Structure of matter and energy scales Subatomic physics deals with objects of the size of the atomic nucleus and smaller. We cannot see subatomic particles directly, but we may obtain knowledge of their structures by observing the effect of projectiles that are scattered from them. The resolution any such probe is limited to of order the de Broglie wavelength, λ = h p. () If we wish to resolve small distances, smaller than the atom, we will need to do so with probes with high momenta. Typical sizes and are given below. Object Size Binding energy Atom 0 0 m ev Nucleus 0 5 m MeV Quark < 0 8 m > TeV We can see that small objects also tend to have high binding energies, and hence probes of large energy will be required in order to excite them or break them up. 2 Decays and the Fermi Golden Rule In subatomic physics we are interested in the decays of unstable particles, such as radioactive nuclei, or cosmic muons. In our quantum mechanics course we used time-dependent perturbation theory to show that that the transition rate of an unstable state into a continuum of other states was given by the Fermi Golden Rule: Γ = 2π V fi 2 dn de f. (2)
The density of states within a cubic box with sides length a can be calculated as follows. The wave-function should be of form x Ψ exp (ik x). If we insist on periodic boundary conditions, with period a, then the values of k x are constrained to the values k x = 2πn/a. Similar conditions hold for k y and k z. The number of momentum states within some range of momentum d 3 p = d 3 k is given by dn = d3 p (2π) 3 V where V = a 3 is the volume of the box. Γ is the rate of the decay V fi is the matrix element of the Hamiltonian coupling the initial and the final states dn de f is the density of final states. Fermi G.R. example: consider the isotropic decay of a neutral spin-0 particle into two massless daughters A B + C. The Fermi G.R. gives the decay rate as Γ = 2π V fi 2 dn de f = 2π V fi 2 4πp2 B (2π) 3 dp B de f V. Since all decay angles are equally probable, the integrals over the angles contributes 4π. The decay products have momentum p B = E f /2 so dp B de f = 2. Normalising to one unstable particle per unit volume gives V =, and results in a decay rate Γ = 2π V fi 2 p 2 B = 8π V fi 2 m 2 A. February 3, 202 2 c A.J.Barr 2009-20.
The number of particles remaining at time t is governed by the decay law Which integrates to give dn dt = ΓN N(t) = N 0 exp ( Γt). We can easily calculate the particles average proper lifetime τ, using the probability that they decay between time t and t + δt p(t) δt = N 0 dn dt δt = Γ exp ( Γt) δt. The mean lifetime is then τ = t t p(t) dt 0 = p(t) dt 0 = Γ Don t forget that if such particles are travelling relativistically, their decay clocks will be time-dilated, and so the average number remaining after some distance x will be N = N 0 exp ( x/l 0 ) where L 0 = βγcτ. The time-energy uncertainty relationship of quantum mechanics tells us that if a state has only a finite lifetime, then it does not have a perfectly-defined energy E t. An exponentially decaying state with lifetime τ has an uncertainty in its lifetime of order τ. We therefore expect that its energy is only defined up to an uncertainty E τ = Γ. Using the fact that Γ = /τ, and moving to natural units to set =, we find that E Γ. (3) February 3, 202 3 c A.J.Barr 2009-20.
In natural units, the uncertainty in the rest-energy of a particle is equal to the rate of its decay. This means that if we take a set of identical unstable particles, and measure the mass of each, we will expect to get a range of values with width of order Γ. We can make our statement about the smearing of the energy more precise. If we require that the probability of finding the particle decays with rate Γ, then the amplitude must satisfy a(t) 2 = exp( Γt) The appropriate form for the amplitude is then a(t) = exp( i(e E 0 )t Γ/2t). When Fourier transformed into the energy domain, the probability density function for finding the particle with energy E is p(e) (E E 0 ) 2 (Γ/2) 2. (4) E is the energy of the system, and E 0 is the characteristic rest-mass of the unstable particle. The probability density function has a Lorentzian, peaked, line shape, with full-width at half max (FWHM) of the peak equal to Γ. The quantity Γ is often called the width of the particle. Long-lived particles have narrow widths and well-defined energies. Short-lived particles have large widths and less well defined energies. When the state is so short-lived that its width Γ is similar to its mass, then the decay is so rapid that it is no longer useful to think of it as a particle. 3 Resonances and the Breit-Wigner formula Unstable states are created and then decay. Consider the process A + B O C + D The initial particles collide to form an unstable intermediate, which then decays to the final state. This could represent a familiar process such as the absorption and then emission of a photon by an atom, with an intermediate excited atomic February 3, 202 4 c A.J.Barr 2009-20.
