparrow \uparrow) \\
\chi\left(M_S\right) & =\sqrt{\frac{1}{6}}(\uparrow \downarrow \uparrow+\downarrow \uparrow \uparrow-2 \uparrow \uparrow \downarrow) \\
\chi\left(M_A\right) & =\sqrt{\frac{1}{2}}(\uparrow \downarrow \uparrow-\downarrow \uparrow \uparrow) .
\end{aligned}
$$

To enumerate the baryons expected in the quark model, we must combine the $S U$ (3) flavor decomposition of (2.59) with the $S U(2)$ spin decomposition of (2.64),
$$
\begin{array}{cccc}
(10+8+8+1), & (4+2+2) \\
s & M_S \quad M_A \quad A & s & M_S \quad M_A .
\end{array}
$$

Considering the product symmetries, we are led to assign the ( $S U(3), S U(2))$ multiplets to the following categories:
$$
\begin{aligned}
S & :(10,4)+(8,2) \\
M_S & :(10,2)+(8,4)+(8,2)+(1,2) \\
M_A & :(10,2)+(8,4)+(8,2)+(1,2) \\
& :(1,4)+(8,2)
\end{aligned}
$$
where, for example, the totally symmetric $(S)$ octet arises from the combination
$$
\begin{gathered}
\sqrt{\frac{1}{2}}[(8,2)+(8,2)] . \\
M_s, M_s \quad M_A, M_A
\end{gathered}
$$

The lowest-mass baryons fit neatly into the symmetric spin- $\frac{3}{2}$ decuplet $(10,4)$ and the spin- $\frac{1}{2}$ octet $(8,2)$ (see Fig. 2.8).

This symmetry of the ground state poses a problem, however. For example, a $\Delta^{++}$of $J_3=\frac{3}{2}$ is described by the symmetric wave function
$$
\mathbf{u} \uparrow \mathbf{u} \uparrow \mathbf{u} \uparrow,
$$
whereas we expect antisymmetry under the exchange of identical fermion quarks. As noted in Chapter 1, the explanation is that the quarks possess an additional attribute, called color, which can take three possible values, R, G, or B. The quarks form a fundamental triplet of an $S U(3)$ color symmetry which, unlike $S U(3)$ flavor, is believed to be exact. All hadrons are postulated to be colorless; that is, they belong to singlet representations of the $S U(3)$ color group. The color wavefunction for a baryon is therefore [compare (2.63)]
$$
(\mathrm{qqq})_{\text {col. singlet }}=\sqrt{\frac{1}{6}}(\mathrm{RGB}-\mathrm{RBG}+\mathrm{BRG}-\mathrm{BGR}+\mathrm{GBR}-\mathrm{GRB}) \text {. }
$$

The required antisymmetric character of the total wavefunction is achieved; it is overall symmetric in space, spin, and flavor structure and antisymmetric in color. As the color structure of (2.70) is common to all baryons, we suppress it from now on, but remember to select only overall symmetric representations of space $\times$ spin $\times$ flavor.

A relevant example of an explicit quark model wavefunction is that for a spin-up proton. From (2.68),
$$
|p \uparrow\rangle=\sqrt{\frac{1}{2}}\left(\mathrm{p}_S \chi\left(M_S\right)+\mathrm{p}_A \chi\left(M_A\right)\right),
$$
where the flavor and spin components are given by $(2.60),(2.62)$, and (2.65). Thus (omitting an irrelevant overall minus sign), we have
$$
\begin{aligned}
|p \uparrow\rangle= & \sqrt{\frac{1}{L H}}[\operatorname{uud}(\uparrow \downarrow \uparrow+\downarrow \uparrow \uparrow-2 \uparrow \uparrow \downarrow)+\mathrm{udu}(\uparrow \uparrow \downarrow+\downarrow \uparrow \uparrow-2 \uparrow \downarrow \uparrow) \\
& +\operatorname{duu}(\uparrow \downarrow \uparrow+\uparrow \uparrow \downarrow-2 \downarrow \uparrow \uparrow)] \\
= & \sqrt{\frac{1}{1 \kappa}}[\mathbf{u} \uparrow \mathbf{u} \downarrow \mathbf{d} \uparrow+\mathbf{u} \downarrow \mathbf{u} \uparrow \mathbf{d} \uparrow-2 \mathrm{u} \uparrow \mathbf{u} \uparrow \mathrm{d} \downarrow+\text { permutations }] .
\end{aligned}
$$

