decays, resonances and scatteringbarra/teaching/resonances.pdf · 2012. 2. 3. · decays,...

Decays, resonances and scattering

1 Structure of matter and energy scales

Subatomic physics deals with objects of the size of the atomic nucleus andsmaller. We cannot see subatomic particles directly, but we may obtain knowl-edge of their structures by observing the effect of projectiles that are scatteredfrom them. The resolution any such probe is limited to of order the de Brogliewavelength,

λ =h

p. (1)

If we wish to resolve small distances, smaller than the atom, we will need to doso with probes with high momenta. Typical sizes and are given below.

Object Size Binding energy

Atom 10−10 m ∼ eV

Nucleus ∼ 10−15 m ∼ MeV

Quark < 10−18 m > TeV

We can see that small objects also tend to have high binding energies, and henceprobes 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, suchas radioactive nuclei, or cosmic muons. In our quantum mechanics course weused time-dependent perturbation theory to show that that the transition rateof an unstable state into a continuum of other states was given by the FermiGolden Rule:

Γ =2π

~|Vfi|2

dN

dEf

. (2)

1


The density of states within a cubic box with sides length a can be calculatedas 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 ofkx are constrained to the values kx = 2πn/a. Similar conditions hold for ky

and kz. The number of momentum states within some range of momentumd3p = d3k is given by

dN =d3p

(2π)3V

where V = a3 is the volume of the box.

• Γ is the rate of the decay

• Vfi is the matrix element of the Hamiltonian coupling the initial and thefinal states

• dNdEf

is the density of final states.

Fermi G.R. example: consider the isotropic decay of a neutral spin-0 particleinto two massless daughters

A→ B + C.

The Fermi G.R. gives the decay rate as

Γ = 2π|Vfi|2dN

dEf

= 2π|Vfi|24πp2

B

(2π)3dpB

dEfV.

Since all decay angles are equally probable, the integrals over the angles con-tributes 4π. The decay products have momentum |pB| = Ef/2 so dpB

dEf= 1

2 .

Normalising to one unstable particle per unit volume gives V = 1, and resultsin a decay rate

Γ =12π|Vfi|2p2

B

=18π|Vfi|2m2

A.

February 3, 2012 2 c© A.J.Barr 2009-2011.


The number of particles remaining at time t is governed by the decay law

dN

dt= −ΓN

Which integrates to give

N(t) = N0 exp (−Γt) .

We can easily calculate the particles’ average proper lifetime τ , using the prob-ability that they decay between time t and t+ δt

p(t) δt = − 1

N0

dN

dtδt = Γ exp (−Γt) δt.

The mean lifetime is then

τ = 〈t〉

=

∫ ∞0t p(t) dt∫ ∞

0p(t) dt

=1

Γ

Don’t forget that if such particles are travelling relativistically, their decayclocks will be time-dilated, and so the average number remaining after somedistance x will be

N = N0 exp (−x/L0)

where L0 = βγcτ .

The time-energy uncertainty relationship of quantum mechanics tells us that if astate 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 oforder τ . We therefore expect that its energy is only defined up to an uncertainty

∆E ∼ ~τ

= ~Γ.

Using the fact that Γ = 1/τ , and moving to natural units to set ~ = 1, we findthat

∆E ∼ Γ. (3)



In natural units, the uncertainty in the rest-energy of a particle is equal to therate 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 widthof order Γ.

We can make our statement about the smearing of the energy more precise. Ifwe require that the probability of finding the particle decays with rate Γ, thenthe amplitude must satisfy

|a(t)|2 = exp(−Γt)

The appropriate form for the amplitude is then

a(t) = exp(−i(E − E0)t− Γ/2t).

When Fourier transformed into the energy domain, the probability density func-tion for finding the particle with energy E is

p(E) ∝ 1

(E − E0)2 − (Γ/2)2. (4)

E is the energy of the system, and E0 is the characteristic rest-mass of theunstable particle. The probability density function has a Lorentzian, peaked, lineshape, with full-width at half max (FWHM) of the peak equal to Γ. The quantityΓ is often called the width of the particle.

