Quarks

Section 9.7 Quarks

Nice as these patterns of the Eightfold Way are, many questions remain. For instance, why are there two particles at the center of the baryon hexagon of Figure 9.4, and three in the center of the meson pattern in [cross-reference to target(s) "fig_meson_SvsQ" missing or not unique]? Why don't other patterns occur?

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The fact that the elementary particles fall into neat patterns is reminiscent of the construction of the periodic table of elements. There it was found that when elements were arranged in (approximate) order of increasing mass number, certain groups of elements (the columns of the table) emerged with similar chemical properties. Once the periodic table was well established, the jump was quickly made to understanding atomic structure, with electrons in shells, subshells, and orbitals.

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This idea that regular patterns of particle properties could indicate a regular underlying structure occurred to Gell-Mann, and in 1964 he proposed the quark model of elementary particle structure. Gell-Mann's quark model states that all hadrons are composed of a small number of constituent particles called quarks. The quarks of Gell-Mann's model come in three different varieties (or flavors) called up, down, and strange. The quarks' properties are summarized in Table 9.5. The vast variety of hadrons comes about as different combinations and configurations of the quarks, just as the hundreds of different isotopes of atoms arise from different combinations of protons, neutrons, and electrons.

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The mass of a hadron is not simply the sum of the masses of its constituent quarks — the energies of the bound quarks can affect the mass of the hadron.¹ Most of our discussion will focus on quarks combined together in ground state configurations. But it is possible for the same set of quarks to be configured into higher energy states (i.e., higher mass particles), which usually decay quickly. (Since Gell-Mann's original model, three more flavors of quarks have been identified, but we will not study them in detail.)

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Let's see how different combinations of quarks can be assembled to make various particles. Consider a neutron, a neutral, non-strange baryon. Whenever we combine quarks to make a particle, the particle's additive properties (

Q,

B,

and

S

) are simply the sum of the constituent quarks' properties. Since quarks have

B = 1 / 3,

we need at least three quarks to make a baryon. let's try to make a neutron from just three. If we use a strange quark, the resulting particle would carry strangeness. So to make a neutron, we should use only ups and downs. But how many of each? We can find out by making the charge come out to zero: the result is one up and two downs. Thus, in symbols,

n = (u d d) .

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Table 9.5. Quark properties

Flavor	Name	Spin	$Q$	$B$	$S$

$u$	up	1/2	$+ 2 / 3$	$1 / 3$	$0$
$d$	down	1/2	$- 1 / 3$	$1 / 3$	$0$
$s$	strange	1/2	$- 1 / 3$	$1 / 3$	$- 1$

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To make a meson is even simpler. Since the baryon number must come out zero, a quark plus an antiquark will do it. Let's make a positive pion, a non-strange meson. Since

S = 0,

we could try

π^{+} = (s \overset{―}{s}),

but then the charge would be

- \frac{1}{3} + \frac{1}{3},

which isn't right. Thus pions are constructed using just ups and downs.

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We can summarize our findings about constructing hadrons out of quarks with the following:

baryons.
contain three quarks ( $q q q$ )
antibaryons.
contain three antiquarks ( $\overset{―}{q} \overset{―}{q} \overset{―}{q}$ )
mesons.
contain a quark-antiquark ( $q \overset{―}{q}$ )

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The quark model is especially convincing because it explains so well the Eightfold Way patterns of particles. Consider again the spin zero mesons. [cross-reference to target(s) "fig_meson_SvsQ" missing or not unique] showed a hexagonal pattern with three particles in the center. When we make a similar plot of the nine possible quark-antiquark pairs, we get the pattern shown in Figure 9.6. There are nine combinations in exactly the same places on the plot as in [cross-reference to target(s) "fig_meson_SvsQ" missing or not unique]. A similar situation occurs when we make three-quark combinations for baryons. In fact, every possible combination of up, down, and strange quarks has been observed as a particle, and every known hadron can be built of quarks.

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Figure 9.6. The nine quark-antiquark combinations on a strangeness vs. charge plot. Note the striking similarity to Figure \`[cross-reference to target(s) "fig_meson_SvsQ" missing or not unique]`.

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Once one accepts the idea of quarks, the strong interaction conservation laws become almost automatic. Consider the reaction

\begin{matrix} (9.16) & π^{-} + p \to Λ + K^{0} \end{matrix}

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By now you could easily check that angular momentum, charge, baryon number, and strangeness are all conserved. But write the reaction in terms of quarks:

\begin{matrix} (9.17) & (d + \overset{―}{u}) + (u + u + d) \to (d + u + s) + (\overset{―}{s} + d) \end{matrix}

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Then simply note that quark conservation implies conservation of all the additive physical quantities, since each quark retains those properties regardless of configuration. By quark conservation, we mean that a quark can be created or destroyed only if its antiquark is created or destroyed simultaneously, as occurs with a

u \overset{―}{u}

pair and the

s \overset{―}{s}

pair in (9.17). You'll see this quark description of reactions more vividly when you learn about reaction diagrams in Chapter 11.

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Example 9.7.

{Quark content of the $Δ^{+ +} .$ } Construct from quarks a baryon with charge $+ 2,$ the $Δ^{+ +} .$

Solution.

Since a baryon contains three quarks, and only an up quark has a charge of $+ 2 / 3,$ it would require three such quarks to obtain a charge of $+ 2 .$ Thus, the $Δ^{+ +}$ must be made of $(u u u) .$

Don't forget the equivalence of energy and mass from relativity.