The decay width can be generalised to a particle which has many different decay modes. The rate of decay into mode i is given Γ i. The total rate of decay is given by Γ = Γ i. i=...n It is the total rate of decay Γ that enters the equation of the line-shape (4). The fraction of particles that decay into final state i, is known as the branching ratio B = Γ i Γ. The quantity Γ i is known as the partial width to final state i, whereas the sum of all partial widths is known as the total width. state. It could well equally represent the formation of a very heavy Z 0 particle from the collision of a high-energy electron with its anti-particle, followed by the decay of that Z 0 particle to a muon and its anti-particle. Under the condition that the transition from A + B to C + D can proceed exclusively via the intermediate state 0, and that the width of the intermediate is not too large (Γ E 0 ), the cross-section for the process is given by the Breit-Wigner formula σ i 0 f = π k 2 The terms in this equation are as follows: Γ i Γ f (E E 0 ) 2 + Γ 2 /4. (5) Γ i is the partial width of the resonance to decay to the initial state A + B E 0 The Breit-Wigner line shape. E Γ f is the partial width of the resonance to decay to the final state C + D Γ is the full width of the resonance E is the centre-of-mass energy of the system E 0 is the characteristic rest mass energy of the resonance k is the wave-number of the incoming projectile in the centre-of-mass frame. The cross-section is non-zero at any energy, but has a sharp peak at energies E close to the rest-mass-energy E 0 of the intermediate particle. Longer lived February 3, 202 5 c A.J.Barr 2009-20.
intermediate particles have smaller Γ and hence sharper peaks. The sharp peak is known as resonance. Scattering that proceeds exclusively or dominantly via a narrow-width intermediate particle will have a cross section shape given by the Breit-Wigner and is known as resonance scattering. We can see where some of the terms in (5) must come from. The energy dependence on the denominator is the same as was found for the line-shape of an unstable particle (4). By the Fermi GR, the factors of Γ i and Γ f are proportional to the mod-square of the matrix-elements V 0i and V f0 that are responsible for the production and decay of the unstable particle, respectively. The /k 2 factor comes from a combination of density-of-states and flux factors. The flux factor When calculating a cross section from a rate, we need to take into account that for scattering from a single fixed target σ = W J where J is the flux density of incoming particles. The flux density itself is given by J = n p v where n p is the number density of projectiles and v is their speed. If we normalise to one incoming particle per unit volume, then n p = and the cross section is simply related to the rate by σ = W v 4 Scattering theory We are interested in a theory that can describe the scattering of a particle from a potential V (x). Our Hamiltonian is H = H 0 + V. where H 0 is the free-particle kinetic energy operator H 0 = p2 2m. Which is just as well, as we are producing and decaying an unstable particle. February 3, 202 6 c A.J.Barr 2009-20.
4. Scattering amplitudes Decays, resonances and scattering In the absence of V the solutions of the Hamiltonian could be written as the freeparticle states satisfying H 0 φ = E φ. These free-particle eigenstates could be written as momentum eigenstates p, but since that isn t the only possibility we hold off writing an explicit form for φ for now. The full Schrödinger equation is H 0 + V φ = E φ. We define these eigenstates of H such that ψ φ as V 0, where φ and ψ have the same energy eigenvalue. (We are able to do this since the spectra of both H and H + V are continuous.) A possible solution is 2 ψ = E H 0 V ψ + φ. (6) By multiplying by (E H 0 ) we can show that this looks fine, other than the problem of the operator /(E H 0 ) being singular. The singular behaviour in (6) can be fixed by making E slightly complex and defining ψ (±) = φ + E H 0 ± iɛ V ψ(±). (7) This is the Lippmann-Schwinger equation. We will find the physical meeting of the (±) in the ψ (±) shortly. 4. Scattering amplitudes To calculate scattering amplitudes we are going to have to use both the position and the momentum basis, because φ is a momentum eigenstate, and V is a function of x. If φ stands for a plane wave with momentum k then the wavefunction can be written eik x x φ =. (2π) 3 2 We can express (7) in the position basis by bra-ing through with x and inserting the identity operator d 3 x x x x ψ (±) = x φ + d 3 x x x x V ψ (±). (8) E H 0 ± iɛ 2 Remember that functions of operators are defined by f(â) = i f(a i) a i a i. The reciprocal of an operator is well defined/ provided that its eigenvalues are non-zero. February 3, 202 7 c A.J.Barr 2009-20.
