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Chapter 1 Interactions And Field

1.3 The Strong Interaction

Interactions and fields

Particle Physics

The Strong Interaction

Fig 1

The strong interaction is responsible for the powerful attraction between nucleons. This force between protons and neutrons holds the nucleus together. The strong or nuclear interaction is the most powerful of the four basic interactions, with a strength of approximately 100 times the electromagnetic interaction. When two particles are separated by more than about 3 X 10-15 meters, the strong interaction between them is negligible. The explanation for the short range for the strong interaction is given by the Yukawa theory (1935), which attributes the force to an exchange of pions.

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Interactions and fields Particle Physics Feynman diagrams illustrating the exchange of pions by nucleons are shown in Fig 2. In Fig 2(a) a proton (p) emits a virtual neutral pion (π0) and a short time later another proton absorbs the pion.

Fig 2(a)

During the time Δt in which the pion exists, energy is not conserved. If the uncertainty in energy ΔE does not exceed that allowed by the uncertainty principle (ΔE Δt ≥ ħ/2π), the process can occur. The overall process does conserve both energy and momentum. The other two diagrams in Fig 2 represent the exchange of a virtual positively charged pion (π+) between a proton and a neutron (n), and the transfer of a virtual negatively charged pion (π-) from a neutron to a proton.

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Interactions and fields Particle Physics To estimate range of the strong interaction, we use the time-energy uncertainty principle ΔE Δt ≥ ħ, where ΔE is the magnitude to violation of conservation of the energy and Δt the length of time for which the violation may continue. If we take the "borrowed" energy to be approximately the rest-mass of the pion (140 MeV), the virtual pion can exist for a minimum time interval

Dividing ħ (6.6 x 10-22 MeV s-1 ) by ΔE (140 MeV) gives the minimum uncertainty in time Δt = 4.7 x 10-24 s. If we assume for simplicity that the pion travels essentially at the speed of light, in time Δt the pion can move no farther than

Fig 2(b)

c Δt or 1.4 x 10-15 m.

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Interactions and fields Although the carrier of the electromagnetic interaction, the photon, has no rest-mass the carrier of the strong interaction have a definite rest-mass. As a result of this finite rest mass, the range of nuclear forces produced by the strong interaction is very limited.

Particle Physics

Fig 2(c)

Because of the strength of the strong-interaction field (102 times the electromagnetic), each vertex in the Feynman diagrams of Fig 2 involves a factor of about 1 instead of 10-2 found in the electromagnetic interaction. For this reason, much more complicated virtual process are equally likely. Since all the processes are of comparable importance, it is difficult to calculate the interaction of particles such as neutrons and protons. Particles that participate in strong interaction are called hadrons. The hadrons are subdivided into mesons and baryons. Some of the more important "elementary" particles will be discussed in the next chapter.

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Interactions and fields Particle Physics Because no reaction has ever been observed in which one baryon disappeared and another one did not appear (unless they were annihilated by their antiparticle), baryons are given a baryon number B. Mesons, which, like photons, can be created and annihilated in arbitrary numbers, obey no such conservation law and are given a baryon number of zero. The particle baryon number is +1 and the antiparticle baryon number -1. In the antiproton annihilation event p + p → π+ + π- + π0, the total baryon number before the interaction is zero, since B = +1 for the proton (p) and -1 for the anti proton (p). Since the pions have B = 0, baryon number is conserved in this event. The particles that account for nuclear structure, the neutron and the proton and the carriers of the nuclear interaction, the pions, are strongly interacting particles. These particle, which are the decay products of particle collisions at very high energy, are not detected directly but by the products of their own decay.

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Interactions and fields

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They live approximately 10-23s and are strong interacting. Because of its great strength, the strong interaction produces reactions even in this short time interval. Electromagnetic reactions, which are 100 times weaker than the strong reactions, take around 100 times longer, or typically 10-21 s. The short lived particle-or resonances, as they are now called-can be considered as excited states of more fundamental elementary particles. There is precedent description of nuclei.






