F R A N C E S C O R E N G A

I N T R O D U C T I O N T O T H EQ U A N T U M T H E O R Y O FPA R T I C L E S C AT T E R I N G

S A P I E N Z A U N I V E R S I TÀ D I R O M AC O R S O D I L A U R E A I N F I S I C AC O R S O D I F I S I C A N U C L E A R E E S U B N U C L E A R E I

Contents

1 The scattering problem in quantum mechanics 7

2 The Lippmann-Schwinger equation 13

3 The Born approximation 17

4 Cross section calculation with examples 23

5 Partial waves and resonances 27

6 Discrete symmetries 29

7 Electromagnetic transitions 31

Introduction

These lectures will give an introduction to the non-relativistic quan-tum-mechanical description of the scattering between particles.

Scattering is the process where particles coming from a distancelong enough to neglect their interactions get near enough to eachother for feeling a force modifying their trajectories. In classicalmechanics, the goal of a scattering theory is to determine the finaltrajectories of particles from their initial ones. In quantum mechan-ics, the Heisenberg indetermination principle prevents to define pre-cisely, at the same time, both the position and the momentum ofparticles. Moreover, typical experimental configurations allow to de-termine the momentum of the particles with fairly good precisionwith respect to the involved energy scales (e.g. the MeV scale for nu-clear physics processes), while the position cannot be measured witha precision comparable to the corresponding distance scale (that isO(hc/1 MeV ∼ 197 fm) in the example above). Finally, in most casesbeams of particles are used in modern experiments, i.e collimatedbunches of many particles of same momentum. Consequently, thescattering problem in quantum mechanics is typically set as the de-termination of the distribution of the particle momenta (magnitudeand direction) and other internal quantum numbers (e.g. the spin)after the scattering of beams of particles into a target (fixed-targetexperiments) or among them (collider experiments), that means, es-sentially, the so-called cross section of the process.

In these notes we will concentrate on the most simple configu-ration, the elastic scattering of two particles without spin, which isindeed more easily treated as the scattering of a beam of particles ina central field of forces, if the interactions among particles within thesame beam are neglected.

In the first chapter we will introduce some basic concepts and for-mulas that will be used through the notes, such as the definitions ofcross section and phase space. In Chapters 2 and 3 we will presentan integral form of the Schrödinger equation, know as the Lippmann-Schwinger equation, that is particularly useful to approach the scat-tering problem in a time-independent form, and its solution in theso-called Born approximation. In Chapter 4 we will show how to usethese tools to calculate scattering cross sections with some relevantexamples. Chapter 5 will introduce the partial wave formalism andthe concept of scattering resonance. We will finally discuss, in Chap-ter 6, discrete symmetries and the consequent conservation laws that

6 francesco renga

apply in scattering problems. The earned knowledge will be usedin Chapter 7 for a specific treatment of electromagnetic transitions,which are relevant in nuclear, atomic and molecular physics.

1The scattering problem in quantum mechanics

1.1 Cross sections

As already mentioned in the introduction to these notes, the maingoal of a quantum scattering theory is the determination of the dis-tribution of the momenta and other internal quantum numbers ofparticles going out from a scattering process. We can have for in-stance an experiment where a beam of particles A hits a fixed targetmade of particles B, producing the reaction:

A + B→ X1 . . . Xn (1.1)

In this experiment we can firstly measure the number of events ofthis kind that are produced per unit time, i.e. the total rate:

Γout =dNout

dt(1.2)

If we define the flow of incoming particles as:

ΦA =dNAdt dΣ

(1.3)

where dΣ is a surface element orthogonal to the beam, the event ratewill be proportional to ΦA and to the number NB of B particles seenby the beam:

dNout

dt= ΦA · NB · σ (1.4)

The proportionality constant σ has the dimensions of a surface, itonly depends on the kinematics and dynamics of the elementaryreaction (1.1) and it is called the cross section of the process. Its mea-surement will be the first result of our experiment. We can then mea-sure the distribution of the directions and energies of the outcomingparticles:

A

B

Xi

dΩ

zθ

Figure 1.1: Schematisation of ascattering process.

dNout

dt dE1 . . . dEn dΩ1 . . . dΩn= ΦA · NB ·

dσ

dE1 . . . dEn dΩ1 . . . dΩn(1.5)

or specifically the distribution of one of them:

dNout

dt dEi dΩi= ΦA · NB ·

dσ

dEi dΩi(1.6)

8 introduction to the quantum theory of particle scattering

The quantity dσ/(dEi dΩi), that we can also extract from our mea-surements, is called differential cross section.

