Introduction to Particle Physics I

Risto Orava Spring 2016

decay rates and cross sections

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outline •  Lecture I: Introduction, the Standard Model •  Lecture II: Particle detection •  Lecture III: Relativistic kinematics •  Lecture IV: Non-relativistic quantum

mechanics •  Lectures V: Decay rates and cross sections •  Lecture VI: The Dirac equation •  Lecture VII: Particle exchange •  Lecture VIII: Electron-positron annihilation

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outline continued... •  Lecture IX: Electron-proton elastic

scattering •  Lecture X: Deeply inelastic scattering •  Lecture XI: Symmetries and the quark

model •  Lecture XII: Quantum Chromodynamics •  Lecture XIII: The Weak Interaction •  Lecture XIV: Electroweak unification •  Lecture XV: Tests of the Standard Model •  Lecture XVI: The Higgs boson

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•  Fermi’s golden rule •  Phase space and wavefunction normalisation

•  Lorentz invariant phase space •  Fermi’s golden rule revisited •  Lorentz invariant phase space

•  Particle decays •  Two body decays

•  Interaction cross sections •  Lorentz invariant flux •  Scattering in the centre-of-mass frame

•  Differential cross sections •  Differential cross section calculations •  Laboratory frame differential cross sections

Lecture V; Decay rates and cross sections

4 Ref. Mark Thomson, Modern particle Physics

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•  in particle physics experiments, collisions between particle states are quantum mechanical transitions - per unit of time - from initial states to a set of final states - transitions - are measured.

•  to 1st order perturbations, the transition probability is given by

•  Tfi is the transition matrix element and ρ(Ει) is the density of final states – number of continuum states per unit energy

•  the transition matrix element is determined by the Hamiltonian for the interaction which causes the transitions, to lowest order:

f f H ' ii

Fermi’s golden rule

Γ fi = 2π Tfi2ρ(Ei ) where, to first order, Tfi = f H ' i

Tfi = f H ' i +f H ' k k H ' i

(Ei −Ek )k≠i∑ ...

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ρ(Ei ) =dndE Ei

Fermi’s golden rule

•  the transition rate depends on the density of states: •  dn is the number of accessible states within E->E+dE. •  using the Dirac delta function, density of states written as an integral over all final state energies: dn

dE Ei

=dndE∫ δ(Ei −E)dE

Γ fi = 2π Tfi∫2δ(Ei −E)dn

transition rates: 1)  transition matrix element –

dynamics

2)  density of accessible states - kinematics

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Phase space and wave function normalisation

•  QM: a è 1 +2, to 1st order, matrix element (Fermi’s golden rule) •  a small perturbation => the initial & final state is represented by a plane wave, A = normalization factor over volume, V •  for probability density ρ = ψ*ψ this reads: •  the normalization constant is given as: •  a particle in a box satisfies the periodic boundary conditions: •  and imply that the momentum components are quantised to (ni’s are integers):

Tfi = Ψ1Ψ2 H ' Ψa = Ψ1*

V∫ Ψ2*H 'Ψad

3x

Ψ(!x, t) = Aei(!p⋅!x−Et )

0

a

∫0

a

∫ Ψ*

0

a

∫ Ψdxdydz =1

A2 =1/ a2 =1/V

Ψ(x + a, y, z) =Ψ(x, y, z),...

(px, py, pz ) = (nx,ny,nz )2πa

V

pz

periodic boundary conditions: wave function zero at boundaries of the box

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Phase space and wave function normalisation

•  allowed momentum states restricted to a discrete set (figure), each state in the momentum space occupies a cubic volume of

•  no. of states dn with p->p+dp, is equal to the momentum space volume of the spherical cell

at momentum p with thickness dp divided by the average volume occupied by a single state and •  the density of states in Fermi’s golden rule is then obtained by: •  corresponds to the no. of momentum states available in a given decay.

p y

px

d3 !p = dpxdpydpz =2πa

⎛

⎝⎜

⎞

⎠⎟3

=(2π )3

V

dn = 4π p2dp× V(2π )3

dndp

=4π p2

(2π )3V

ρ(E) = dndE

=dndp

dpdE

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Phase space and wave function normalisation