4. Scattering amplitudes Decays, resonances and scattering The solution to the Green s function defined by is G ± (x, x ) 2 x x 2m E H 0 ± iɛ G ± (x, x ) = e ±ik x x 4π x x. Using this result we can see that the amplitude of interest simplifies to x ψ (±) = x φ 2m 4π 2 d 3 x e±ik x x x x V (x ) x ψ (±) (9) where we have also assumed that the potential is local in the sense that it can be written as x V x = V (x )δ 3 (x x ). The wave function (9) is a sum of two terms. The first is the incoming plane wave. For large x the spatial dependence of the second term is e ±ikr /r. We can now understand the physical meaning of the ψ (±) states; they represent outgoing (+) and incoming ( ) spherical waves respectively. We are interested in the outgoing (+) spherical waves the ones which have been scattered from the potential. We want to know the amplitude of the outgoing wave at a point x. For practical experiments the detector must be far from the scattering centre, so we may assume x x. We define a unit vector in the direction of the observation point ˆr = x x and also a wave-vector for particles travelling in the direction ˆx, k = kˆr. Far from the scattering centre we can write x x = r 2 2rr cos α + r 2 = r 2 r r 2 cos α + r r 2 r ˆr x x x February 3, 202 8 c A.J.Barr 2009-20.
4.2 Born Decays, resonances and scattering where α is the angle between the x and the x directions. It s safe to replace the x x in the denominator in the integrand of (9) with just r, but the phase term will need to be replaced by r ˆr x. So we finally simplify the wave function to x ψ (+) r large x k 2m e ikr 4π 2 r which we can write as x ψ (+) = (2π) 3 2 d 3 x e ik x V (x ) x ψ (+) ] [e ik x + eikr r f(k, k ). This makes it clear that we have a sum of an incoming plane wave and an outgoing spherical wave with amplitude f(k, k) given by f(k, k) = 2m (2π)3 4π k V 2 ψ(±). (0) We will ignore the interference between the first term which represents the original plane wave and the second term which represents the outgoing scattered wave. x φ e i k x / x Wave function of an outgoing spherical wave. So we find that the partial cross-section dσ the number of particles scattered into a particular region of solid angle per unit time divided by the incident flux 3 is given by dσ = r2 j scatt j incid dω = f(k, k) 2 dω. This means that the differential cross section is given by the simple result dσ dω = f(k, k) 2. The differential cross section is the mod-square of the scattering amplitude. 4.2 The Born approximation If the potential is weak we can assume that the eigenstates are only slightly modified by V, and so we can replace ψ (±) in (0) by k. f () (k, k) = 2m (2π)3 4π k V 2 k. () 3 Remember that the flux is given by j = 2im [ψ ψ ψ ψ ]. February 3, 202 9 c A.J.Barr 2009-20.
This is known as the Born approximation. Within this approximation we have found the nice simple result f () (k, k) k V k. Up to some constant factors, the scattering amplitude is found by squeezing the perturbing potential V between incoming and the outgoing momentum eigenstates of the free-particle Hamiltonian. Expanding out () in the position representation (by insertion of a couple of completeness relations d 3 x x x ) we can write f () (k, k) = 2m d 3 x e i(k k ) x V (x ). 4π 2 This result is telling us that scattering amplitude is proportional to the 3d Fourier transform of the potential. By scattering particles from targets we can measure, and hence infer the functional form of V (r). dσ dω 5 Virtual Particles One of the insights of subatomic physics is that at the microscopic level forces are caused by the exchange of force-carrying particles. For example the Coulomb force between two electrons is mediated by excitations of the electromagnetic field i.e. photons. There is no real action at a distance. Instead the force is transmitted between the two scattering particles by the exchange of some unobserved photon or photons. The mediating photons are emitted by one electron and absorbed by the other. It s generally not possible to tell which electro emitted and which absorbed the mediating photons all one can observe is the net effect on the electrons. Other forces are mediated by other force-carrying particles we shall meet examples later on. In each case the messenger particles are known as virtual particles. Virtual particles are not directly observed, and have properties different from real particles which are free to propagate. To illustrate why virtual particles have unusual properties, consider the elastic scattering of an electron from a nucleus, mediated by a single virtual photon. We can assume the nucleus to be much more massive than the electron so that it is approximately stationary. Let the incoming electron have momentum p e γ e e e February 3, 202 0 c A.J.Barr 2009-20.