We will be then eager to use our results to test some theory forthe interaction that produces the process (1.1). So, the main goal ofa quantum scattering theory will be the calculation of cross sections,to be compared with the results of our experiments, starting fromHamiltonians describing the interactions under study.

1.2 Elastic scattering

Let us consider the scattering between two particles A and B. Weassume that the interaction between the two particles goes to zero astheir distance r → ∞, so that in the initial state (incoming particlecoming from a large distance) and in the final one (both particlesgoing far from the interaction point) we can assume free particlestates.

We consider specifically processes where the particle species arenot changed by the interaction. This is what is called, in particlephysics, an elastic scattering, as opposed to the inelastic scattering,where the outcoming particles differ with respect to the incomingones, for instance when the target particle is broken into fragmentsor brought to an excited state.1 1 It is worth noticing how these terms

are used differently here with respect tothe theory of classical collisions, whichare called inelastic if the mechanical en-ergy is not conserved. In particle scat-tering, the total energy of the system(kinetic energies plus masses) is alwaysconserved, also for inelastic processes.

For an elastic scattering A + B → A + B, the conservation of thetotal energy of the system implies:

KA + mA + mB = K′A + mA + K′B + mB −→ KA = K′A + K′B (1.7)

where K is the kinetic energy, m the mass and primed quantitiesrefer to the final state. The total kinetic energy is hence conserved inelastic scatterings. The total momentum will be also conserved:

~pA = ~p ′A + ~p ′B (1.8)

and the quantity ~q = ~pA − ~p ′A is called momentum transfer.

1.3 Two-particle scattering as a potential scattering

As in classical mechanics, the two-body problem can be reduced inquantum mechanics to two independent, one-body problems.

By symmetry, the potential describing the interaction between twoparticles not subject to any external force can only depend on theirrelative position ~r = ~xA − ~xB, their relative velocity ~r and internalquantum numbers ~nA, ~nB:

V = V(~r,~r,~nA,~nB) (1.9)

For cleanliness of the notation, in the following we will indicate ex-plicitly only the dependence on~r. The time-independent Schrödingerequation for the system can be written as:[

−h2 ∇2A

2mA− h2 ∇2

B2mB

+ V(~r)

]Ψ(~xA,~xB) =

= E Ψ(~xA,~xB) (1.10)

the scattering problem in quantum mechanics 9

If we also define:

~R =mA~xA + mB~xB

mA + mB(1.11)

the Schrödinger equation becomes:[−h2∇2

r2µ− h2∇2

R2M

+ V(~r)

]Ψ(~r, ~R) =

= E Ψ(~r, ~R) (1.12)

where M = mA + mB is the total mass of the system and the reducedmass µ is defined as:

µ =mAmB

mA + mB(1.13)

The equation can be separated in two by writing:

Ψ(~r, ~R) = χ(~R)ψ(~r) , E = Er + ECM (1.14)

so that:

−h2∇2R

2Mχ(~R) = ECM χ(~R) (1.15)

is an uninteresting free-particle Schrödinger equation for the centerof mass of the system, while:[

−h2∇2r

2µ+ V(~r)

]ψ(~r) = Er ψ(~r) (1.16)

contains all the dynamics of the problem.In the next section we will see how the scattering problem can be

treated as a standing wave scattering. So, we will need to solve thetime-independent Schrödinger equation for a two-body system, andwe have just shown that it can be reduced to the problem of a singleparticle of mass µ subject to the potential V(~r). The boundary con-dition is dictated by the fact that the particle is free for r → ∞, andso the wave function at large distance from the scattering center hasto reduce to a plane wave. This is typically referred as the problemof the scattering of a particle by a potential (potential scattering).

If we also neglect the effect of the particles’s spin and other inter-nal quantum numbers, there is no preferred direction for the interac-tion between the two particles, which consequently can only dependon their distance r = |~r|, i.e. there is a central potential V(~r) = V(r).The equation to solve is simply:[

−h2∇2

2µ+ V(r)

]ψ(~r) = E ψ(~r) (1.17)

where we dropped the subscripts r for cleanliness of the notation.

1.4 Time-independent treatment of quantum scattering

Let us consider the elastic scattering of a beam of incoming particles(projectiles) of mass µ and same momentum ~p into fixed scattering

10 introduction to the quantum theory of particle scattering

centers generating a potential V(~r). Based on the discussion made inthe previous section, the results we are going to extract can be easilygeneralised to the problem of a two-particle scattering.