Long-lived particles have narrow widths and well-defined energies. Short-livedparticles have large widths and less well defined energies. When the state is soshort-lived that its width Γ is similar to its mass, then the decay is so rapid thatit 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 decaysto the final state. This could represent a familiar process such as the absorptionand then emission of a photon by an atom, with an intermediate excited atomic



The decay width can be generalised to a particle which has many differentdecay modes. The rate of decay into mode i is given Γi. The total rate ofdecay is given by

Γ =∑

i=1...n

Γi.

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 thebranching ratio

B =Γi

Γ.

The quantity Γi is known as the partial width to final state i, whereas thesum of all partial widths is known as the total width.

state. It could well equally represent the formation of a very heavy Z0 particlefrom the collision of a high-energy electron with its anti-particle, followed by thedecay of that Z0 particle to a muon and its anti-particle.

Under the condition that the transition from A + B to C + D can proceedexclusively via the intermediate state ‘0’, and that the width of the intermediateis not too large (Γ E0), the cross-section for the process is given by theBreit-Wigner formula

σi→0→f =π

k2

ΓiΓf

(E − E0)2 + Γ2/4. (5)

The terms in this equation are as follows:

G

E0 E

Σ

The Breit-Wigner lineshape.

• Γi is the partial width of the resonance to decay to the initial state A+B

• Γf 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

• E0 is the characteristic rest mass energy of the resonance

• k is the wave-number of the incoming projectile in the centre-of-massframe.

The cross-section is non-zero at any energy, but has a sharp peak at energiesE close to the rest-mass-energy E0 of the intermediate particle. Longer lived



intermediate particles have smaller Γ and hence sharper peaks. The sharp peakis known as resonance. Scattering that proceeds exclusively or dominantly viaa narrow-width intermediate particle will have a cross section shape given by theBreit-Wigner and is known as resonance scattering.

We can see where some of the terms in (5) must come from. The energydependence on the denominator is the same as was found for the line-shapeof an unstable particle (4) 1. By the Fermi GR, the factors of Γi and Γf areproportional to the mod-square of the matrix-elements V0i and Vf0 that areresponsible for the production and decay of the unstable particle, respectively.The 1/k2 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 accountthat for scattering from a single fixed target

σ =W

J

where J is the flux density of incoming particles. The flux density itself isgiven by

J = npv

where np is the number density of projectiles and v is their speed. If wenormalise to one incoming particle per unit volume, then np = 1 and thecross 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 froma potential V (x). Our Hamiltonian is

H = H0 + V.

where H0 is the free-particle kinetic energy operator

H0 =p2

2m.

1Which is just as well, as we are producing and decaying an unstable particle.


4.1 Scattering amplitudes Decays, resonances and scattering

In the absence of V the solutions of the Hamiltonian could be written as the free-particle states satisfying H0|φ〉 = E|φ〉. These free-particle eigenstates could bewritten as momentum eigenstates |p〉, but since that isn’t the only possibility wehold off writing an explicit form for |φ〉 for now. The full Schrodinger equationis

H0 + 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 spectraof both H and H + V are continuous.)

A possible solution is2

|ψ〉 =1

E −H0

V |ψ〉+ |φ〉. (6)

By multiplying by (E − H0) we can show that this looks fine, other than theproblem of the operator 1/(E − H0) being singular. The singular behaviour in(6) can be fixed by making E slightly complex and defining

|ψ(±)〉 = |φ〉+1

E −H0 ± iεV |ψ(±)〉 . (7)

This is the Lippmann-Schwinger equation. We will find the physical meetingof the (±) in the |ψ(±)〉 shortly.

4.1 Scattering amplitudes

To calculate scattering amplitudes we are going to have to use both the positionand the momentum basis, because |φ〉 is a momentum eigenstate, and V isa function of x. If |φ〉 stands for a plane wave with momentum ~k then thewavefunction can be written

〈x|φ〉 =eik·x

(2π)32

.

