The strong and weak force

by Roman Zwicky – University of Edinburgh

These notes follow the text in the MOOC-video, largely, word for word. A few additional explanationshave been added (often as footnotes). Table of content:

1. Prologue – the two (missing) microscopic forces

2. Strong Interactions ⌘ quantum chromodynamics (QCD)

2.1. Quark flavour as the key to many bound states2.2. Confinement and asymptotic freedom2.3. Lines of forces of QED and QCD2.4 Jets (= energetic colour neutral packets of quarks and gluons)

Figure: The four fundamental forces of nature. The opening point of this lecture is summarised in a figure. Thepotential V (F

orce

= � @V@r

) between two particles is responsible for whether or not the force acts on a macroscopicscale.

1. Prologue – the two (missing) microscopic forces

As a matter of fact there are four fundamental forces in nature, of which we have seen two sofar: Gravitation, corresponding to attraction of masses (as seen in part 2), and electromagnetism

corresponding to attraction of charges (as seen in part 3). These are well-known in our macroscopicworld: for example through the orbiting of planets and the propagation of light respectively. In theadvent of the age of particle physics it became clear that there ought to be more forces.

(1) The nucleus made out of protons (p) and neutrons (n) is held together very strongly.Gravitation is too weak and electromagnetism is repelling as protons carry positive charge.Hence there has to be a strong force, binding the nucleus together, which operates on themicroscopic or nuclear scale.

(2) �-decay: n ! p+ ... The neutron decays into a proton plus other particles with very smallprobability. It therefore seemed reasonable to assume that this decay is governed by a weakforce.

In the following lectures we are going to discuss the nature of the strong and the weak force, what theforce carriers are, (analogous to the photon in electromagnetism), and why they are not experiencedin the macroscopic world.Concerning the latter, I shall remind you that the potential (denoted by V ) in electromagnetism andgravitation is proportional to one over the distance r :

V ⇠1

r, gravitation & electromagnetism ,

⇣Force = �

@V

@r⇠

1

r2

⌘(1)

This is due to the force carriers, the photon (�) and the graviton, having no rest mass. By virtue ofEinstein’s famous formula E = mc2 = 0 no virtual energy needs to be created and the force carrierscan propagate freely.

We will see that the potential of the strong force is proportional to the distance and therefore radicallydi↵erent:

Vstrong

⇠ r , Vweak

⇠e�M

weak

r

r. (2)

The potential of the weak force is exponentially suppressed by a mass associated with the weak forcecarriers.a The Higgs mechanism has got everything to do with this mass.The way the theory of strong interaction was uncovered is rather subtle. Information came from aseemingly never ending series of discoveries of new particles.b Quotes that summarise the spirit of thetime are: “One should give a prize to the physicist who does not discover a new particle” (anonymous)and “Had I foreseen that, I would have gone into botany,” (Wolfgang Pauli).Eventually the idea that the discovered particles are bound states

c, made of more elementary con-stituents called quarks, emerged. We will see that the proliferation of bound states can be understood,as coming from di↵erent arrangements of these quarks.For the sake of clarity we shall from now on mostly explain the facts, since the full story of thediscovery of the strong interactions is rather intricate.

aThe weak force carriers are the so-called W - and Z -bosons to be discussed in the next lecture.bIt led people into all kind of models and ideas. One of the most extreme approaches, so-called S-matrix

approach, gave up the search for a microscopic theory trying to fix predictions from experimental data and generalprinciples such as conservation of probability alone.

cElementary states bound together by forces are called bound states. A famous example is the hydrogen atomwhere the electron and proton are bound together by the electromagnetic force. Another example is the nucleonitself where a certain number of neutrons and protons are bound together by the, to be discussed, strong force.

2. Strong Interactions ⌘ quantum chromodynamics (QCD)

Quantum electrodynamics, or QED for short, can be understood as the theory of interactions betweenelectrically charged objects such as the electron and the electromagnetic force carrier the photon. Inanalogy the strong force can be understood as the interaction of particles called quarks carrying colourcharge and the strong force carrier the gluon.a The quarks are matter particles of spin 1/2b and thegluons are the force carriers of spin 1. The most important di↵erence to QED is that the gluon alsocarries colour charge and therefore can interact with itself, which I have drawn in a set of so-calledFeynman-graphs.

Figure: QED-interactions vs QCD-interactions. Novel element: self-interactions of gluons. The latter is responsiblefor asymptotic freedom which is discussed later on.

This is due to the colour charge being more elaborate. More precisely the colour charge can have threedi↵erent qualities which are called red, green and blue. The analogue of ± for the electric chargemay be thought of as red and anti-red. All these “colourful words” led the particle physicist to namethe theory of strong interactions: quantum chromodynamics, or QCD for short. The terms stronginteraction and QCD shall be used interchangeably from now on.

aThe name gluon alludes to the fact that the strong force carrier binds (glues) the quarks into bound states.bRecall the spin and the mass, modulo their interactions, characterise a particle.