For a set of particles, we define the probabilty current ~J in such away that the probability of a particle to be found inside a volume τ

closed by the surface Σ changes with time according to:

dPdt

= −∫

Σ~J · n dΣ (1.18)

on the other hand, if ϕ is the wave function of a particle in the beam:

τn

J

ΣdΣ

Figure 1.2: Definition of theprobability current.

P =∫

τ|ϕ|2 dτ (1.19)

and so, from the Gauss theorem:

ddt

∫τ|ϕ|2 dτ =

∫τ

∂

∂t|ϕ|2 dτ = −

∫Σ~J · n dΣ = −

∫τ

~∇ ·~J dτ (1.20)

−→ −~∇ ·~J = ∂

∂t|ϕ|2 (1.21)

By exploiting the Schrödinger equation:

ih∂ϕ

∂t= − h2

2µ∇2 ϕ + Vϕ (1.22)

and the identity:

~∇ · (ψ~∇ϕ) = ψ∇2 ϕ + ~∇ψ · ~∇ϕ (1.23)

one gets:

~J = − ih2µ

(ϕ∗~∇ϕ− ϕ~∇ϕ∗

)(1.24)

For a set on N particles, the particle flow through the surface is re-lated to the probability current by:

Φ =d2Ndt dΣ

= Nd2P

dt dΣ= N(~J · n) (1.25)

Now, let us consider the incoming beam of particles. Each particlecan be described by a standing plane wave:2 2 It is well known that a plane wave can

be properly normalized only within alimited volume. For instance, for a cu-bic box of side L, one gets A = L−3/2.Since the choice of the normalizationbox is arbitrary, it is important thatphysics observables do not depend onthe normalization constant A. For thisreason, most textbooks omit the nor-malization constant, that is equivalentto the use of a cubic box of side L = 1.

ϕ(~r) = A ei~k·~r (1.26)

with~k = ~p/h. The probability current is:

~Jin =h~kµ

A2 (1.27)

and for a beam of N particles, the particle flow through a surfaceorthogonal to the beam will be:

Φin = N|~Jin| = Nhkµ

A2 (1.28)

Under the action of this plane wave, in analogy with the Huygensprinciple, a scattering center becomes a source of spherical standing

the scattering problem in quantum mechanics 11

waves. The total wave function will be the sum of a plane wave rep-resenting the beam and a spherical wave representing the scatteredparticles. If we also consider feeble interactions that make only asmall fraction of the incoming beam to be scattered out, so that thewave function of the beam is approximately the same before and af-ter the scattering, i.e. for negative and positive ~k ·~r, we expect thetotal wave function to be in the form:

ψ(~r) = A

[ei~k·~r + f (θ, φ)

eikr

r

](1.29)

where f (θ, φ) is called scattering amplitude. The flow of scatteredparticles is given by the probability current of the second term:

A eik·r

A f(θ,ϕ) eikr/r

k

Figure 1.3: Scattering wavefunction as the superpositionof a mostly preserved planewave representing the incom-ing beam (continuous wave-fronts) and a scattered sphericalwave (dashed wavefronts).

Φout = N|~Jout| = Nhkµ

| f (θ, φ)|2r2 A2 (1.30)

Once the scattering amplitude f (θ, φ) is found, it can be used tocalculate the differential cross section for the scattering process. Inparticular, since we are considering the elastic scattering by fixedscattering centers, the kinetic energy of the projectiles will be con-served in the process, so the energy distribution of the outcomingparticles will be trivially a Dirac δ, and we will be only interestedon their angular distribution. Following the definition of the crosssection, we have:

d2Nout

dt dΩ= Φin NB

dσ

dΩ(1.31)

where Nout is the numbers of scattered particles and NB is the num-ber of targets. If we consider a single scattering center, NB = 1. At adistance r, the rate of scattered particles is connected to the flow by:

d2Nout

dt dΩ= r2 d2Nout

dt r2 dΩ= r2 d2Nout

dt dΣ= r2Φout (1.32)

so, from Eq. (1.28), (1.30) and (1.31):

dσ

dΩ= r2 Φout

Φin= | f (θ, φ)|2 (1.33)

Since the process is treated as a steady one, the problem we aregoing to solve consists in demonstrating that the solution of the time-independent Schrödinger equation (1.16) can be put in the form ofEq. (1.29), and solving it to find the f (θ, φ) function, and so the crosssection. For a central potential, with the beam directed along thez axis, the cylindrical symmetry of the problem will also imposef (θ, φ) = f (θ).