We can express (7) in the position basis by bra-ing through with 〈x| and insertingthe identity operator

∫d3x′ |x′〉〈x′|

〈x|ψ(±)〉 = 〈x|φ〉+

∫d3x′

⟨x∣∣∣ 1

E −H0 ± iε

∣∣∣x′⟩〈x′|V |ψ(±)〉. (8)

2Remember that functions of operators are defined by f(A) =∑

i f(ai)|ai〉〈ai|. Thereciprocal of an operator is well defined/ provided that its eigenvalues are non-zero.


4.1 Scattering amplitudes Decays, resonances and scattering

The solution to the Green’s function defined by

G±(x,x′) ≡ ~2

2m

⟨x∣∣∣ 1

E −H0 ± iε

∣∣∣x′⟩is

G±(x,x′) = − 1

4π

e±ik|x−x′|

|x− x′|.

Using this result we can see that the amplitude of interest simplifies to

〈x|ψ(±)〉 = 〈x|φ〉 − 1

4π

2m

~2

∫d3x′

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 canbe written as

〈x′|V |x′′〉 = V (x′)δ3(x′ − x′′).

The wave function (9) is a sum of two terms. The first is the incoming planewave. For large |x| the spatial dependence of the second term is e±ikr/r. Wecan now understand the physical meaning of the |ψ(±)〉 states; they representoutgoing (+) and incoming (−) spherical waves respectively. We are interestedin the outgoing (+) spherical waves – the ones which have been scattered fromthe potential.

x

x´

We want to know the amplitude of the outgoing wave at a point x. For practicalexperiments the detector must be far from the scattering centre, so we mayassume |x| |x′|.

We define a unit vector in the direction of the observation point

r =x

|x|

and also a wave-vector for particles travelling in the direction x,

k′ = kr.

Far from the scattering centre we can write

|x− x′| =√r2 − 2rr′ cosα+ r′2

= r

√1− 2

r′

rcosα+

r′2

r2

≈ r − r · x′


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) withjust r, but the phase term will need to be replaced by r − r · x′. So we finallysimplify the wave function to

〈x|ψ(+)〉 r large−−−−→ 〈x|k〉 − 1

4π

2m

~2

eikr

r

∫d3x′ eik′·x′V (x′)〈x′|ψ(+)〉

which we can write as

〈x|ψ(+)〉 =1

(2π)32

[eik·x +

eikr

rf(k,k′)

].

This makes it clear that we have a sum of an incoming plane wave and anoutgoing spherical wave with amplitude f(k′,k) given by

f(k′,k) = − 1

4π(2π)3 2m

~2〈k|V |ψ(±)〉. (10)

〈x|φ〉 ∝ ei|k||x|/|x|

Wave function of anoutgoing spherical wave.

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.

So we find that the partial cross-section dσ – the number of particles scatteredinto a particular region of solid angle per unit time divided by the incident flux3

– is given by

dσ =r2|jscatt||jincid|

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 slightlymodified by V , and so we can replace |ψ(±)〉 in (10) by |k′〉.

f (1)(k′,k) = − 1

4π(2π)3 2m

~2〈k|V |k′〉. (11)

3Remember that the flux is given by j = ~2im [ψ∗∇ψ − ψ∇ψ∗].



This is known as the Born approximation. Within this approximation we havefound the nice simple result

f (1)(k′,k) ∝ 〈k|V |k′〉.

Up to some constant factors, the scattering amplitude is found by squeezing theperturbing potential V between incoming and the outgoing momentum eigen-states of the free-particle Hamiltonian.

Expanding out (11) in the position representation (by insertion of a couple ofcompleteness relations

∫d3x′ |x′〉〈x′|) we can write

f (1)(k′,k) = − 1

4π

2m

~2

∫d3x′ei(k−k′)·x′V (x′).

This result is telling us that scattering amplitude is proportional to the 3d Fouriertransform of the potential. By scattering particles from targets we can measuredσdΩ

, and hence infer the functional form of V (r).

5 Virtual Particles

One of the insights of subatomic physics is that at the microscopic level forces arecaused by the exchange of force-carrying particles. For example the Coulombforce between two electrons is mediated by excitations of the electromagneticfield – i.e. photons. There is no real ‘action at a distance’. Instead the force istransmitted between the two scattering particles by the exchange of some unob-served photon or photons. The mediating photons are emitted by one electronand absorbed by the other. It’s generally not possible to tell which electro emit-ted and which absorbed the mediating photons – all one can observe is the neteffect on the electrons. e

−

e−

e−

γ

e−

Other forces are mediated by other force-carrying particles – we shall meet exam-ples later on. In each case the messenger particles are known as virtual particles.Virtual particles are not directly observed, and have properties different from ‘realparticles’ which are free to propagate.