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04:03

Problem 13

Construct the quark model wavefunctions $|\mathrm{p} \downarrow\rangle,|\mathrm{n} \uparrow\rangle$. and $|\mathrm{n} \downarrow\rangle$. The charge operator is defined as $Q=\sum Q_{\text {, where }} Q$, are the charges of the quarks in units of the proton charge $e$. The sum is over the constituent quarks of the hadron. Show that
$$
\begin{aligned}
& \langle\mathrm{p} \uparrow|Q| \mathrm{p} \uparrow\rangle=\langle\mathrm{p} \downarrow|Q| \mathrm{p} \downarrow\rangle=1 \\
& \langle\mathrm{n} \uparrow|Q| \mathrm{n} \uparrow\rangle=\langle\mathrm{n} \downarrow|Q| \mathrm{n} \downarrow\rangle=0 .
\end{aligned}
$$

Robert Zaballa
Robert Zaballa
Numerade Educator
01:23

Problem 14

Express the $\pi^*$-wavefunction in terms of the spin, flavor. and color of the component quarks.

Carlos Henrique De Lima
Carlos Henrique De Lima
Numerade Educator
02:21

Problem 15

Convince yourself that the photon is a $U$-spin scalar: that is, $U=0$. By inspection of Fig. 2.8, show that if $S U(3)$ flavor symmetry were exact, the electromagnetic decay $\Sigma^*(1385)^{-} \rightarrow \Sigma^{-} \gamma$ is forbidden, whereas $\Sigma^*(1385)^{+} \rightarrow \Sigma^{+} \gamma$ is allowed.

Ashwin Banarsee
Ashwin Banarsee
Numerade Educator
03:21

Problem 16

The $L=1$ baryons are most easily identified as resonances in $\pi \mathrm{N}$ elastic scattering. Show that the relative orbital angular momentum, $L^{\prime}$, between the $\pi$ and the $\mathrm{N}$ is a good quantum number, and that it is even for resonances of negative parity. Use the quark model to list the isospin, spin, and $L^{\prime}$ of the $\pi \mathrm{N}$ states expected in the first excited level. Identify these resonances in the particle data tables.

Linda Winkler
Linda Winkler
Numerade Educator
01:43

Problem 17

Determine the magnetic moments of the other members of the $J^P=\frac{1}{2}^{+}$baryon octet in terms of $\mu_P$ and compare with the measured values.

Carlos Henrique De Lima
Carlos Henrique De Lima
Numerade Educator

Problem 18

The spin-flavor wavefunctions of the ground-state baryons are symmetric, and color was invoked to recover the required antisymmetric character. You should notice, however, and some people did, that we can construct a totally antisymmetric proton wavefunction, for example,
$$
|\mathrm{p} \uparrow\rangle=\sqrt{\frac{1}{2}}\left[\mathrm{p}_A \chi\left(M_S\right)-\mathrm{p}_S \chi\left(M_A\right)\right]
$$
and forget about color! Write this function in an explicit form, comparable to (2.71). Obtain $|\mathrm{n} \uparrow\rangle$, and hence show that
$$
\frac{\mu_n}{\mu_p}=-2 .
$$

So this option is ruled out by experiment. In fact, glancing at your derivation, you will notice that $\mu_p$ is negative. It is measured to be positive. Long live color.

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02:31

Problem 19

Prove that the quark model relations for the magnetic moments of the $\rho^{ \pm}$mesons are
$$
\mu_{p^{+}}=-\mu_p=\mu_p
$$

Zhaojie Xu
Zhaojie Xu
Numerade Educator
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Problem 20

Use the quark model to calculate the amplitude for the radiative decay $\omega \rightarrow \pi^0 \gamma$. The $\omega$ and $\pi^0$ belong to the $J^P=1^{-}, S=1$ and the $J^P=0^{-}, S=0$ nonets, respectively. We therefore require a quark spin flip (magnetic dipole) transition. This will involve the quark magnetic moment operator.