1.5 Density of states

An important concept that will be used later in this notes is that ofdensity of states ρ(E). For a quantum system, we define ρ(E) dE asthe number of states of energy between E and E + dE.

In general, in a N-dimensional system, the energy is a functionof N quantum numbers (n1, ..., nN) related to the coordinate system

12 introduction to the quantum theory of particle scattering

is use, e.g. (nx, ny, nz) in a 3-dimensional system in cartesian coor-dinates. For an isotropic system the relation has to be spherically

symmetric, so for instance E ∝ n =√

n2x + n2

y + n2z in the example

above, and ρ(E) dE = dnx dny dnz.

If we take in particular the case of a plane wave, ϕ(~r) = A ei~k·~r,normalized on a cubic box of side L (i.e. A = L−3/2), we know that:

E =h2k2

2m, ~k =

(2πnx

L,

2πny

L,

2πnz

L

)(1.34)

so:

ρ(E) dE = dnx dny dnz =L3

(2π)3 d3~k (1.35)

that is often written, for a unit-volume box (L = 1) in natural units(h = c = 1) as:

ρ(E) dE =d3~p(2π)3 (1.36)

and also:

ρ(E) dE =L3

(2π)3 k2 dk dΩ (1.37)

ρ(E) =L3

(2π)3 k2 dkdE

dΩ =m k L3

8π3 h2 dΩ (1.38)

2The Lippmann-Schwinger equation

In Chapther 1 we have reduced the problem of the elastic scatter-ing between two particles to the resolution of a time-independentSchrödinger equation, with the wave function in the form of Eq. (1.29).Here we will demonstrate that the solution of the Schrödinger equa-tion for a scattering problem assumes indeed this form, and we willextract a formal expression for the scattering amplitude f (θ, φ). Inthe next chapter, a series expansion of this formal expression will begiven and the first-order solution will be discussed.

2.1 Integral form of the Schrödinger equation

In the scattering problem we have to solve the Schrödinger equationfor a particle: [

−h2∇2

2µ+ V(~r)

]ψ(~r) = E ψ(~r) (2.1)

with the boundary condition that the wave function is a plane waveat large distances from the scattering center. First of all, let us writethe equation in the form:[

∇2 + k2]

ψ(~r) =2µ

h2 V(~r)ψ(~r) (2.2)

The general solution of this equation can be written as:

ψ(~r) = ϕ(~r) + ψSC(~r) (2.3)

where ϕ(~r) is the general solution of the corresponding homoge-neous equation, that is the Helmotz equation

[∇2 + k2] ϕ(~r) = 0,

while ψSC(~r) is a particular solution of the inhomogeneous equation.Such a solution can be found with the Green’s function method. TheGreen’s function G0(~r,~r ′) for the Helmotz operator, i.e. the solutionof: [

∇2 + k2]

G0(~r,~r ′) = δ(~r−~r ′) (2.4)

is given by:

G0(~r,~r ′) = − 14π|~r−~r ′| e

ik|~r−~r ′ | (2.5)

A particular solution of Eq. (2.2) is hence:

ψSC(~r) =∫

G0(~r,~r ′)2µ

h2 V(~r ′)ψ(~r ′) d3~r ′ (2.6)

14 introduction to the quantum theory of particle scattering

and finally:

ψ(~r) = ϕ(~r)− 14π

2µ

h2

∫ d3~r ′

|~r−~r ′| eik|~r−~r ′ | V(~r ′)ψ(~r ′) (2.7)

This is an integral form of the Schrödinger equation, known as theLippmann-Schwinger equation.

2.2 The Lippmann-Schwinger equation at large distances

We are going to compare the results of our calculations with exper-iments where the outcoming particles are detected at large distancefrom the scattering centers, where V(~r) → 0 and we observe againfree particles, with the same energy of the incoming ones (the scat-tering is elastic) and only a different direction, represented by thewave vector~k ′ = k (~r/r).

zθk

k'r

D

V(r)

Figure 2.1: A particle with in-coming wave vector~k, scatteredelastically by a potential V(~r)acting over a limited regionof space, and observed withoutcoming wave vector ~k ′ =

k (~r/r) by a detector D at posi-tion~r .