To illustrate why virtual particles have unusual properties, consider the elasticscattering 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 thatit is approximately stationary. Let the incoming electron have momentum p



and the outgoing, scattered electron have momentum p′. For elastic scattering,the energy of the electron is unchanged E ′ = E. The electron has picked upa change of momentum ∆p = p′ − p from absorbing the virtual photon, butabsorbed no energy. So the photon must have energy and momentum

e−

e−

γ

Eγ = 0

pγ = ∆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,E2

γ 6= p2γ. The fact that Eγ = 0 is special to the case we have chosen. However,

the general result is that for any virtual particle there is an energy-momentuminvariant which is not equal to the square of its mass

P · P = E2 − p · p 6= m2.

Such virtual particles do not satisfy the usual energy-momentum invariant andare said to be ‘off mass shell’.

Note that we would not have been able to escape this conclusion if we hadtaken the alternative viewpoint that the electron had emitted the photon andthe nucleus had absorbed it. In that case the photon’s momentum would havebeen pγ = −∆p. The square of the momentum would be the same, and thephoton’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-mass-shell photons. It is interesting to realise that all of classical electromagnetism isactually 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 scat-tering, which has the form (in natural units)

V (r) =g2e−µr

r. (12)

This is known as the Yukawa potential. When µ = 0 this has the familiar 1/rdependence of the electrostatic and gravitational potentials. When µ is non-zero,



The electromagnetic force is mediated by excitations of the electromagneticforce, i.e. photons. The photon is massless so the electrostatic potential fallsas 1/r with no exponential. Other forces are mediated by heavy particles andso are only effective over a short range. An example is the weak nuclear force,which is mediated by particles with µ close to 100 GeV and so is feeble atdistances larger than about 1/(100 GeV) ≈ (197 MeV fm)/(100 GeV) ∼10−18 m.

the potential also falls off exponentially with r, with a characteristic length of1/µ.

To understand the meaning of the µ term it is useful to consider the relativisticwave equation known as the Klein-Gordan equation(

∂2

∂t2−∇2 + µ2

)ϕ(r, t) = 0. (13)

This is the relativistic wave equation for spin-0 particles. The plane-wave solu-tions to (13) are

φ(X) = A exp (−iP ·X)

= A exp (−iEt+ ip · x) .

These solutions require the propagating particles to be of mass µ =√E2 − p2.

The Klein-Gordon equation is therefore describing excitations of a field of particleseach of mass µ. The Yukawa potential is another solution to the field equation(13). The difference is that the Yukawa potential describes the static solutiondue to virtual particles of mass µ created by some source at the origin.

The constant g2 tells us about the depth of the potential, or the size of the force.It is equivalent to the factor of Q1Q2/(4πε0) in electrostatics.

The scattering amplitude of a particle bouncing off a Yukawa potential is foundto be

〈k′|VYukawa|k〉 = − g2

(2π)3

1

µ2 + |∆k|2. (14)

We can go some way towards interpreting this result as the exchange of a virtualparticle as follows. We justify the two factors of g as coming from the pointswhere a virtual photon is either created or annihilated. This vertex factor g is ameasure of the interaction or ‘coupling’ of the exchanged particle with the otherobjects. There is one factor of g the point of creation of the virtual particle, andanother one at the point where it is absorbed.



In electromagnetism we want each vertex factor to be proportional to thecharge of the particle that the (virtual) photon interacts with. We are inter-ested in a (µ = 0) Yukawa potential of the form

VEM =q1q2e

2

4πε0r.

For scattering from a Coulomb potential we can therefore use the Yukawaresult (14) by making the substitution

g2 ⇒ q1q2e2

4πε0.

In a general scattering process we will want a vertex factor g ∝ qe at eachvertex.

