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