In other words, we are interested to the form of the wave functionat |~r| much larger than the distances |~r ′| for which V(~r ′) is signifi-cantly different from zero. It means that we can consider the integralin Eq. (2.7) to get significant contributions only over a volume where|~r ′| |~r|. Under this assumption:

|~r−~r ′| = (r2 + r′2 − 2~r ·~r ′) 12 =

= r

(1 +

(r′

r

)2

− 2~r ·~r ′r2

) 12

∼

∼ r(

1− 2~r ·~r ′r2

) 12

∼

∼ r(

1−~r ·~r ′r2

)= r−~r ·~r ′

r(2.8)

1|~r−~r ′| ∼

1r

(2.9)

and Eq. (2.7) becomes:

ψ(~r) = ϕ(~r)− 14π

2µ

h21r

eikr∫

e−i kr~r·~r ′ V(~r ′)ψ(~r ′)d3~r ′ (2.10)

The second term goes to zero as r → ∞. So, the boundary conditionof having a plane incoming wave at large distances can be satisfiedby choosing, among the solutions of the Helmotz equation, the planewave:

ϕ(~r) = A ei~k·~r (2.11)

The solution of the Lippmann-Schwinger equation for the scatteringproblem at large distances is then in the form:

ψ(~r) = A ei~k·~r − 14π

2µ

h21r

eikr∫

e−i~k ′ ·~r ′ V(~r ′)ψ(~r ′)d3~r ′ (2.12)

= A

[ei~k·~r + f (θ, φ)

eikr

r

](2.13)

with:f (θ, φ) = − µ

2πh2 A

∫e−i~k ′ ·~r ′ V(~r ′)ψ(~r ′)d3~r ′ (2.14)

the lippmann-schwinger equation 15

We demonstrated that, when observed at large distances from thescattering center, the wave function is indeed in the form of Eq. (1.29).Anyway, this is still an integral equation to be solved. We will discussan approximate solution in the next chapter.

3The Born approximation

3.1 First-order Born approximation

Let us consider feeble interactions, such that only a small fractionof the beam is scattered. We expect the total wave function ψ(~r) todiffer only slightly from the incoming wave ϕ(~r). So, at a first orderof approximation, we can replace ψ(~r) with ϕ(~r) in Eq. (2.7):

ψ(~r) = ϕ(~r)− 14π

2µ

h2

∫ d3~r ′

|~r−~r ′| eik|~r−~r ′ | V(~r ′) ϕ(~r ′) (3.1)

This is called Born approximation and, at large distances, it allows toget a solution of the scattering problem as:

ψ(~r) = A ei~k·~r − A4π

2µ

h21r

eikr∫

e−i~k ′ ·~r ′ V(~r ′) ei~k·~r ′d3~r ′ (3.2)

If we write:

f (θ, φ) = − µ

2πh2

∫ei~q·~r ′ V(~r ′) d3~r ′ ≡ V(~q) (3.3)

where h~q = h(~k−~k ′) is the momentum transfer, we see that the scat-tering amplitude is simply the Fourier transform of the potential,defined in the momentum-transfer domain. It triggers an interestinganalogy with the use of the Fourier transform in signal or image pro-cessing. If one wants to reconstruct a time series from its frequencyspectrum, higher the frequencies that are used finer the time inter-vals that can be studied. Similarly here, if one wants to study thestructure of the potential at smaller distances, one has to measurethe scattering amplitude (i.e. the cross section) at larger momentumtransfer, that implies in turns a higher momentum of the projectiles.This is the reason why, if we want to explore smaller distances withscattering experiments, we need beams with higher energies. Therule of thumb is that, for projectiles of momentum p, one cannot getany information about distances lower than h/p. This result can bealso seen as a consequence of the Heisenberg uncertainty principle.To give a quantitative clue, since the size of nuclei is on the scale ofthe femtometer (1 fm = 10−15 m), in order to study the nuclear struc-ture with electron scatterings one needs energies significantly largerthan hc/(1 fm) ∼ 200 MeV.

18 introduction to the quantum theory of particle scattering

3.2 The Born series

The Born approximation can be regarded as the first term of a series(the Born serires) built by replacing iteratively ψ in the integral termof Eq. (2.7) with its expression from the equation itself.