First, assume (2.54) and obtain the spin-flavor wavefunctions for an $\omega$ with $M_J=1$ and for a $\pi^0$. If the $z$ axis is chosen as in Fig. 2.10, show that the required amplitude is
$$
\begin{aligned}
\sum_{i=1,2}\left\langle\pi^0\left|\mu_i \sigma_i \cdot \varepsilon_R^*\right| \omega\left(M_J=1\right)\right\rangle & =-\sqrt{2} \sum_{i=1,2}\left\langle\pi^0\left|\mu_i\left(\sigma_{-}\right),\right| \omega\left(M_J=1\right)\right\rangle \\
& =\mu_d-\mu_u
\end{aligned}
$$
where $\varepsilon_R \equiv-\sqrt{\frac{1}{2}}(1, i, 0)$ is the polarization vector of the emitted (helicityone) photon and $\sigma_{-} \equiv \frac{1}{2}\left(\sigma_1-i \sigma_2\right)$ is the operator which "steps down" or "flips" the quark spin.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
01:54

Problem 21

Assuming (2.54), show that the quark model forbids the decay $\phi \rightarrow \pi^0 \gamma$, and predicts that
$$
\frac{\operatorname{Rate}\left(\omega \rightarrow \pi^0 \gamma\right)}{\operatorname{Rate}\left(\rho \rightarrow \pi^0 \gamma\right)}=\left(\frac{\mu_d-\mu_u}{\mu_d+\mu_u}\right)^2=9
$$

Carlos Henrique De Lima
Carlos Henrique De Lima
Numerade Educator

Problem 22

According to weak interaction theory, the dominant hadronic weak decay proceeds via the quark transmutations $\mathrm{c} \rightarrow \mathrm{s}$ and/or $\mathrm{u} \leftrightarrow \mathrm{d}$ (see Chapter 12). For example, an allowed charmed meson decay is $\mathrm{cu} \rightarrow \mathrm{s} \overline{\mathrm{d}}(\mathrm{u} \overline{\mathrm{u}})$.
Assuming that these, and only these, transmutations can occur, show that
$$
\mathrm{D}^0 \rightarrow \mathrm{K}^{-} \pi^{+} \quad \text { and } \quad \mathrm{K}^{-} \pi^{+} \pi^{+} \pi^{-}
$$
are allowed decay modes, but that
$$
\mathrm{D}^0 \rightarrow \pi^{+} \pi^{-}, \mathrm{K}^{+} \mathrm{K}^{-}, \mathrm{K}^{+} \pi^{-}, \quad \text { and } \mathrm{K}^{+} \pi^{-} \pi^{+} \pi^{-}
$$
are all forbidden, Further, show that $\mathrm{D}^{+} \rightarrow \mathrm{K}^{-} \pi^{+} \pi^{+}$is an allowed weak decay, but that $\mathrm{D}^{+} \rightarrow \mathrm{K}^{+} \pi^{+} \pi^{-}$is forbidden. This distinctive feature of $\mathrm{D}^{+}$ decays was in fact convincing evidence in the first ever observation of a charmed particle in 1976 , some 18 months after the revolutionary discovery of the "hidden" charm state $\psi(\mathrm{c} \tilde{\mathrm{C}})$.

Each meson multiplet contains a state. cē. of "hidden" charm. For the $J^P=0$ and $1^{-}$multiplets, it is $\eta_c(2.98)$ and the original $\psi(3.1)$, respectively. The states of the bound cẽ system can be compared with those of positronium $\mathrm{e}^{+} \mathrm{e}^{-}$. We speak of "charmonium." It is a particularly clean system to study and has revolutionized meson spectroscopy. States with $J^{P C}=1^{--}$can be directly produced $\left(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow\right.$ virtual $\left.\gamma \rightarrow \mathrm{c} \overline{\mathrm{c}}\right)$; and, via their decays, other charmonium states can be identified. The observed states are shown in Fig. 2.13, labeled in the conventional spectroscopic manner ${ }^{2 S+1} L_J$, where $S, L$, and $J$ are, respectively, the total intrinsic spin, orbital angular momentum, and total angular momentum of the ce system. This is, of course, a nonrelativistic classification; it is the heavy mass of the $\mathrm{c}$ quark which makes it possible to use a nonrelativistic picture. We also show the $J^{P C}$ values of the states and note that the observations coincide with quark model expectations. The six $J^{P C}$ values listed in Table 2.2 are reproduced, except that the $1^{+-}\left(\right.$or $\left.{ }^1 P_1\right)$ state still awaits discovery. As in positronium, radial as well as orbital excitations are expected. In fact, the $2{ }^3 S$ and $3{ }^3 S$ excitations are seen directly as resonances in the cross section for $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow$ hadrons (see Fig. 2.13).

