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1
Introduction Free particles Cross section 2 → 2 Decay width
FYSH300, fall 2011
Tuomas [email protected]
Office: FL249. No fixed reception hours.
kl 2011
Part 3: Scattering theory
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Introduction Free particles Cross section 2 → 2 Decay width
Basic idea
Particle physics experiment:Know incoming particle(s) — measure outgoing particle properties:
Momenta (|p| and direction) Charges (electric, other conserved), masses, spins/polarizations
=⇒ identify particle type Repeat experiment and measure number/frequency of events
=⇒ collect statistics , form spectra (jakauma) dN/ d3p, correlations Compare to theory to test/learn about fundamental interactions.
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Introduction Free particles Cross section 2 → 2 Decay width
Terminology IWhat goes in
Beam Consists of projectile (ammus) particles a (e.g.e±, µ±, ν, p, π, . . . ).
∼ monoenergetic & collimated (|∆p|/|p| ≪ 1). Sufficient (1011 . . . 1013 particles/s) but not too high
intensity : want many scatterings, but no interactionsbetween beam particles
May be polarized or not Consists of bunches (LHC: 7.4cm long, 7.5m = 25ns
apart)
Target Macroscopic sample at rest in lab (kohtio)
Large number of particle b. distance between scattering centers db ≫ λa = h/pa:
independent scatterings Thin, so that each a scatters most once.
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Introduction Free particles Cross section 2 → 2 Decay width
Terminology IIWhere it collides
Collider experiment: two beams collide ; LHC, Tevatron, LEP,+ Achieve higher energy/$ (
√s = E∗
a + E∗b ,
$ ∼ Ebeam ∼ √s)
- Hard to align beams: less events(Not always symmetric: e.g. HERA 30GeV e− + 920GeV p )
Fixed target experiment: beam+target+ Can make target big enough so every beam particle
scatters =⇒ better statistics- Lower energy/$ (
√s ≈
p
2mbETRFa , $ ∼ Ebeam ∼ s )
An experimentalist cannot boost the lab to v = 0.9999 but a theorist can, sowe can do the following derivations in the TRF (= fixed target lab frame).
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Introduction Free particles Cross section 2 → 2 Decay width
Terminology IIIWhat comes out
Exclusive experiment measure and identify all final state particles.Examples
π− + p → π+ + π− + n e+ + e− → µ+ + µ−
Inclusive experiment: only care about/measure certain particles(rest denoted as “X ”). Completely random examples:
p + p → X π− + p → π+ + X p + p → µ+ + µ− + X
Elastic scattering: a + b → a + b (initial and final particles same).
Inelastic scattering: final state particles 6= initial state particles.Examples:
e+ + e− → e+ + e−, elastic e+ + e− → µ+ + µ−, inelastic e+ + e− → µ+ + µ− + γ, inelastic p + p → X , with X 6= p + p inelastic (total inelastic cross
section)
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Introduction Free particles Cross section 2 → 2 Decay width
Luminosity and cross section
Number of events in scattering a + b → X depends on Experimental setup: Intensity of incoming projectile particles a + number
of target particles b encountered Properties of fundamental interaction
Number of events per unit time
dNa+b→Xev
dt= σa+b→X
Lz |
ΦaNb, where
Φa is the flux of particle a (particles/(time*area))
Nb is the number of target particles b “under” the beam
L is the luminosity
σ is the cross section : this encodes everything about theparticle interaction.
Dimensions:»
dNdt
–
=1s
= m2 1sm2
=⇒ Cross section is an area.
(Remember we chose to work in TRF, but you can always boost.)
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Introduction Free particles Cross section 2 → 2 Decay width
Some interpretationCross section
σ can be thought of as size of particle; but “size” depends on interaction . . .σ of billiard ball scattering is π(2R)2, (scatter if centers less than 2R apart.)
— but σ of neutrino-billiard ball scattering is much much less.
Loosely cross section = area × probability to scatter
Probability is dimensionless — Billiard balls have P = 1 for distance < 2R.
Cross section is Lorentz-invariantIt does not change in boost along the beam axis direction(unlike luminosity, because of time dilation!)
σ usually is differential, e.g.dNa+b→c+X
ev
dt d3pc(pc) =
dσa+b→c+X
d3pc(pc)
Lz |
ΦaNb
Number of scattering events where final state has particle c in momentumspace cube between pc = (px , py , pz) and (px + ∆px , py + ∆py , pz + ∆pz)— divided by time interval and ∆px∆py∆pz .