ψ(~r) = ϕ(~r) +2µ

h2

∫G0(~r,~r ′)V(~r ′)ψ(~r ′) d3~r ′

= ϕ(~r) +2µ

h2

∫G0(~r,~r ′)V(~r ′) ϕ(~r ′) d3~r ′ +

+2µ

h2

∫∫G0(~r,~r ′)V(~r ′)

2µ

h2 G0(~r ′,~r ′′)V(~r ′′) ϕ(~r ′′) d3~r ′ d3~r ′′ + . . . (3.4)

and, at large distance:

ψ(~r) = A ei~k·~r − A4π

2µ

h21r

eikr∫

e−i~k ′ ·~r ′ V(~r ′) ei~k·~r ′d3~r ′

− A4π

2µ

h21r

eikr∫∫

e−i~k ′ ·~r ′ V(~r ′)2µ

h2 G0(~r ′,~r ′′)V(~r ′′) ei~k·~r ′′ d3~r ′ d3~r ′′ + . . . (3.5)

The scattering amplitude becomes:

f (θ, φ) = − µ

2πh2 A·[∫

e−i~k ′ ·~r ′ V(~r ′) ei~k·~r ′d3~r ′ +

+∫∫

e−i~k ′ ·~r ′ V(~r ′)2µ

h2 G0(~r ′,~r ′′)V(~r ′′) ei~k·~r ′′ d3~r ′ d3~r ′′ + . . .]

(3.6)

Due to the assumed weakness of the interaction, terms with more Vterms are smaller and smaller, and the series can be truncated whenthe contributions become smaller than the required accuracy of thecalculation.

3.3 An introduction to Feynman diagrams

V(r')

V(r'')

V(r')exp(-ikr'') exp(-ik'r')

exp(ikr') exp(-ik'r')

Figure 3.1: Graphical represen-tation of the Born series.

We can give an intuitive graphical description of the series (3.6), seethe diagrams in Fig. 3.1. The first term of the series describes acase without interaction (top diagram); the second term describescases where there is a single interaction at point ~r ′, and one sumsover all possible values of~r ′ by taking the integral (middle diagram);the third term describes cases where there are two interactions, at~r ′ and ~r ′′, and again one integrates over all possible values (downdiagram), and so on. For each graph there are the incoming andoutcoming plane waves contributing with ei~k·~r terms (external legs),the interaction vertices contributing with factors V (couplings) and2µ

h2 G0 factors that "propagate" the particle through different internalpoints (propagators).

A similar formalism is commonly used in relativistic quantummechanics, where the amplitude for a process can be described andcalculated in terms of Feynman diagrams. For each kind of force thereis a set of rule to build diagrams connecting external legs that rep-resent the incoming and outcoming particles. Once the diagram is

the born approximation 19

built, one can associate to each element (external legs, propagators,vertices) a term in the amplitude expression, and calculate it. Therules to convert diagrams into mathematical expressions in relativis-tic quantum mechanics go beyond the scope of these notes. Anyway,it can be useful to introduce the rules to build the diagrams them-selves, because they provide a useful tool to understand what canhappen in the different interactions of particles.

γ

X± X±

Figure 3.2: The electromagneticvertex for a charged particleX±.

Feynman diagrams are drawn in a two-dimensional frame, whereone axis (usually the horizontal one) represents the time and theother axis represents space coordinates. Instead of couplings withpotentials, only couplings between particles are considered, and ananti-particle moving forward in time is equivalent to a particle mov-ing backward in time, and vice versa.

For electromagnetic interactions, the only admitted coupling isthe vertex formed by a charged particle with a photon γ (Fig. 3.2).If one consider for instance the elastic scattering between an electronand a muon in Fig. 3.3, the simplest diagram that can be consideredis (a), where the interaction happens through the exchange of onephoton. Adding more vertices, one can consider the set of diagrams(b), and so on. In the scattering between an electron and a positron(Bhabha scattering), more diagrams arise because of the equivalenceof positrons with electrons moving in the opposite time direction(Fig. 3.4). Each vertex contributes to the amplitude with a factorproportional to the fine structure constant α ∼ 1/137, so that thecontribution of diagrams with more vertices are suppressed. Hence,the simplest diagrams give the leading order approximation, andmore complex diagrams give higher order contributions.

µ− µ−

γ

e− e−

(a)

µ− µ−

γ

e− e−γ

µ− µ−γ

γ

e− e−

µ− µ−

γ

γ

e− e−

µ− µ−

γ

e− e−

γ

µ− µ−

γ

e− e−

γ

(b)

Figure 3.3: Leading order (a)and next-to-leading order (b)diagrams for the e−µ− → e−µ−

scattering by electromagneticinteractions.