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02:00

Problem 23

The decay
$$
\psi^{\prime}(3.7) \rightarrow \psi(3.1)+\text { hadrons }
$$
is observed. What are the hadrons?

Salamat Ali
Salamat Ali
Numerade Educator
03:48

Problem 24

Mark on Fig. 2.13 the expected radiative transitions between the levels, indicating which are electric and which are magnetic dipole transitions.

Justify that the rates for the radiative transitions from $\psi^{\prime}(3.7)$ to the three $\chi$-levels, ${ }^3 P_J$, with $J=2,1,0$, are proportional to $(2 J+1) k^3$, where $k$ is the momentum of the emitted photon. Hence, show that the branching ratios of these decay modes of $\psi^{\prime}$ are approximately equal.

Suzanne W.
Suzanne W.
Numerade Educator

Problem 25

The leptonic decay of neutral vector $\left(J^{P C}=1^{--}\right)$mesons can be pictured as proceeding via a virtual photon,
$$
\mathrm{V}(\mathrm{q} \overline{\mathrm{q}}) \rightarrow \gamma \rightarrow \mathrm{e}^{+} \mathrm{e}^{-} .
$$

The technique for calculating such amplitudes will be explained in succeeding chapters. Here, it suffices to note that the $\mathrm{V}-\gamma$ coupling is proportional to the charge of the quark q. Neglecting a possible dependence on the vector meson mass, show that the leptonic decay widths are in the ratios
$$
\rho: \omega: \phi: \psi=9: 1: 2: 8 \text {. }
$$

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01:29

Problem 26

Comment on the rate you would expect for the $\mathrm{e}^{+} \mathrm{e}^{-}$ decay mode of the ${ }^3 D_1$ state as compared to the $\psi^{\prime}(3.7)$ state. Can these two states mix?

Nick Johnson
Nick Johnson
Numerade Educator

Problem 27

The hadronic decay widths of $\eta_c$ and $\psi(3.1)$ are estimated using
$$
\begin{aligned}
& \eta_c(\mathrm{c} \overline{\mathrm{c}}) \rightarrow n \mathrm{~g} \rightarrow \text { hadrons } \\
& \psi(\mathrm{c} \overline{\mathrm{c}}) \rightarrow n^{\prime} \mathrm{g} \rightarrow \text { hadrons, }
\end{aligned}
$$
where $\mathrm{g}$ is a gluon and $n$ and $n^{\prime}$ are integers. These are QCD analogues of the QED process of (2.80). Show that the minimum values of $n$ and $n^{\prime}$ are 2 and 3 , respectively.

Properties of the potential between the $\mathrm{c}$ and $\overline{\mathrm{c}}$ can be inferred from the charmonium spectrum. In Chapter 1 , we noted that at small $\mathrm{c}$ and $\overline{\mathrm{c}}$ separations, QCD predicts a Coulomb-type potential $-\alpha_s / r$, but that at large separation $r$, we expect a confining potential which increases with $r$. A glance at the $1 S, 2 S$, and the "center of gravity" of the $P$ levels of Fig. 2.13 shows that the potential is in fact somewhere between Coulomb (which has $2 S$ and $P$ degenerate) and an oscillator potential $V \sim r^2$ (which has the $P$ level halfway between $1 S$ and $2 S$ ). A naive potential, which is phenomenologically rather satisfactory, is
$$
V(r)=-\frac{4}{3} \frac{\alpha_s}{r}+a r
$$
where $a$ is a constant parameter and $\frac{4}{3}$ is the color factor associated with the quark-gluon coupling $\alpha_s$ [see (2.98)].