The other important factor in the scattering amplitude (14) is associated withthe momentum and mass of the exchanged particle:

− 1

µ2 + |∆k|2

In general it is found that if a virtual particle of mass µ and four-momentum Pis exchanged, there is a propagator factor

1

P · P− µ2(15)

in the scattering amplitude. This relativistically invariant expression is consistentwith our electron-scattering example, where the denominator was:

P · P− µ2 = E2 − p2 − µ2

= 0− |∆k|2 − µ2

= −(µ2 + |∆k|2

)Note that the propagator (15) becomes singular as the particle gets close to itsmass shell. i.e. as P · P → µ2. It is only because the exchanged particles are offtheir mass-shells that the result is finite.

The identification of the vertex factors and propagators will turn out to be veryuseful when we later try to construct more complicated scattering processes.In those cases we will be able to construct the most important features of thescattering amplitude merely by writing down:



• an appropriate vertex factor each time a particle is either created or anni-hilated and

• a propagator factor for each virtual particle.

By multiplying together these factors we capture the most important propertiesof the scattering amplitude.



Key concepts

• The leading Born approximation to the scattering amplitude is

f (1)(k′,k) ∝ 〈k′|V |k〉 .

• The scattering amplitude is proportional to the 3d Fourier transform ofthe potential.

• The differential cross-section is given in terms of the scattering amplitudeby

dσ

dΩ= |f(k′,k)|2

• Forces are transmitted by virtual mediating particles which are off-mass-shell:

P · P = E2 − p · p 6= m2.

• The Yukawa potential for an exchanged particle of mass µ and couplingg is

V (r) =g2e−µr

r. (16)

• The scattering amplitude contains a vertex factors g for any point whereparticles are created or annihilated

• The scattering amplitude contains a propagator factor

1

P · P− µ2.

for each virtual particle.

• The rate of an interaction is given by the Fermi Golden Rule

Γ =2π

~|Vfi|2

dN

dEf

.

• The Breit-Wigner formula for resonant scattering is

σ(n, γ) =π

k2

ΓnΓγ

(E − E0)2 + Γ2/4.



• In natural units, ~ = c = 1 and

[Mass] = [Energy] = [Momentum] = [Time]−1 = [Distance]−1

• A useful conversion constant is

~c ≈ 197 MeV fm

• The cross section is defined by:

σi =Wi

nJ δx(17)

• The differential cross section is the cross section per unit solid angle

dσi

dΩ

• Cross sections for sub-atomic physics are often expressed in the unit ofbarns.

1 barn = 10−28 m2



A Natural units

We have been used to using units in which times are measured in seconds anddistances in meters. In such units the speed of light takes the value close to3× 108 ms−1.

We could instead have chosen to use unit of time such that c = 1.4 Doing thisallows us to leave c out of our equations, provided we are careful to rememberthe units we are working in. Such units are useful in relativistic systems, sincenow the relativistic energy-momentum-mass relations are

E = γm

p = γmv

E2 − p2 = m2.

So for a relativistic system setting c = 1 means that energy, mass and momentumall have the same dimensions.

Since we are interested in quantum systems, we can go further and look for unitsin which ~ is also equal to one. In such units the energy of a photon will beequal to its angular frequency

E = ~ω = ω.

What quantities does ~ relate? Remember the time-energy uncertainty relation-ship ∆E∆t ∼ ~. Setting ~ = 1 means that time (and hence distance) musthave the same dimensions as E−1.

So in our system natural units we have have that

[Mass] = [Energy] = [Momentum] = [Time]−1 = [Distance]−1

We are going to use units of energy for all of the quantities above. The nuclearenergy levels have typical energies of the order of 106 electron-volts, so we shallmeasure energies and masses in MeV, and lengths and times in MeV−1. At theend of a calculation how can we recover a “real” length from one measured inMeV−1? We can use the conversion factor

~c ≈ 197 MeV fm

which tells us that one of our MeV−1 length units corresponds to 197 fm where1 fm = 10−15 m.

4You will already have used units in which time is measured in seconds and distance inlight-seconds. In those units c = 1, since the speed of light is one light-second per second.