All nuclei that contain more than one nucleon can exist at a variety of energy levels, comprising a ground state and many excited states. These levels correspond to different degrees of binding energy among the nucleons. The binding energy of nucleons, of course, is the responsibility of the strong interaction. An example of a "resonance" particle decay is (Λ*)0 → K - + p, Where Λ*, with a rest-mass energy of approximately 1520 MeV, is thought to be an excited state of the low lying state of the Λ.

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Interactions and fields

Particle Physics

Quantum Chromodynamics (QCD) QCD is the quantum description of the strong force. QED


quantum theory of the electromagnetic interactions

quantum theory of the strong interactions

mediated by the exchange of virtual photons

mediated by the exchange of gluons

acts on all charged particles

acts on quarks only

couples to electrical charge

couples to colour charge

coupling strength ∝ e ∝ √α

coupling strength ∝ gs ∝ √αs

Table 1

Basic Strong Interaction The hadrons are composite objects consisting of quarks which carry, in addition to electric charge, a ‘strong charge’, or colour, which plays a role in the strong interactions analogues to the role of electric charge in the electromagnetic interactions. We neglect, for the time being, the fact that there are three colours. The field quanta, or gauge bosons, of the strong interactions are called gluons which, like the quarks, carry colour. Dayalbagh Educational Institute


Interactions and fields

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Colour Colour charge is the charge associated with QCD interactions. Three colours: red, blue, green. Like eLectric charge, it is a conserved quantum number. Quarks always have a colour charge: r, g or b Anti-quarks always have an anti-colour charge: r, b or g Leptons and bosons for other forces (γ, W, Z) don’t carry colour charge. Mesons are colour neutral; colour charges are: (r r), (b b) or (g g) Baryons are colour neutral; coLour charges are: (r g b) Anti-baryons (r g b) Formally different colours of quarks are fundamental particles. 1st generation is: e-, υe , ur, ub ug, dr, db dg

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Interactions and fields

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Gluons Gluons are massless, spin-1ħ bosons. They propagate the strong force: exchange momentum between quarks. We draw gluons as curly-wurly lines:

Fig 3

Gluons also carry colour charge.

Fig 4

Colour charged is always conserved. Number of gluons: there are eight different gluons. Symmetry of the strong interaction tell us these are: rb rg bg br gr gb (rr – gg)/√2 (rr + gg — 2bb)/√6

One big difference between QED and QCD QED propagated by photons: photons no electric charge QCD propagated by gluons: gluons have colour charge Fig 5

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Interactions and fields

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Quark a Gluon Interactions Quark-anti-quark scattering

Short distance potential:

Fig 6

describes a meson: e.g. π- = du strong force is responsible for holding meson together. Gluons carry colour charge. They also feel the strong force → gluons can interact with other gluons! 4-gluon vertex g

3-gluon vertex g

Fig 7

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Interactions and fields Particle Physics The fundamental coupling for the strong interactions is then between quarks and gluons figure 8(a) with a strength proportional to √αs, where αs is the strong coupling constant, and a quark-quark interaction proceeds via the exchange of a virtual intermediate gluon. The gluons have zero mass and this would appear to contradict the fact that the strong or colour force is a short-range force. Fig 8(a): The basic strong coupling between coloured quarks and gluons Fig 8(b): A Feynman diagram for quark- quark scattering via the exchange of a virtual intermediate gluon The analogy with the electromagnetic interactions is not as close as would appear: since the photon does not carry electric charge whereas the gluons are coloured. This implies that when two quarks interact, there is a flow of colour between them. In the electromagnetic interactions electric charge is conserved not only in the overall process but at each vertex in a Feynman diagram.

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Interactions and fields Particle Physics Similarly, the colour quantum number must be conserved at each vertex in the strong interaction diagrams. This implies that the gluons are in fact bi-coloured. Let us assume that the incoming quark in figure 8(a) is red. It cannot get rid of its colour simply by emitting a red gluon since conservation of colour would then imply that the outgoing quark is colourless and colourless quarks do not exist. If on the other hand it gets rid of its redness by emitting a red-anti-blue gluon, colour can be conserved if the outgoing quark is blue, as shown in figure 9(a).