This is the QCD kinetic minus potential term,

LQCD

=X

q(uarks)

q̄�µDµq

| {z }analogue of Dirac equation

�1

4Gµ⌫G

µ⌫

| {z }analogue Maxwell’s equations

(3)

in so-called Lagrangian forma, which I merely write down because it fits on one line. The first termcorresponds to the analogue of the Dirac equation for QED (as seen in part 3). The second termgives rise to the generalisation of Maxwell’s equations in vacuum (as seen in part 2). You are notsupposed to be familiar with this equation for the remainder of the course. It is remarkable thoughthat one, if not the, most complicated theory in nature, can be written on one line and yet is veryhard to solve as such. For completeness we mention that the quarks also carry electric charges.

aFrom a Lagrangian one can derive the equations of motions. I.e. the equation whose solutions describe thetheory.

2.1. Quark flavour as the key to many bound states

In QED we have seen that there are electrons (e), muons (µ) and taus (⌧), where the muon and tauscan be thought of as heavy electrons. Similarly the quarks come in di↵erent types called flavours.Actually there are six flavours: up (u), down (d), strange (s), charm (c), beauty (b) and top (t) inincreasing weight.a The last quark, the top, was only directly confirmed in 1995 at Fermilab nearChicago.

Figure: Chronological timeline of discovery of the quarks.

It is time to return to the beginning of this lecture and fit the protons and the neutrons back into thenew picture: They correspond to (uud) and (udd) bound states. The up quark and down quark carryQu = 2/3 and Qd = �1/3 units of electric charge respectively. Therefore the proton made out of(uud) caries 2/3 + 2/3 � 1/3 = 1 unit of electric charge and the neutron made out of (udd) caries2/3 � 1/3 � 1/3 = 0 units of electric charge.

Qp(roton) = 2Qu + Qd = 2 (2/3) + (�1/3) = 1 (unit electric charge) ,

Qn(eutron)= Qu + 2Qd = (2/3) + 2 (�1/3) = 0 (unit electric charge) .

aThe names of the quarks reflect to some extent the attitude of the community towards them at various stages.

The quarks are bound together into the proton by continuously exchanging gluons. Let us add someterminology: Bound states of so-called quarks are called hadrons. This explains why the CERNexperiment is called large hadron collider as two hadrons (namely protons) are collided at high energy.

Figure: Bound states of quarks (hadrons). From left to right: proton, neutron and ⇤-particle.

The large range of particles discovered by the physicists in the 1950s can therefore be understood asvariations thereof interchanging the light flavours of up, down and strange quarks. For example thebound state made out of (uds) is known as the ⇤-particle and weighs approximately 1.2 times theproton mass. This scheme can be thought of as the analogue of the periodic table of particle physics.This is a nice picture but there are still paradoxical aspects. QED as well as QCD are gauge theoriesa

and, with the caveat of the yet to be discussed Higgs-mechanism, imply that the force carriers haveno mass and therefore should act at the macroscopic scale. Here is the paradox: How come the strongforce only operates on the microscopic scale? This is related to our next topic of confinement andasymptotic freedom.

aGauge theories are specific quantum field theories. As far as we know all particle physics is described at thefundamental level by gauge theories. Why this is the case is certainly one of the biggest mysteries of physics.

2.4 Confinement and asymptotic freedom

Possibly the most puzzling aspect of QCD is that mathematically it is defined in terms of quarks andgluons, but neither have ever appeared in a particle detector (in isolated form). That this can neverhappen, is known as the confinement-hypothesis.Whereas a hydrogen atom can be split into its two constituents the electron and the proton, thesame is not possible for the hadrons! This was the real puzzle. It is therefore not surprising thatit was largely thought, that quarks had no reality, but were merely convenient mathematical objects(bringing some order into the world of hadrons). In 1969 experiments in deep inelastic scattering gavethe community new results. In those experiments energetic photons probed the proton. More generally,particles of high energies can probe the short distance structure of other (composite) particles. Thelatest example of this principle is the Large Hadron Collider at CERN in Geneva, which collides highlyenergetic protons, probing the unknown subatomic structure of 10�18m.

Figure: Deep inelastic scattering. An energetic electron (e) emits a photon (�) that probes the proton (p) structure .From these experiments particle physicists gained empirical knowledge about the nature of the quarks. E.g. that thespin is equal to 1/2 and that the quarks act as free particles within the proton. The latter finds its explanation in thephenomenon of asymptotic freedom a few years later (to be discussed).

The best interpretation was that the quarks could be thought of as free particles within the proton.a

By free we mean that no force acts on them.

aThis is known as the parton model for which James Bjorken and Richard Feynman are credited for.