and the outgoing, scattered electron have momentum p. For elastic scattering, the energy of the electron is unchanged E = E. The electron has picked up a change of momentum p = p p from absorbing the virtual photon, but absorbed no energy. So the photon must have energy and momentum e γ e E γ = 0 p γ = p = p p. The exchanged photon carries momentum, but no energy. This sounds odd, but is nevertheless correct. What we have found is that for this virtual photon, Eγ 2 p 2 γ. The fact that E γ = 0 is special to the case we have chosen. However, the general result is that for any virtual particle there is an energy-momentum invariant which is not equal to the square of its mass P P = E 2 p p m 2. Such virtual particles do not satisfy the usual energy-momentum invariant and are said to be off mass shell. Note that we would not have been able to escape this conclusion if we had taken the alternative viewpoint that the electron had emitted the photon and the nucleus had absorbed it. In that case the photon s momentum would have been p γ = p. The square of the momentum would be the same, and the photon s energy would still have been zero. These exchanged, virtual, photons are an equally valid solution to the (quantum) field equations as the more familiar travelling-wave solutions of real on-massshell photons. It is interesting to realise that all of classical electromagnetism is actually the result of very many photons being exchanged. 6 The Yukawa Potential There is a type of potential that is of particular importance in subatomic scattering, which has the form (in natural units) V (r) = g2 e µr. (2) r This is known as the Yukawa potential. When µ = 0 this has the familiar /r dependence of the electrostatic and gravitational potentials. When µ is non-zero, February 3, 202 c A.J.Barr 2009-20.
The electromagnetic force is mediated by excitations of the electromagnetic force, i.e. photons. The photon is massless so the electrostatic potential falls as /r with no exponential. Other forces are mediated by heavy particles and so are only effective over a short range. An example is the weak nuclear force, which is mediated by particles with µ close to 00 GeV and so is feeble at distances larger than about /(00 GeV) (97 MeV fm)/(00 GeV) 0 8 m. the potential also falls off exponentially with r, with a characteristic length of /µ. To understand the meaning of the µ term it is useful to consider the relativistic wave equation known as the Klein-Gordan equation ( ) 2 t 2 2 + µ 2 ϕ(r, t) = 0. (3) This is the relativistic wave equation for spin-0 particles. The plane-wave solutions to (3) are φ(x) = A exp ( ip X) = A exp ( iet + ip x). These solutions require the propagating particles to be of mass µ = E 2 p 2. The Klein-Gordon equation is therefore describing excitations of a field of particles each of mass µ. The Yukawa potential is another solution to the field equation (3). The difference is that the Yukawa potential describes the static solution due to virtual particles of mass µ created by some source at the origin. The constant g 2 tells us about the depth of the potential, or the size of the force. It is equivalent to the factor of Q Q 2 /(4πɛ 0 ) in electrostatics. The scattering amplitude of a particle bouncing off a Yukawa potential is found to be k V Yukawa k = g2 (2π) 3 µ 2 + k. (4) 2 We can go some way towards interpreting this result as the exchange of a virtual particle as follows. We justify the two factors of g as coming from the points where a virtual photon is either created or annihilated. This vertex factor g is a measure of the interaction or coupling of the exchanged particle with the other objects. There is one factor of g the point of creation of the virtual particle, and another one at the point where it is absorbed. February 3, 202 2 c A.J.Barr 2009-20.