Let us now repeat the steps of constructing baryons, but this time include the $\mathrm{c}$ quark. Combining three basic quark multiplets, we find that the analogue of $(2.59)$ is
$$
4 \otimes 4 \otimes 4=20 \oplus 20 \oplus 20 \oplus \overline{4} .
$$
$$
s \quad M_s \quad M_A \quad \text { A }
$$

Rather than to derive this decomposition, it is better at this stage to use the elegant techniques of group theory (Young tableaux); see, for example, Close (1979). Including spin, (2.64), we can as before form the required symmetric spin-flavor ground state in two ways: either the symmetric 20 with a symmetric spin $\frac{3}{2}$ or a mixed-symmetry 20 with spin $\frac{1}{2}$ constructed in exact analogy to (2.68). Extracting the flavor multiplets from a superposition of three basic (quark) tetrahedra leads to the ground-state baryons of Fig. 2.14a.

The spin- $\frac{1}{2}$ multiplet can be viewed as three $S U(3)$ octets propping each other up and based on the edges of a fourth $S U(3)$ octet. In fact, we do not need the elegance of group theory to enumerate the states. For example, the $C=1 \mathrm{spin}-\frac{1}{2}$ baryons are cqq composites with $\mathrm{q}=\mathrm{u}, \mathrm{d}$, or $\mathrm{s}$. The qq decomposition is given in (2.58), namely,
$$
3 \otimes 3=6 \oplus \overline{3},
$$
and the states are shown in Fig. 2.14b. The lightest charmed baryons are the $\Sigma_c$ isospin triplet and $\Lambda_c^{+}$. The observed masses are
$$
m\left(\Lambda_c\right)=2.28 \mathrm{GeV}, \quad m\left(\Sigma_c\right)=2.44 \mathrm{GeV} .
$$

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13:51

Problem 28

Determine the flavor wavefunctions of the $\Lambda_{\mathrm{c}}$ and $\Sigma_{\mathrm{c}}$ baryons. Give an observable decay sequence of $\Sigma_{\mathrm{c}}^{++}$.

The $\mathrm{c}$ quark was desired theoretically. The same cannot be said of the b quark. Evidence for this fifth quark came in a replay of the charmonium phenomenon in the $\mathrm{e}^{+} \mathrm{e}^{-}$energy region around $10 \mathrm{GeV}$. Four $\mathrm{e}^{+} \mathrm{e}^{-}$resonances were quickly identified: $\Upsilon(1 S), \Upsilon(2 S), \Upsilon(3 S)$, and $\Upsilon(4 S)$ with masses of 9.46, 10.02, 10.35, and $10.57 \mathrm{GeV}$, respectively. The first three states are narrow and the fourth is much wider. The lightest meson (bū or b $\bar{d}$ ) with explicit beauty is therefore expected to have mass $m\left(\mathrm{D}_{\mathrm{b}}\right) \approx 10.4 / 2=5.2 \mathrm{GeV}$ [cf. (2.77)].

Robert Zaballa
Robert Zaballa
Numerade Educator
08:10

Problem 29

Verify that the spin 1 level $\left({ }^3 S_1\right)$ is higher than the spin 0 level $\left({ }^1 S_0\right)$.

The QED result, (2.88), can be taken over directly to $\mathrm{QCD}$, provided we replace the electromagnetic coupling $e_1 e_2$ by the product of color charges. For mesons and baryons, the substitutions are
where $\frac{4}{3}$ and $\frac{2}{3}$ are the appropriate color factors. We show how to compute these factors in a moment.

We can now make a model for the ground-state hadron masses. We assume (1) that quark confinement, which is operative at large separations, is independent of the spins and of the masses of the quarks; (2) that at near-separation, $\alpha_s$ is small enough for QCD hyperfine splitting to be relevant; and (3) that the only symmetry breaking arises from the different constituent masses assigned to the quarks of different flavors. In this scheme, the meson and baryon masses are therefore
$$
\begin{aligned}
m\left(\mathrm{q}_1 \overline{\mathrm{q}}_2\right) & =m_1+m_2+\left[a\left(\sigma_1 \cdot \boldsymbol{\sigma}_2\right) / m_1 m_2\right] \\
m\left(\mathrm{q}_1 \mathrm{q}_2 \mathrm{q}_3\right) & =m_1+m_2+m_3+\left[\frac{a^{\prime}}{2} \sum_{i>j}\left(\boldsymbol{\sigma}_i \cdot \boldsymbol{\sigma}_j\right) / m_j m_j\right]
\end{aligned}
$$
where $a$ and $a^{\prime}$ are positive constants [see (2.88)-(2.90)].