Fig 9(a): The basic colour interaction. The incoming red quark qR is absorbed at the vertex and a bi-coloured gluon gRB and a blue quark qB are created colour is conserved at vertex.

We recall that in a Feynman diagram if the direction of a line is reversed the particle must be replaced by its antiparticle. Dayalbagh Educational Institute


Interactions and fields

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The same is true for colour: on reversing a colour line we replace it by its anti-colour and arive at the colour flow diagram (figure 3(b)).

Fig 9 (c): A colour flow diagram for a quark-quark interaction

Fig 9 (b): The flow of colour in (a)

The Colour flow diagram corresponding to the quarkquark interaction of figure 8(b) is shown in figure 9(c), where it can be seen that the net result is that the quarks have exchanged colour. The three colours and three anti-colours give nine different bi-coloured combinations consisting of a colour and an anti-colour, namely

Fig 10

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Interactions and fields Particle Physics The colour wavefunctions of the gluons can in general be linear combinations of these states and the totally symmetric combination R R + G G + B B is a colour singlet, i.e. it is colourless, and therefore plays no part in the strong interactions. Thus in contrast to the electromagnetic case which there is only one type of field quantum, the photon, there are eight coloured gluons. Furthermore since the gluons themselves are coloured they can couple to each other and the Feynman diagrams for colour interactions may contain triple gluon vertices as shown in figure 11.

Fig 11: A quark-quark interaction showing three gluon coupling. Dayalbagh Educational Institute


Interactions and fields

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Colour Confinement Experimentally we do not see free quarks: quarks are confined within hadrons Gluons attract each other: they self interact Gluon-gluon interaction pulls the colour field lines into a narrow tube. Colour Electric field lines

Electric field lines Fig 18

Potential increases linearly with distance: V(r) = kr Infinite energy is required to separate two quarks. COLOUR CONFINEMENT Colour confinement is a direct consequence of gluon selfinteractions Total potential:

Force required to separate quarks:

At large distances F ≈ k ≈ 100 GeV/fm = 160,000 N !!! Fig 19 Dayalbagh Educational Institute


Interactions and fields

Particle Physics

Hadronisation What happens when we try to pull apart two quarks? At LHC production of energetic quarks is common e.g. qq → g → qq. qq produced at same point in space. q and q have very large momentum → they fly apart.

Fig 20 Fig 21

The energy between the qq increases as they move apart E ≈ V(r)≈kr

Fig 22

When E>2mqc2...

Fig 23

As the kinetic energy decreases ... the hadrons freeze out

Fig 24

This process is known as hadronisation. Dayalbagh Educational Institute


Interactions and fields

Particle Physics

QCD Coupling Constant In QCD interaction is αs – also not really a constant. Quark emit gluons: which can form virtual quark – anti quark pairs. However the gluons themselves also carry colour charge, which effects the screening.

Fig 27

Fig 28

Fig 29

αs decreases at high energies! ⇔ αs increases at large distances! At low energies the coupling constant becomes large, αs~1. We cannot use perturbation theory to calculate cross sections! The understanding of this phenomena won the Nobel prize in 2004.

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Interactions and fields

Particle Physics

QCD Summary QCD: Quantum Chromodynamics is the quantum description of the strong force.

Gluons are the propagator of the strong force

Quarks and gluons carry colour charge. Gluons selfinteract:

Electromagnetic coupling constant Îą decreases as a charged particles get further apart. Strong coupling constant increases as further apart quarks become.

Hadrons can be described as consisting of partons: quarks and gluons, which interact independently

Only quarks feel the strong force.

Colour Confinement Quarks and gluons produced Energy required to separate in collisions hadronise: quarks quarks are confined hadrons are produced. to hadrons The decay products of the hadrons appear in the detector as jets. Table 2

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Interactions and fields

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Problem 1 a) The Δ++ resonance has a full width of Γ = 120 MeV. How far an average would such a particle of energy 200 GeV travel before decaying?