•  the calculated decay rate does not depend on the normalisation volume; the volume dependence is cancelled by the factors of V in the wavefunction normalisations (in the square of the matrix element). •  normalisation to one particle per unit volume: V=1, the number of available states is: •  for a particle decay into N particle final states, N-1 independent momenta, and •  by including the momentum space volume element for the Nth particle, and the Kronecker δ-

function: •  pa is the momentum 3-vector of the decaying particle; the non-relativistic N-body phase space is:

dni =d3 !pi(2π )3

dn = dnii=1

N−1

∏ =d3 !pi(2π )3i=1

N−1

∏

dn = d3 !pi(2π )3i=1

N−1

∏ δ3!pa −

!pii=1

N

∑⎡

⎣⎢

⎤

⎦⎥d3!pN

dn = (2π )3 d3 !pi(2π )3i=1

N

∏ δ3!pa −

!pii=1

N

∑⎡

⎣⎢

⎤

⎦⎥

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Lorentz invariant phase space

•  relativistic particles: the box is Lorentz-contracted by 1/γ in the direction of motion => original normalisation changes by γ = E/m particles per unit volume in the Lorentz-boosted case

•  Lorentz-invariant wavefunction normalised to E particles per unit volume: increase in energy accounts for the Lorentz contraction – 2E particles per unit volume adopted for normalisation

•  è è

•  wavefunctions are normalised to one particle per unit volume:

•  in general, for a+b+...=> 1 + 2 + ... the Lorentz-invariant matrix element is defined as:

•  the Lorentz-invariant matrix element is related to the transition matrix element of Fermi’s golden rule (all initial and final state particles are included on the right hand side):

a

a/γ

v=βc

Ψ*V∫ Ψd3x =1 changed to Ψ*

V∫ Ψd3x = 2E and Ψ ' = (2E)1/2Ψ

M fi = Ψ1'Ψ2

' ⋅ ⋅ ⋅ H ' Ψa' Ψb

'

M fi = Ψ1'Ψ2

' ⋅ ⋅ ⋅ H ' Ψa' Ψb

' = (2E1 ⋅2E2 ⋅ ⋅ ⋅2Ea ⋅2Eb ⋅ ⋅⋅)1/2Tfi

proton gets Lorentz-flattened as well

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Fermi’s golden rule once again..

•  for a two-body decay a -> 1 + 2

•  and

•  by using the relation for the Lorentz-invariant matrix element

•  when using Lorentz-invariant normalisation, the phase space integral over d3p/(2π)3 gets replaced by d3p/(2π)32E – the Lorentz-invariant phase space factor

•  prove this!

Γ fi = 2π Tfi∫2δ(Ea −E1 −E2 )dn

Γ fi = (2π )4 Tfi∫2δ(Ea −E1 −E2 )δ3( !pa −

!p1 −!p2 ) d

3 !p1

(2π )3d3 !p2

(2π )3

Γ fi =(2π )4

2Ea

M fi∫2δ(Ea −E1 −E2 )δ3( !pa −

!p1 −!p2 ) d3 !p1

(2π )32E1

d3 !p2

(2π )32E2

with M fi2= (2Ea2E12E2 ) Tfi

2

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Lorentz-invariant phase space

dLIPS = d3 !pi(2π )32Eii=1

N

∏

δ(Ei2∫ −!pi2 −m2 )dEi =

12Ei

⋅ ⋅ ⋅dLIPS∫ = ⋅ ⋅ ⋅ (2π )−3i=1

N

∏∫ δ(Ei2 −!pi2 −mi

2 )d3 !pidEi

⋅ ⋅ ⋅dLIPS∫ = ⋅ ⋅ ⋅ (2π )−3i=1

N

∏∫ δ( !pi2 −mi

2 )d 4pi

Γ fi =(2π )4

2Ea

(2π )−6∫ M fi2δ 4 (pa − p1 − p2 )δ(p1

2 −m12 )δ(p2

2 −m22 )d 4p1d

4p2

•  decay rate for a particle a decaying into N particles: a -> 1+2+...+N - the phase space integral has •  dLIPS is known as the element of Lorentz-invariant phase space (LIPS); by imposing energy-

momentum conservation •  the integral over Lorentz-invariant phase space can be written as: •  in terms of the four-momenta of the final state particles: •  the transition rate for the two-body decay: a -> 1 + 2, is: •  the integral is over all values of the energies and momenta in the final state.