20 introduction to the quantum theory of particle scattering

e+

e−

γ

e+

e−

(a)

e+

e−

γ

e+

e−

γ

e+

e−

γ

e+

e−

γ

e+

e−

γ γ

e+

e−

e+

γ

e+

e−

γ

e−

e+

γ

e+

e−

γ

e−

(b)

Figure 3.4: Additional dia-grams contributing to the thee+e− → e+e− scattering byelectromagnetic interactions atleading order (a) and next-to-leading order (b).

For weak interactions, the vertices with the W± and Z0 bosons areintroduced (Fig. 3.5). Notice that conservation laws (e.g charge con-servation) apply to any single vertex. The W± vertex can change theflavour of the quarks, not the flavour of the leptons. The Z vertex donot change the flavour. The coupling in the W± and Z0 vertices is ofthe same order of the electromagnetic coupling, but the W± and Z0

propagators produce factors O(q2/m2W,Z). Since mW ∼ 80 MeV/c2

and mZ ∼ 91 MeV/c2, the amplitudes are highly suppressed if themomentum transfer is . 100 GeV.

W

u (`) d (ν`)

Z

f f

W

W Z(γ)

W

W W (Zγ)

W (Zγ)

Figure 3.5: The weak vertices.The label u indicates a up-likequark (u, c, t), d a down-likequark (d, s, b), ` a charged lep-ton (e, µ, τ) ν` the correspond-ing neutrino, and f a genericfermion.

the born approximation 21

Finally, strong interactions involve the vertex between a quark anda gluon g and between gluons (Fig. 3.6). It should be noticed that, inthis case, the coupling constant that plays the same role of α is largerthan 1 if the momentum transfer is below ΛQCD ∼ 300 MeV/c. Itmeans that in the low-energy regime the approach of interruptingthe series at some order does not work and Feynman diagrams areonly useful to get a clue of the process.

g

q q

g

g g

g

g g

g

Figure 3.6: The strong vertices.The label q indicates a genericquark.

4Cross section calculation with examples

4.1 The Fermi’s golden rule

From Eq. (3.2) we can write:

f (θ, φ) = − µ

2πh21

A2

∫Ae−i~k ′ ·~r ′ V(~r ′) Aei~k·~r ′ d3~r ′ ≡

≡ − µ

2πh21

A2

⟨ϕ f

∣∣∣V∣∣∣ϕi

⟩(4.1)

whereM f i ≡⟨

ϕ f

∣∣∣V∣∣∣ϕi

⟩is called the transition matrix element of the

interaction V between the initial and final states ϕi and ϕ f . Hence:

dσ

dΩ= | f (θ, φ)|2 =

µ2

4π2h41

A4

∣∣∣M f i

∣∣∣2 (4.2)

If we consider the scattering by a single target particle, from thedefinition of cross section and from Eq. (1.33) we get:

dNout

dt dΩ= Φin

dσ

dΩ= N

h|~k|µ

A2 µ2

4π2h41

A4

∣∣∣M f i

∣∣∣2 =

= N2π

h

(m k L3

8π3 h2

) ∣∣∣M f i

∣∣∣2 = N2π

hρ(E)dΩ

∣∣∣M f i

∣∣∣2 (4.3)

So, the scattering rate per incoming particle, into the solid angle ele-ment dΩ, is:

Γ f i = dΩ1N

dNout

dt dΩ=

2π

h

∣∣∣M f i

∣∣∣2 ρ(E)

This equation, that is know as the Fermi’s Golden Rule, has indeeda wider validity, that emerges when quantum transitions from aninitial to a final state are studied in a time-dependent quantum per-turbation theory. In general, one can say that the probability per unittime for a quantum transition from an initial state |i〉 to the final state | f 〉as an effect of the interaction Hamiltonian HI is given by:

Γ f i =2π

h

∣∣∣M f i

∣∣∣2 ρ(E) =2π

h| 〈 f |HI |i〉|2 ρ(E) (4.4)

24 introduction to the quantum theory of particle scattering

4.2 Scattering by the Yukawa potential and the Rutherfordcross section

As a first example, let us consider the scattering of a particle by apotential in the Yukawa form:

V(r) = V0e−αr

r(4.5)