Let us summarise the situation: Experiment suggests that the quarks interact weakly at short distances(known as asymptotic freedom) and very strongly at large distances (confinement), binding colouredobjects such as quarks and gluons into bound states. The challenge for particle physicists was to showthat these two properties are contained with the QCD equations (derivable from the Lagrangian (3)).This is easily said but totally counterintuitive as in QED it works just the other way around. Moreprecisely charges are known to be screened in QED. Imagine a charge placed into the vacuum. If avirtual pair of opposite charges is created from the vacuum then the initial charge will attract the oneof opposite charge. This is analogous to a polarised medium. The net e↵ect is that from a distanceit will look like there is less charge. To see this we place a charge into such a configuration and wesee that the virtual charges counteract the e↵ect of the initial charge (c.f. figure). Therefore theelectromagnetic force gets stronger at short distances due to quantum e↵ects. The idea of asymptoticfreedom in QCD is to turn this picture upside down!

Figure: Charge screening in QED. Left: classical picture where the test charge (red) is exposed to the potential forceof the original charge (black). The potential is V ⇠ 1/r where r is the distance. Right: quantum picture wherevirtual pairs of electrons and positrons (=anti-electron) can exist for a short time by virtue of Heisenberg’s uncertaintyprinciple. The virtual charges (green) place themselves such as to counteract the e↵ect of the original charge (black).The test charge (red) is repelled less strongly in the quantum picture than in the classical picture. We therefore saythat the charge is screened in QED. (In fact the e↵ect is such as to modify the potential by a logarithmic term:Vclassical ⇠ 1/r ! Vquantum ⇠ ln(r)/r .)

In 1973 it was indeed shown mathematically, using the QCD equations (3), that the force doesbecome smaller at short distances. The discovery of asymptotic freedom in QCD led to the Nobelprize for David Politzer, David Gross and Frank Wilczek in 2004. Possibly we should add that theword asymptotic refers to small distances. The new key feature, as with respect to QED, is theself interacting nature of the gluons, which turns the screening picture, described for QED, upsidedown. The fact that the force is small at short distances complies with the force being strong at largedistances. Using the QCD equations it has been shown through elaborate computer simulations, asperformed at Edinburgh University, that the force is indeed strong at large distances and leads to theconfinement of the quarks inside the hadrons. An ab initio (from first principles) analytical approachto confinement is one of the great outstanding problems of particle physics.

Figure: E↵ective charge as a function of the distance r in QED (left) and QCD (right). More precisely Politzer, Grossand Wilczek determined that the slope (which is known as the �-function in quantum field theory) of the e↵ectivecharge is negative (asymptotic freedom) in QCD. I shall add that in the literature there were statements that this couldnot be realised within a quantum field theory.

2.3 Lines of forces of QED and QCD

At the qualitative level the lines of forces for two QED and QCD charges look as follows

Figure: Flux lines of QED (left) as compared with QCD (right). The former spread into the three dimensional spacewhereas the latter are squeezed into one dimension. This is connected with the phenomenon of confinement.

The photon flux lines spread out into the entire space and therefore weaken with distance. The gluonflux lines are squeezed into one dimension and do not weaken.a

The force is therefore independent of the distance. This implies that the potential scales linearly withthe distance as announced at the beginning of the lecture c.f. Eq. (2). Let us summarise the pictureof confinement: One cannot isolate coloured particles! Only colour neutral particles can propagateover long distances. This is the explanation of why the strong interactions are of short range:

The force carrier of strong interactions, the gluon has colour charge andcan therefore not act at a long range; despite having no rest mass!

aThe electromagnetic potential in d dimension is V ⇠ r2�d . For QED d = 3 gives rise to V ⇠ 1/r . For QCDthe dimension e↵ectively reduces to d = 1 (as depicted in the figure) and thus V ⇠ r c.f. Eq. (2).

2.4 Jets (= energetic colour neutral packets of quarks and gluons)

Let us mention, as it is of importance for the Higgs discovery plots, that when a quark pair is pulledapart to a certain distance, then it is energetically favourable, to break up into two colour neutralquark pairs. This leads to the formation of so called jets of colour neutral particles in high energyexperiments.

Figure: When a pair of quarks is pulled apart strongly it becomes energetically favourable to break o↵ into two pairsof quarks (known as string breaking). This is connected with the formation of jets.

The puzzle of why the weak force is not of long range nature has a very di↵erent solution and has goteverything to do with the Higgs mechanism! To be discussed in forthcoming lectures.

End of the lecture.

On the next page you find an overview of the elementary particles of the Standard Model(thus all elementary particles that we know of)

Elementary particle chart

Figure: Chart of elementary particles. (left) matter particles which are all spin 1/2 and therefore fermions. (right)h (higgs) boson of spin 0. Three force carriers of spin 1: g (gluon = strong force) � (photon = electromagnetism)

W±, Z (W -,Z -bosons = weak force). The fourth force carrier the graviton is of spin 2 and it is apart, as a consistentand testable theory of quantum gravity is one of the outstanding problems of physics. The underlined colours indicatewith which force carrier the particles interact. The graviton interacts with all of them. On the left of the charts (withbrown) we have indicated the unit of electric charge of the particles. A noticeable feature is that the matter particlescome in three families, which is only partially explained and motivates the search for physics beyond the StandardModel.