In electromagnetism we want each vertex factor to be proportional to the charge of the particle that the (virtual) photon interacts with. We are interested in a (µ = 0) Yukawa potential of the form V EM = q q 2 e 2 4πɛ 0 r. For scattering from a Coulomb potential we can therefore use the Yukawa result (4) by making the substitution g 2 q q 2 e 2 4πɛ 0. In a general scattering process we will want a vertex factor g qe at each vertex. The other important factor in the scattering amplitude (4) is associated with the momentum and mass of the exchanged particle: µ 2 + k 2 In general it is found that if a virtual particle of mass µ and four-momentum P is exchanged, there is a propagator factor P P µ 2 (5) in the scattering amplitude. This relativistically invariant expression is consistent with our electron-scattering example, where the denominator was: P P µ 2 = E 2 p 2 µ 2 = 0 k 2 µ 2 = ( µ 2 + k 2) Note that the propagator (5) becomes singular as the particle gets close to its mass shell. i.e. as P P µ 2. It is only because the exchanged particles are off their mass-shells that the result is finite. The identification of the vertex factors and propagators will turn out to be very useful when we later try to construct more complicated scattering processes. In those cases we will be able to construct the most important features of the scattering amplitude merely by writing down: February 3, 202 3 c A.J.Barr 2009-20.
an appropriate vertex factor each time a particle is either created or annihilated and a propagator factor for each virtual particle. By multiplying together these factors we capture the most important properties of the scattering amplitude. February 3, 202 4 c A.J.Barr 2009-20.
Key concepts The leading Born approximation to the scattering amplitude is f () (k, k) k V k. The scattering amplitude is proportional to the 3d Fourier transform of the potential. The differential cross-section is given in terms of the scattering amplitude by dσ dω = f(k, k) 2 Forces are transmitted by virtual mediating particles which are off-massshell: P P = E 2 p p m 2. The Yukawa potential for an exchanged particle of mass µ and coupling g is V (r) = g2 e µr. (6) r The scattering amplitude contains a vertex factors g for any point where particles are created or annihilated The scattering amplitude contains a propagator factor for each virtual particle. P P µ 2. The rate of an interaction is given by the Fermi Golden Rule Γ = 2π V fi 2 dn de f. The Breit-Wigner formula for resonant scattering is σ(n, γ) = π k 2 Γ n Γ γ (E E 0 ) 2 + Γ 2 /4. February 3, 202 5 c A.J.Barr 2009-20.
In natural units, = c = and [Mass] = [Energy] = [Momentum] = [Time] = [Distance] A useful conversion constant is c 97 MeV fm The cross section is defined by: σ i = W i n J δx (7) The differential cross section is the cross section per unit solid angle dσ i dω Cross sections for sub-atomic physics are often expressed in the unit of barns. barn = 0 28 m 2 February 3, 202 6 c A.J.Barr 2009-20.
A Natural units We have been used to using units in which times are measured in seconds and distances in meters. In such units the speed of light takes the value close to 3 0 8 ms. We could instead have chosen to use unit of time such that c =. 4 Doing this allows us to leave c out of our equations, provided we are careful to remember the units we are working in. Such units are useful in relativistic systems, since now the relativistic energy-momentum-mass relations are E = γm p = γmv E 2 p 2 = m 2. So for a relativistic system setting c = means that energy, mass and momentum all have the same dimensions. Since we are interested in quantum systems, we can go further and look for units in which is also equal to one. In such units the energy of a photon will be equal to its angular frequency E = ω = ω. What quantities does relate? Remember the time-energy uncertainty relationship E t. Setting = means that time (and hence distance) must have the same dimensions as E. So in our system natural units we have have that [Mass] = [Energy] = [Momentum] = [Time] = [Distance] We are going to use units of energy for all of the quantities above. The nuclear energy levels have typical energies of the order of 0 6 electron-volts, so we shall measure energies and masses in MeV, and lengths and times in MeV. At the end of a calculation how can we recover a real length from one measured in MeV? We can use the conversion factor which tells us that one of our fm = 0 5 m. c 97 MeV fm MeV length units corresponds to 97 fm where 4 You will already have used units in which time is measured in seconds and distance in light-seconds. In those units c =, since the speed of light is one light-second per second. February 3, 202 7 c A.J.Barr 2009-20.