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 constantwithin that target, the rate of any reaction Wi will be proportional to the flux ofincoming projectiles J (number per unit time) the number density of scatteringcentres n (number per unit volume), and the width of the target

Wi = σinJ δx. (18)

The constant of proportionality σi has dimensions of area. It is known as thecross section for process i and is defined by

σi =Wi

nJ δx(19)

We can get some feeling for why this is a useful quantity if we rewrite (18) as

Wi = (nA δx)︸︷︷︸Ntarget

Jσ

A︸︷︷︸Pscatt

where A is the area of the target, which shows that the cross section can beinterpreted as the effective area presented to the beam per target for which aparticular reaction can be expected to occur.

The total rate of loss of beam is given by W = ΣWi, and the correspondingtotal cross section is

σ =∑

i

σi.

We could chose to quote cross sections in units of e.g. fm2 or GeV−2, howeverthe most common unit used in nuclear and particle physics is the so-called barnwhere

1 barn = 10−28 m2

We can convert this to MeV−2 units using the usual ~c conversion constant asfollows

1 barn = 10−28 m2

= 100 fm2/(197 MeV fm)2

= 0.00257 MeV−2.



C Luminosity

In a collider – a machine which collides opposing beams of particles – therate of any particular reaction will be proportional to the cross section for thatreaction and on various other parameters which depend on the machine set-up.Those parameters will include the number of particles per bunch, their spatialdistribution, and the frequency of the collision of those bunches.

We call the constant of proportionality which encompasses all those machineeffects the luminosity

L =W

σ.

It has dimensions of [L]−2[T ]−1 and is useful to factor out if you don’t care aboutthe 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 countof the events of any type

Nevents, i = σi

∫L dt .

For a machine colliding opposing bunches containing N1 and N2 particles at ratef , you should be able to show that the luminosity is

L =N1N2f

A,

where A is the cross-sectional area of each bunch (perpendicular to the beamdirection).

We’ve assumed above that the distributions of particles within each bunch isuniform. If that’s not the case (e.g. in most real experiments the beams haveapproximately Gaussian profiles) then we will have to calculate the effective over-lap area A of the bunches by performing an appropriate integral.

D Beyond Born: propagators in non-relativistic scatter-ing theory

Non-examinable



Previously we examined the leading Born term in the non-relativistic scatteringtheory. 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 forT

T |φ〉 = V |φ〉+ V1

E −H0 + iεT |φ〉.

Since this is to be true for any |φ〉, the corresponding operator equation mustalso be true:

T = V + V1

E −H0 + iεT.

This operator is defined recursively. It is is exactly what we need to find thescattering amplitude, since from (10), the amplitude is given by

f(k′,k) = − 1

4π

2m

~2(2π)3〈k′|T |k〉.

We can now find an iterative solution for T :

T = V + V1

E −H0 + iεV + V

1

E −H0 + iεV

1

E −H0 + iεV + . . . (20)

We can interpret this series of terms as a sequence of the operators correspondingto the particle interacting with the potential (operated on by V ) and propagatingalong for some distance (evolves according to 1

E−H0+iε).

Vk

k´

V

k

k´

V

1/(E-H +i )0 e

The operator1

E −H0 + iε(21)

is the non-relativistic version of the propagator. The propagator can be seen tobe a term in the expansion (20) which is giving a contribution the amplitudefor 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 inthe position representation (8).

We are now in a position to quantify what we meant by a ‘weak’ potential earlieron. From the expansion (20) we can see that the first Born approximation (11)will be useful if the matrix elements of T can be well approximated by its firstterm V .



When is this condition likely to hold? Remember that the Yukawa potentialwas proportional to the square of a dimensionless coupling constant ∝ g2. Ifg2 1 then successive applications of V introducing higher and higher powersof g and can usually be neglected. This will be true for electromagnetism, sincethe dimensionless coupling relevant for electromagnetism is related to the finestructure constant

g2 = α =e2

4πε0~c≈ 1

137.

Since α 1, we can usually get away with just the first term of (20) for electricinteractions (i.e. we can use the Born approximation).


decays, resonances and scatteringbarra/teaching/resonances.pdf · 2012. 2. 3. · decays,...

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