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Introduction Free particles Cross section 2 → 2 Decay width
Some cross sections
Unit of cross section1barn = 10−28m2
E.g. proton-proton totalinelastic σ at√
s = 200GeV is∼ 40mb = (2fm )2.
Named by Enrico Fermi:( The area corresponding to a
barn is the approximate area of
a typical atomic nucleus which
has a size of 10−12cm. For
most sub-nuclear processes,
this is a very large
cross-section, so Fermi
suggested ”it was as big as a
barn”. )
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Introduction Free particles Cross section 2 → 2 Decay width
More cross sections I
10
-8
10-7
10
-6
10-5
10
-4
10
-3
10-2
1 10 102
σ[m
b]
ω
ρ
φ
ρ′
J/ψ
ψ(2S)Υ
Z
σ(e+e− → hadrons) vs.√
s/GeV :
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Introduction Free particles Cross section 2 → 2 Decay width
More cross sections II
10
10 2
10-1
1 10 102
103
104
105
106
107
108
⇓
Plab GeV/c
Cro
ss s
ectio
n (m
b)
10
10 2
10-1
1 10 102
103
104
105
106
107
108
⇓
Plab GeV/c
Cro
ss s
ectio
n (m
b)
√s GeV
1.9 2 10 102 103 104
p p
p−p
total
elastic
total
elastic
pp and pp total and elastic cross sections
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Introduction Free particles Cross section 2 → 2 Decay width
Collider/experiment luminosity
The luminosity is the “figure of merit” of a particle accelerator/collider.Bigger luminosity is better, because you can see rarer events.LHC design luminosity ∼ 1034cm−2s−1 is very big for a collider.Often quoted is
Integrated luminosityZ
dtL
Has units 1/m2: inverse of cross section.
Interpretation:
When LHC has delivered 1fb−1 of integrated lumi, a process with crosssection σ = 1fb should have happened one time (± statistical fluctuations).(LHC design luminosity would give ∼ 300fb−1 in full year of nonstop running.)
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Introduction Free particles Cross section 2 → 2 Decay width
ATLAS measured integrated luminosity at CERN/LHC
Day in 2011
27/02 29/03 29/04 30/05 30/06 31/07
]-1
Tot
al In
tegr
ated
Lum
inos
ity [f
b
00.20.40.60.8
11.2
1.4
1.61.8
22.2
Day in 2011
27/02 29/03 29/04 30/05 30/06 31/07
]-1
Tot
al In
tegr
ated
Lum
inos
ity [f
b
00.20.40.60.8
11.2
1.4
1.61.8
22.2 = 7 TeVs ATLAS Online Luminosity
LHC Delivered
ATLAS Recorded
-1Total Delivered: 1.75 fb-1Total Recorded: 1.66 fb
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Introduction Free particles Cross section 2 → 2 Decay width
Quantum mechanical scattering theory
We now want to relate the cross section to a quantum mechanical matrixelement that we can compute from the Hamiltonian (in practice the Lagrangian) ofa given particle physics theory.
For this we need the state vectors |i〉 and |f 〉 corresponding to the initialand final particle content.These are many particle states created from the vacuum |0〉 by particlecreation operators that should be familiar if you have taken QMII.We will, however, not go into the second quantization formalism here; youneed it to derive the Feynman rules, but we will in this course not do that.
Relativistic plane waves
Contrary to many QM courses we want to have a completely relativisticnormalization (So some things may differ from nonrelativistic treatment QM I/II, conventions in
atomic physics, quantum optics etc.)
We will just assume that the in and out-states are described by a completeset of orthonormal Fock (=multiparticle) states |n〉.We now need to state our normalization of the wave functions (plane wavestates) that correspond to these multiparticle states.