B Cross sections Many experiments take the form of scattering a beam of projectiles into a target. Provided that the target is sufficiently thin that the flux is approximately constant within that target, the rate of any reaction W i will be proportional to the flux of incoming projectiles J (number per unit time) the number density of scattering centres n (number per unit volume), and the width of the target W i = σ i nj δx. (8) The constant of proportionality σ i has dimensions of area. It is known as the cross section for process i and is defined by σ i = W i n J δx (9) We can get some feeling for why this is a useful quantity if we rewrite (8) as σ W i = (n A δx) J }{{}}{{} A N target P scatt where A is the area of the target, which shows that the cross section can be interpreted as the effective area presented to the beam per target for which a particular reaction can be expected to occur. The total rate of loss of beam is given by W = ΣW i, and the corresponding total cross section is σ = σ i. i We could chose to quote cross sections in units of e.g. fm 2 or GeV 2, however the most common unit used in nuclear and particle physics is the so-called barn where barn = 0 28 m 2 We can convert this to follows MeV 2 units using the usual c conversion constant as barn = 0 28 m 2 = 00 fm 2 /(97 MeV fm) 2 = 0.00257 MeV 2. February 3, 202 8 c A.J.Barr 2009-20.
C Luminosity In a collider a machine which collides opposing beams of particles the rate of any particular reaction will be proportional to the cross section for that reaction and on various other parameters which depend on the machine set-up. Those parameters will include the number of particles per bunch, their spatial distribution, and the frequency of the collision of those bunches. We call the constant of proportionality which encompasses all those machine effects the luminosity L = W σ. It has dimensions of [L] 2 [T ] and is useful to factor out if you don t care about the details of the machine and just want to know the rates of various processes. The time-integrated luminosity times the cross section gives the expected count of the events of any type N events, i = σ i L dt. For a machine colliding opposing bunches containing N and N 2 particles at rate f, you should be able to show that the luminosity is L = N N 2 f A, where A is the cross-sectional area of each bunch (perpendicular to the beam direction). We ve assumed above that the distributions of particles within each bunch is uniform. If that s not the case (e.g. in most real experiments the beams have approximately Gaussian profiles) then we will have to calculate the effective overlap area A of the bunches by performing an appropriate integral. D Beyond Born: propagators in non-relativistic scattering theory Non-examinable February 3, 202 9 c A.J.Barr 2009-20.
Previously we examined the leading Born term in the non-relativistic scattering theory. To see how things develop if we don t want to rashly assume that ψ ± φ it is useful to define a transition operator T such that V ψ (+) = T φ Multiplying the Lippmann-Schwinger equation (7) by V we get an expression for T T φ = V φ + V E H 0 + iɛ T φ. Since this is to be true for any φ, the corresponding operator equation must also be true: T = V + V E H 0 + iɛ T. This operator is defined recursively. It is is exactly what we need to find the scattering amplitude, since from (0), the amplitude is given by f(k, k) = 2m 4π 2 (2π)3 k T k. We can now find an iterative solution for T : T = V + V E H 0 + iɛ V + V E H 0 + iɛ V E H 0 + iɛ V +... (20) k We can interpret this series of terms as a sequence of the operators corresponding to the particle interacting with the potential (operated on by V ) and propagating along for some distance (evolves according to E H 0 +iɛ ). k V The operator E H 0 + iɛ (2) is the non-relativistic version of the propagator. The propagator can be seen to be a term in the expansion (20) which is giving a contribution the amplitude for a particle moving from an interaction at point A to another at point B. Mathematically it is a Green s function solution to the Lippmann-Schwinger in the position representation (8). k V k /(E-H 0+i ) V We are now in a position to quantify what we meant by a weak potential earlier on. From the expansion (20) we can see that the first Born approximation () will be useful if the matrix elements of T can be well approximated by its first term V. February 3, 202 20 c A.J.Barr 2009-20.
When is this condition likely to hold? Remember that the Yukawa potential was proportional to the square of a dimensionless coupling constant g 2. If g 2 then successive applications of V introducing higher and higher powers of g and can usually be neglected. This will be true for electromagnetism, since the dimensionless coupling relevant for electromagnetism is related to the fine structure constant g 2 = α = e2 4πɛ 0 c 37. Since α, we can usually get away with just the first term of (20) for electric interactions (i.e. we can use the Born approximation). February 3, 202 2 c A.J.Barr 2009-20.