Solution (a):

is a distance which is much less than the resolution obtainable by available techniques. The best resolution obtained is only 1μm. Dayalbagh Educational Institute


Interactions and fields

Particle Physics

b) Given that the width for W-boson decay is less than 6.5MeV, estimate the limit for the corresponding life time.

Solution b):

According to Uncertainty principle:


Problem 2: The particles X and Y can be produced by strong interaction K−+p→K++X K − + p → π0 + Y Identify the particles X (1,321 MeV) and Y (1,192MeV) and deduce their quark content. If their decay schemes are X → Λ + π− and Y → Λ + γ, give a rough estimate of their lifetime.

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Solution 2:

From the conservation laws for strong interactions the quantum numbers for X are B=+1, Q/e =−1, S =−2 It is Xi hyperon (Ξ−) and decays weakly with lifetime τ∼10−13s. Its quark structure is dss; The quantum numbers for Y are B =+1, Q/e = 0, S=−1. It is a sigma hyperon (Σ0) which decays electromagnetically with lifetime of the order of 10−20 s. Its quark structure is uds.

Problem 3: Explain why at the same energy the total crosssections σ( π− + p ) ≅ σ( π++ n) , while σ( K− + p ) ≠ σ( K++ n )

Solution 3: The reaction channels for the pion reactions π− + p → π− + p and π− + p → π0 +n are charge symmetric to the reactions, π+ + n → π+ + n and π+ + n → π0 + p.

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Interactions and fields Particle Physics We can therefore expect the reaction cross sections with pions to be nearly identical. On the other hand, the reactions with kaons are not charge symmetric because K+ and K− do not constitute an iso spin doublet but are particle–antiparticle. Typical reactions with K+ are K+ + n → K+ + n and K+ + n → K0 + p However for K− many more channels are open. K− + p → K − + p → K0 + n → Σ+ + π− → Σ0 + π0 → Σ− + π+ So, there is no reason for σ(K− + p) to be equal to σ(K++ n).

Problem 4 How can the neutral K-mesons, K 0 and K 0 be distinguished?

Solution 4

From the CPT (Charge, Parity and time) theorem, K 0 and K 0 which are particle and antiparticle, cannot be distinguished by their decay modes, but can be identified through their distinct strong interaction process. Thus


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Interactions and fields

K 0 + p → Σ+ + π 0 K 0 + n → K− + p K 0 + p → Σ+ + π 0 K +n→K +p 0


Particle Physics

forbidden because of strangeness nonconservation allowed


Problem 5 Calculate the ratio of the cross sections for the reactions π−p → π−p and π−p → π0n. Assume that the two I spin amplitudes are equal in magnitude but differ in phase by 30 ◦.

Solution 5

Pions have T = 1 and nucleons T = 1/ 2 so that the resultant isospin both in the initial state and the final state can be I = 1/2 or 3/2. From the Clebsch – Gordon Coefficients of 1 x 1/2, we can write

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Interactions and fields

Particle Physics


The ratio of cross sections is


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With ϕ=±300


Interactions and fields

Particle Physics

Definitions and Keywords Quarks and anti-quarks: Spin ½ fermions and antifermions now believed to be the constituents of hadrons. Gluons: Spin-1 bosons which, by exchange, are responsible for the forces among quarks and antiquarks. Quantum chromodynamics: The interaction of quarks and gluons.




Gauge bosons: These particles are the carriers of the interaction between the fundamental fermions and are the modern version of Yukawa's boson as carrier of a force. We know 12 gauge bosons: 8 gluons, the photons, the W+, W-, and the Z0. Confinement: This is the term describing the observation that it appears impossible to liberate quarks, or anti-quarks, from hadrons. Strong interaction: This term covers the interactions that exists between members of a whole class of particles, including the pions and nucleons, that are believed to be composed of quarks and anti-quarks. Hadron: The generic name for all particles composed of quarks and anti-quarks. Lepton: The generic name for all fermions, charged or uncharged, which do not experience the strong interaction. Dayalbagh Educational Institute


Chapter 1.3