In the Born approximation, from Eq. (3.2) and taking the polar axisfor spherical coordinates in the primed space parallel to ~q, we get:

f (θ, φ) = − µ

2πh2

∫ 2π

0dφ′

∫ 1

−1d cos θ′

∫ ∞

0dr′ r′2 eiqr′ cos θ′V(r′) =

= −2µ

h2

∫ ∞

0dr′ r′2V(r′)

eiqr′ − e−iqr′

iqr′= − 2µ

h2q

∫ ∞

0V(r′) sin

(qr′)r′dr′ = (4.6)

= −2µV0

h2q

∫ ∞

0e−αr′ sin qr′dr′ = −2µV0

h21

q2 + α2 (4.7)

On the other hand, q2 = |~k−~k ′|2 = 2k2(1− cos θ), so:

dσ

dΩ= | f (θ, φ)|2 = | f (θ)|2 =

(2µV0

h2

)1

[2k2(1− cos θ) + α2]2 (4.8)

Notice that Eq. (4.6) is a general form for the scattering amplitudein a central potential, f (θ) (notice also the dependence on the sole θ

angle).The Coullomb potential is a special case of Yukawa potential, with

α = 0 and V0 = zZe2/(4πε0). In this case we get:

(dσ

dΩ

)Ruth

= | f (θ, φ)|2 = | f (θ)|2 =

(zZe2

4πε0

)2 1

16(

h2k2

2µ

)2sin4 θ

2

= (4.9)

=

(zZe2

4πε0

)2 116E2 sin4 θ

2

(4.10)

that is the well known Rutherford cross section for the scattering of acharged particle by a point-like charged target.

4.3 Scattering by a charge distribution: the form factor

The Rutherford cross section can be generalise d considering anextended target with a known electric charge density distributionZeρ(~r). In this case, the potential is given by:

V(~r) =zZe2

4πε0

∫ρ(~r ′′)|~r−~r ′′| d3~r ′′ (4.11)

and we get:

f (θ, φ) = − µ

2πh2zZe2

4πε0

∫d3~r ′ei~q·~r ′

∫d3~r ′′

ρ(~r ′′)|~r ′ −~r ′′| = (4.12)

= − µ

2πh2zZe2

4πε0

∫d3~r ′′ρ(~r ′′) ei~q·~r ′′

∫d3~r ′

ei~q·(~r ′ ~r ′′)

|~r ′ −~r ′′| (4.13)

cross section calculation with examples 25

With a change of variables, the second integral is the same that wesolved in Eq. (4.7) with α = 0. It means that the scattering amplitudeis the same as the Rutherford one, multiplied by:

F(q2) =∫

d3~r ′′ρ(~r ′′) ei~q·~r ′′ (4.14)

that is called form factor and corresponds to the Fourier transform ofthe charge distribution. Notice the dependence on the sole magni-tude of q, that is evident if one writes the integral in spherical coor-dinates with the polar axis parallel to ~q. For a spherically symmetriccharge distribution, it simplifies to:

F(q2) =4π

q

∫r′′ρ(r′′) sin

(qr′′)dr′′ (4.15)

It follows that, for an extended charge distribution:

dσ

dΩ=

(dσ

dΩ

)Ruth

∣∣∣F(q2)∣∣∣ (4.16)

i.e. the cross section for the scattering by a point-like charge, multi-plied by the squared modulus of the form factor.

If we consider for instance a charge uniformly distributed over asphere of radius R:

ρ(~r) =1

4πR3/3for r ≤ R ; ρ(~r) = 0 for r > R (4.17)

the form factor, with some calculation starting from Eq. (4.15), is:

F(q2) = 3(

1qR

)3

[qR cos(qR)− sin(qR)] (4.18)

This form factor, and hence the cross section, goes to zero as tan(qR) =qR. Since q is related to the scattering angle by q = 2k sin θ/2, itmeans that scattering probability has a sequence of maxima and min-ima as a function of θ, as in the diffraction of a classical wave by asperical obstacle. For a given q, the position of the minima gives ameasurement of the radius R. If the scattering of charged particlesby nuclei is considered, the charge distribution does not have a sharpedge as in this example. Nonetheless, the diffraction-like structure isstill observable, as shown in Fig. 4.1, and can be used to define thecharge radius of a nucleus.

Figure 4.1: Differential crosssection for the scattering of757 MeV electrons by 40Ca and48Ca. The diffraction-like struc-ture is visible. The curves areobtained assuming an uniformcharge distribution extendingup to a suitable radius R, witha smoothed edge around R.

5Partial waves and resonances

5.1 Partial wave analysis

5.2 Resonances

6Discrete symmetries

7Electromagnetic transitions