The standard model, quarks and their conservation laws

In this article Carl briefly discusses quarks, the conservation laws that involve them and how we can use the Standard Model to check whether certain particle interactions are allowed or prohibited…

Quarks, we believe, are fundamental particles. This means they cannot be broken down any further into other particles; they are a type of the smallest building blocks of everything we see around us. It is thought that all the fundamental particles, except the Higgs Boson, have been observed to date, which includes three generations of quarks, with each generation having two types of quark. These are up ( $u$ ) and down ( $d$ ), charm ( $c$ ) and strange ( $s$ ) and top ( $t$ ) and bottom ( $b$ ) quarks, in ascending generation order. Like many things in physics, quantities must be conserved and quarks obey these rules just like anything else. Today, we explore quarks and their conservation laws.

The Standard Model

The Standard Model (SM) of elementary particles suggests there are six quarks, six leptons and four bosons. Like the quarks mentioned above, there are six leptons in three generations: the electron ( $e$ ) and the electron neutrino ( $\nu_e$ ), the muon ( $\mu$ ) and the muon neutrino ( $\nu_{\mu}$ ) and the tau ( $\tau$ ) and the tau neutrino ( $\nu_{\tau}$ ) – spot any patterns?. The bosons are particles that ‘carry’ the fundamental forces of nature: the photon ( $\gamma$ ) mediates the electromagnetic force, the gluon ( $g$ ) mediates the strong interaction and the $Z$ and $W$ bosons mediate the weak interaction. For the purposes of this article, this is all you really need to know about the SM.

Conservation in Physics

Conservation is an important part of physics. Many quantities are required to be conserved – be the same before and after an event – to make sure everything that was there before is there after and that nothing can just disappear; after all we don’t sit at our desks and observe our lamps disappearing into thin air. You may be familiar with conservation of energy, momentum and charge but the standard model requires more than this to be conserved. A few of the quantities include, but are not limited to: energy, colour, generation, lepton number, baryon number and charge conjugation (the laws of physics should be the same for a particle and its anti-particle). The conservation of energy, momentum and angular momentum arise from the fact that the laws of physics do not depend on the origin of time, spacial position or spacial orientation.

Quarks and Conservation

If you have just joined us at this point in the article, quarks are not an endangered species but rather a fundamental particle. If this comes as a shock, start from the top.

Firstly, we need to introduce baryons. These are heavy particle comprised of three any quarks and the number of baryons in a reaction has to be conserved – if there is one baryon before, there has to be one after as well. Some examples of baryons include the neutron ( $n$ ) which is comprised of two down and one up quark ( $ddu$ ) and the proton ( $p$ ) which is made up of two up and one down quark ( $uud$ ). Other examples include: $\Delta^{-}$ particle ( $ddd$ ) and the $\Lambda_0$ particle ( $dus$ ).

Now, we introduce mesons. These chaps are comprised of one quark and one anti-quark. Whilst mesons do not have to be conserved in any reactions, it’s nice to know a few and what they are made of. Examples include the $K^0$ meson ( $s\overline{d}$ ) and the $\pi^{+}$ meson ( $u\overline{d}$ ).

As mentioned earlier, there are six types, or flavours, of quark. Quark flavours must be conserved in any reaction as well. Quantities called up-ness, down-ness, strangeness, charm, truth and beauty relate to the up, down, strange, charmed, top and bottom quarks respectively. These quantities must be conserved, in the exact same way the baryon and lepton numbers must be. Firstly, strangeness is defined as:

$S \equiv -N_S = -(N(s) - N(\overline{s}))$

where, hopefully you’ll notice the minus sign, $N_s$ is the total number of strange quarks, $N(s)$ is the number of ‘normal’ strange quarks and $N(\overline{s})$ is the number of strange anti-quarks. Similarly, charm is defined by

$C \equiv N_C= (N(c) - N(\overline{c}))$

and the rest follow on in the same form and pattern of positive and negative. Beauty is given by $\widetilde{B} \equiv -N_B = -(N(b) - N(\overline{b}))$ with a line over the B to distinguish it from the baryon number, also B. Truth is $T \equiv N_T = (N(t) - N(\overline{t}))$ . From this, we can form an expression for the baryon number:

$B = \frac{1}{3}(N_U+N_D+N_S+N_C+N_B+N_T)$

where we have added in the total number of up and down quarks as well. This quantity is conserved in all interactions. As was said earlier, total charge $Q$ is also conserved. Knowing that the up, charm and top quarks all have charges of +2/3 and the down, strange and bottom quarks all have a charge of -1/3, we can say that total charge is given by

$Q = \frac{2}{3}(N_U+N_C+N_T) - \frac{1}{3}(N_D+N_S+N_B)$

Using the conservation of these two quantities, we can test to see whether or not the standard model allows or forbids certain reactions. Take beta decay, where a neutron decays into a proton, electron and an electron anti-neutrino.

$n \rightarrow p + e^{-} + \overline{\nu}_{e}$

which can be written in terms of quarks as

$udd \rightarrow uud + e^{-} + \overline{\nu}_{e}$

In this process, both B and Q are conserved even though $N_U$ and $N_D$ are not. This is because there is a weak interaction involved (look at the Feynman diagram for beta decay). As B and Q are conserved, this process is allowed by the standard model (as we would expect because we observe beta decay in nature). Where weak processes are involved, B, Q and the Lepton number are conserved.

Test Your Knowledge

Using your knowledge of the conservation laws discussed above, why not have a go at figuring out which of the following processes are allowed and which are forbidden by the standard model.

$n \rightarrow p + e^{-} + \overline{\nu}_{e}$
$\nu_{\mu} + p \rightarrow \mu^{+} + n$
$n + \nu_{e} \rightarrow p + e^{-}$

The answers are available at the bottom of the article.

Further Your Knowledge

Particle physics is an exciting and ongoing area of physics with experiments creating the conditions immediately after the big bang to try and discover the fundamental building blocks of our universe. The search is on for the Higgs boson as well as testing other theories that could potentially double and number of particles we currently know of. The following topics may be of interest:

Higgs boson
Large Hadron Collider (LHC)
Supersymmetry (SUSY)
Weak interaction

Test Your Knowledge Answers: 1) allowed – L_e = 0 on both sides, B = 1 on both sides, Q = 0 on both sides ; 2) forbidden – L_μ not conserved, B = 1 on each side, Q = 1 on both sides ; 3) allowed – B = 1 on both sides, L_e = 1 on both sides, Q = 0 on both sides