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particle decays

•  the transition rate for a given decay mode of a particle calculated by using Fermi’s golden rule. •  individual transition rates Γj are called partial decays rates or partial widths. •  the total decay rate = sum of the partial widths; for N particles, the number decaying within δt is

given by the sum of the numbers of decays into each decay channel: •  the total decay rate per unit time, Γ, is the sum of individual decay rates •  the number of particles remaining after a time, t, is given as: •  the lifetime of the particle in its rest frame, τ, is particle’s proper lifetime, given as: •  the relative frequency of a particular decay mode is called the branching ratio:

δN = −NΓ1δt − NΓ2δt −⋅⋅ ⋅= −N Γ jj∑ δt = −NΓδt

Γ = Γ jj∑

N(t) = N(0)e−Γt = N(0)exp −tτ

⎛

⎝⎜

⎞

⎠⎟

τ =1Γ

Br( j) =Γ j

Γ

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particle decays

•  an example: τ-lepton decay modes – transition rates given by Fermi’s golden rule for each final state = partial decay width/rate

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two body decays

1

2

a

!p*

−!p*

Θ z

-  2-body decay: in the cms-frame: Ea = ma and pa = 0; daughter particles back-to-back -  decay rate:

-  using the cms condition

-  in spherical polar coordinates

Γ fi =1

8π 2ma

M fi2δ(ma −E1 −E2∫ )δ3( !p1 +

!p2 )d3 !p12E1

d3 !p22E2

Γ fi =1

8π 2ma

M fi2 14E1E2

δ(ma −E1 −E2∫ )d3 !p1

d3 !p1 = p12dp1 sinθdθdφ = p1

2dp1dΩ

Γ fi =1

8π 2ma

M fi2δ(ma − m1

2 + p12 − m2

2 + p22∫ ) p1

2

4E1E2dp1dΩ

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two body decays...

Γ fi =1

8π 2ma

M fi2δ(ma − m1

2 + p12 − m2

2 + p22∫ ) p1

2

4E1E2

dp1dΩ

Γ fi =1

8π 2ma

M fi2g(p1)δ( f (p1∫ ))dp1dΩ where g(p1) = p1

2

4E1E2

and

f (p1) =ma − m12 + p1

2 − m22 + p2

2

M fi2g(p1)δ( f (p1∫ ))dp1dΩ = M fi

2g(p*) df

dp1 p*

−1

dfdp1 p*

−1

=p1

m12 + p1

2+

p2

m22 + p2

2= p1

E1 +E2

E1E2

⎛

⎝⎜

⎞

⎠⎟

•  from the previous page:

•  this eq. has the following functional form:

•  the Dirac delta function facilitates energy conservation, is non-zero for p1=p*, only, where p* is the solution of f(p*)=0 – by evaluating the integral over dp1: •  the derivative df/dp1 is given by

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g(p*) dfdp1 p1=p*

−1

=p*2

4E1E2⋅

E1E2p*(E1 +E2 )

=p*4ma

M fi2g(p1)δ( f (p1∫ ))dp1 =

p*4ma

M fi2

M fi2δ(ma −E1 −E2∫ )δ3( !p1 +

!p2 )d3 !p12E1

d3 !p22E2

=p*4ma

M fi2

∫ dΩ

Γ fi =p*

32π 2ma2 M fi

2∫ dΩ

p*= 12ma

(ma2 − (m1 +m2 )

2⎡⎣ ⎤⎦ ma2 (m1 −m2 )

2⎡⎣ ⎤⎦

two body decays... •  combined with the expression for g(p1) (previous slide), gives:

•  the integral then becomes

•  and

•  the general expression for any decay width of a two-body process now is

•  the cms momentum of the final state particles is given as:

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interaction cross sections

rb =σφa

(va + vb )δt

δP = δNσA

=nb(va + vb )Aσδt

A= nbvσδt

particle a particle b

•  for interaction rates, evaluate the initial flux φa of particles a, crossing a target volume with nb particles of type b per unit time:

•  dynamics of the process is in the cross section, σ, having dimensions of area •  in the figure above: a single particle of type a (blue) traverses a volume with particles of type b •  particle a moves with velocity va to the right, particles b with a velocity vb to the left •  in a time interval δt, particle a crosses a region containing δN=nb(va+vb)Aδt particles of type b •  the interaction probability is obtained from the effective total cross sectional area of the δN

particles divided by the cross sectional area A •  this can be taken as the probability that the incident particle (a) passes through a region of area

σ drawn around each of the δN target particles (b) •  the interaction probability δP is (v=va+vb):

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interaction cross sections...

ra =dPdt

= nbvσ

rate = ranaV = (nbvσ )naV

rate = (nav)(nbV )σ = φNbσ

-  the interaction rate for incident particle of type a:

-  for a beam of a-particles, with number density na, within volume V, the interaction rate is:

-  i.e.

-  the total interaction rate then is:

rate = flux × number of target particles × cross section -  cross section for an interaction process is defined as:

σ = (number of interactions per unit time per target particle)/(incident flux)

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Lorentz invariant flux

-  consider two-by-two scattering (figure) of particles a and b: a + b è 1 + 2 in the centre-of-mass frame of reference of the incident particles a and b •  the interaction rate in volume V is:

•  by normalising the wavefunctions to one particle within V, na = nb = 1/V, for which

•  the normalisation factor, V , is cancelled by the wave function normalisations and phase space •  considering one particle per unit volume, the cross section vs. the transition rate is:

•  the transition rate is given by Fermi’s golden rule as:

•  expressed in Lorentz-invariant form:

rate = φanbVσ = (va + vb )nanbσV where φa = na (va + vb )

Γ fi =(va + vb )V

σ

σ =Γ fi

(va + vb )

σ =(2π )4

(va + vb )Tfi∫

2δ(Ea +Eb −E1 −E2 )δ3( !pa +

!pb −!p1 −!p2 ) d

3 !p1

(2π )3d3 !p2

(2π )3

σ =(2π )−2

4EaEb(va + vb )M fi∫

2δ(Ea +Eb −E1 −E2 )δ3( !pa +

!pb −!p1 −!p2 ) d

3 !p1

(2π )3d3 !p2

(2π )3

where M fi = (2E12E2 2E32E4 )1/2Tfi

1

2

b a va vb

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Lorentz invariant flux...

•  quantity F = 4EaEb(va+vb) is known as the Lorentz-invariant flux factor •  to validate the Lorentz-invariance of F:

•  when the incident particle momenta are collinear:

•  one obtains:

•  the Lorentz-invariant flux factor can now be expressed as:

F = 4EaEb(va + vb ) = 4EaEbpaEa

+pbEb

⎛

⎝⎜

⎞

⎠⎟= 4(Eapb +Ebpa )

⇒ F 2 =16(Ea2pb

2 +Eb2pa

2 + 2EaEbpa pb )

(pa ⋅ pb ) = (EaEb + papb )2 = Ea

2Eb2 + pa

2pb2 + 2EaEbpa pb

F 2 =16 (pa ⋅ pb )2 − (Ea

2 − pa2 )(Eb

2 − pb2 )⎡⎣ ⎤⎦

F = 4 (pa ⋅ pb )2 −ma

2mb2⎡⎣ ⎤⎦1/2

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scattering in the centre-of-mass frame

•  the interaction cross section is a Lorentz invariant => it can be calculated in any frame •  the centre-of-mass (cms) system is most convenient since there: pa=-pb =p*i, and p1=-p2=p*f •  the cms energy is given by √s=(E*a+E*b) •  in the cms, the Lorentz-invariant flux factor is:

•  using the cms constraint: pa+pb=0:

•  using the previous derivations (replace ma by √s):

•  the cross section for any 2-to-2 scattering is given as:

F = 4Ea*Eb

*(va* + vb

*) = 4Ea*Eb

* pi*

Ea* +

pi*

Eb*

⎛

⎝⎜

⎞

⎠⎟= 4pi

*(Ea* +Eb

*) = 4pi* s

σ =1

(2π )21

4pi* s

M fi∫2δ( s −E1 −E2 )δ

3( !p1 +!p2 )

d3 !p12E1

d3 !p22E2

σ =1

16π 2pi* s

×pf*

4 sM fi∫

2dΩ*

σ =1

64π 2spf*

pi* M fi∫

2dΩ*

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differential cross sections

e-

e-

p

θ

dΩ

dσ/dΩ = (no. of particles into dΩ per unit time per tgt particle)/(incident flux)

σ =dσdΩ∫ dΩ

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differential cross sections...

dσ =1

64π 2spf

*

pi* M fi

2dΩ*

dσdΩ

=1

64π 2spf

*

pi* M fi

2

t = (p1* − p3

*)2 = p1*2 + p2

*2 − 2p1* ⋅ p3

* =m12 +m3

2 − 2(E1*E3

* −!p1

* ⋅!p3

*) =m1

2 +m32 − 2E1

*E3* + 2p1

*p3* cosθ *

-  differential cross section – dσ

-  for the colliding beams frame = lab frame

- Lorentz transformations between different frames: use Lorentz-invariants

y y

e-

e-

p4

z

p3 p1

p p

θ e- p*1

p*4

p*3

p*2 z θ*

LAB CMS

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differential cross sections...

•  in the cms, the magnitude of the momenta & energies of the final state particles are fixed by energy-momentum conservation

•  the cms scattering angle θ* is the only free parameter, and

•  replacing 1 and 3 by i (initial state) and f (final state), get:

•  assuming no azimuthal angle dependence, can integrate over φ* to get a factor 2π, and

•  the magnitude of the initial state particles in the cms is:

•  since σ, s, t and the matrix element squared are all Lorentz-invariants, dσ/dt is Lorentz-invariant, as well

dt = 2p1*p3

*d(cosθ *)

dΩ* ≡ d(cosθ *)dφ* = dtdφ*

2p1*p3

*

dσ =1

128π 2spi*2 M fi

2dφ*dt

dσdt

=1

64πspi*2 M fi

2

pi*2 =

14s

s− (m1 +m2 )2⎡⎣ ⎤⎦ s− (m1 −m2 )

2⎡⎣ ⎤⎦

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lab frame differential cross section

- example: e-p è e-p elastic scattering in the laboratory frame

p1 ≈ (E1, 0, 0,E1)p2 = (mp, 0, 0, 0)p3 ≈ (E3, 0,E3 sinθ,E3 cosθ ) andp4 = (E4,

!p4 )

pi*2 ≈

(s−mp2 )2

4s where s = (p1 + p2 )2 = p1

2 + p22 + 2p1 ⋅ p2 ≈ mp

2 + 2p1 ⋅ p2

pi*2 =

E1*2mp

2

s

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lab frame differential cross section...

•  the differential cross section vs. the electron scattering angle in the lab frame:

dσdΩ

=dσdt

dtdΩ

=1

2πdt

d(cosθ )dσdt

where t = (p1 − p3)2 ≈ −2E1E3(1− cosθ )

t = (p2 − p4 )2 = 2mp2 − 2p2 ⋅ p4 = 2mp

2 − 2mpE4 = −2mp(E1 −E3)

E3 =E1mp

mp +E1 −E1 cosθ

dtd(cosθ )

= 2mpdE3

d(cosθ )

dE3

d(cosθ )=

E12mp

(mp +E1 −E1 cosθ )2 =E3

2

mp

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lab frame differential cross section...

•  the differential dt/d(cosθ) = 2E32 , and we get for the Lorentz-invariant differential cross section:

•  by eliminating the initial state particle momenta

•  by expressing the scattered electron energy, E3, in terms of the scattering angle:

dσdΩ

=12π2E3

2 dσdt

=E32

64π 2spi*2 M fi

2

dσdΩ

=1

64π 2E3mpE1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

M fi2

dσdΩ

=1

64π 21

mp +E1 −E1 cosθ

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

M fi2