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
08:10

Problem 30

For the $\pi$ (spin 0) and the $\mathrm{K}^*$ (spin 1), show that (2.91) gives
$$
\begin{aligned}
m(\pi) & =m_u+m_d-\left(3 a / m_u m_d\right) \\
m\left(\mathrm{~K}^*\right) & =m_u+m_s+\left(a / m_u m_s\right) .
\end{aligned}
$$
Calculate the masses of all the members of the $0^{-}$and $1^{-}$meson multiplets (Fig. 2.12) using
$$
m_u=m_d=0.31, \quad m_s=0.48, \quad m_c=1.65, \quad a / m_u^2=0.16,
$$
all in units of GeV. Compare your predictions with the meson masses listed in the particle data tables.
Check that
$$
(\rho-\pi)=\frac{m_s}{m_u}\left(\mathrm{~K}^*-\mathrm{K}\right)=\frac{m_c}{m_u}\left(\mathrm{D}^*-\mathrm{D}\right)=\frac{m_c m_s}{m_u^2}\left(\mathrm{~F}^*-\mathrm{F}\right),
$$
where the meson names are used to denote their masses.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
04:56

Problem 31

Show that the model gives the $\Delta$ heavier than the nucleon. Further, show that if $a=a^{\prime}$ in (2.91) and (2.92), then
$$
m(\Delta)-m(\mathrm{~N})=\frac{1}{2}[m(\rho)-m(\pi)] .
$$

Keshav Singh
Keshav Singh
Numerade Educator
01:09

Problem 32

Use (2.92) to study the relative masses of the $\Lambda, \Sigma, \Sigma *$, $\Lambda_c, \Sigma_c$, and $\Sigma_c^*$ baryons of Figs. 2.8 and 2.14. Each baryon is a qqQ composite, where $\mathrm{q}=\mathrm{u}$ or $\mathrm{d}$ and $\mathrm{Q}=\mathrm{s}$ or $\mathrm{c}$. For the $\Lambda$ and $\Sigma$ baryons, show that the qq have isospin $I=0$ and $I=1$, respectively, and hence spin 0 and spin 1 , respectively. Use this result to evaluate $\mathbf{S}_1 \cdot \mathbf{S}_2$, where $\mathbf{S}_i=\frac{1}{2} \boldsymbol{\sigma}_i$. By considering $\left(\mathbf{S}_1+\mathbf{S}_2+\mathbf{S}_3\right)^2$, show that
$$
\left(\mathbf{S}_1+\mathbf{S}_2\right) \cdot \mathbf{S}_3=0,-1 \text {, and }+\frac{1}{2} \quad \text { for } \Lambda, \Sigma \text {, and } \Sigma^* \text {, respectively. }
$$

Thus, confirm that (2.92) gives
$$
\begin{aligned}
& m\left(\Lambda_{\mathrm{Q}}\right)=m_0-\frac{3 a^{\prime}}{2 m_u^2} \\
& m\left(\Sigma_{\mathrm{Q}}\right)=m_0+\frac{2 a^{\prime}}{m_u^2}\left(\frac{1}{4}-\frac{m_u}{m_Q}\right) \\
& m\left(\Sigma_{\mathrm{Q}}^*\right)=m_0+\frac{a^{\prime}}{m_u^2}\left(\frac{1}{2}+\frac{m_u}{m_Q}\right)
\end{aligned}
$$
where $m_0=2 m_u+m_Q$. Verify that $[\sec (2.84)]$
$$
\left[m\left(\Sigma_c\right)-m\left(\Lambda_c\right)\right]=\frac{m_s}{m_c} \frac{\left(m_{c_c}-m_u\right)}{\left(m_s-m_u\right)}[m(\Sigma)-m(\Lambda)] \simeq 0.16 \mathrm{GeV},
$$
The masses of the other $\frac{1}{2}^{+}$and $\frac{3}{2}^{+}$baryons can also be calculated from (2.92) in terms of $a^{\prime}$ and the quark masses.

Considering the crude nature of the model, the quantitative agreement between the predictions and the observed masses is impressive. Indeed, all the observed features are reproduced. It is straightforward to enlarge the calculation to include hadrons containing $b$ quarks.

Keshav Singh
Keshav Singh
Numerade Educator