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Introduction Free particles Cross section 2 → 2 Decay width
S-matrix
Consider a scattering process a + b → 1 + 2 + · · · + nInitial state |i〉 is system of two free particles (a + b) at t → −∞.Final state |f 〉 is system of n free particles at t → ∞
All information on scattering in formally encoded into S-operator.Matrix elements of S are transition amplitudes
Sfi ≡ 〈f |S|i〉,in terms of which the transition probability is
Pfi = |Sfi |2 = 〈i |S†|f 〉〈f |S|i〉.Probability is conserved, sum over all final states is 1 (I=identity operator)
1 =X
f
Pfi = 〈i |S†
=I,complete setz |
X
f
|f 〉〈f | S|i〉 = 〈i |S†S|i〉 = 1,
i.e. S is unitary .Generally S-matrix contains also the trivial part |i〉 = |f 〉, subtracting this weget the “T -matrix” (T like transition) :
S = 1 + i T , i.e. for components Sfi = δfi + iTfi ,
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Introduction Free particles Cross section 2 → 2 Decay width
Klein-Gordon equation and plane waves
Remember from nonrelativistic QM: E → i∂0, p → −i∇.Nonrelativistic dispersion relation is
E =1
2mp2
=⇒ Schrodinger i∂0ψ(t , x) = − 12m
∇2ψ(t , x)
Relativistic dispersion relation
E2 = p2+m2=⇒ Klein-Gordon eq (i∂0)
2ϕ(t , x) =h
(−i∇)2 + m2i
ϕ(t , x)
In covariant notation (recall ¤ = ∂µ∂µ = ∂20 − ∇2) relativistic plane wave
(¤ + m2)ϕ(x) = 0, solution Ne−ip·x = Ne−iEp t+ip·x
(E2, not E , because √ of ∇ awkward. Physically: antiparticles, E < 0 solutions.)
Note: Ep =p
p2 + m2, the energy corresponding to momentum p.
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Introduction Free particles Cross section 2 → 2 Decay width
Normalization and box density of states
Problem: want to work with plane waves (exact ) momentum eigenstates.Heisenberg uncertainty relation =⇒ completely delocalized in space andunnormalizable .
Way around: assume we are in a large box of size V = L3, periodicboundary cond’s.
Normalization choice for plane waves
N =1√V
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Introduction Free particles Cross section 2 → 2 Decay width
Box density of states
Periodic boundary allows us to count discrete states:
ϕ(x , y , z) = ϕ(x + L, y , z) = ϕ(x , y + L, z) = ϕ(x , y , z + L)
ei(p1x+p2y+p3z) = ei(p1(x+L)+p2y+p3z) = ei(p1x+p2(y+L)+p3z) = ei(p1x+p2y+p3(z+L))
1 = eip1L = eip2L = eip3L
only true if (p1, p2, p3) = 2πL (n1, n2, n3); ni ∈ Z.
NowX
states
=X
n1,n2,n3
≈V→∞
Z
d3n =
Z
V(2π)3
d3p
(P
n f (n) ≈R
dnf (n), when f (n) varies slowly.R n+1/2
n−1/2 dx = 1.
“Slowly varying” requires V → ∞ because 2πL (n1, n2, n3) close to 2π
L (n1 + 1, n2, n3).)
Recall normalization choice for plane waves, now it is
N =1√V
=⇒ 〈p|q〉 =
Z
d3x(Ne−ip·x)∗(Ne−iq·x) = δp,q ,
(Discrete Kronecker delta for discrete momenta =⇒ Now well defined matrices S and T . )
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Introduction Free particles Cross section 2 → 2 Decay width
Conserved particle number current, nonrelativistic
In nonrelativistic theory:Particle density ρNR = ψ∗(t , x)ψ(t , x), current jNR = − i
2m (ψ∗∇ψ − ψ∇ψ∗).Probability conservation = continuity equation: ∂tρNR + ∇ · jNR = 0.For a plane wave ψ(t , x) = Ne−iEp t+ip·x
these are ρNR = |N |2; jNR = |N |2p/m.
This does not directly work in relativistic case
A relativistic current is jµ = (ρ, j)It should also have a continuity equation ∂µjµ = 0.But it should be a proper 4-vector;
thus for a plane wave we should have jµ ∼ pµ
In particular: Density of free particles in a box should be proportional tothe energy in a Lorentz-covariant formulation.(Thought experiment: Start from fixed box V , 1 particle in rest, energy m, density 1/V .Boost to another frame, γ = 1/
p
1 − v2; box size contracted to V/γ,
particle energy now mγ, density γ/V .)
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Introduction Free particles Cross section 2 → 2 Decay width
Conserved current, relativistic case
Definition
jµ ≡ i(ϕ∗∂µϕ − ϕ∂µϕ∗), for plane wave ϕ = Ne−ip·x=⇒ jµ = 2pµ|N |2
In particular, for our choice N = 1√V
the particle density is ρ ≡ j0 = 2EV
The density now has funny dimensions (of energy density, not number density) , butthis is just a question of normalization of the (unnormalizable) plane wave states.We could have chosen E/m in stead of E (some books do this) , but then wewould need a separate treatment for massless particles.
Density of states: now “ 2E” particles in V
The number of states per particle isZ
V2Ep(2π)3
d3p = VZ
d4pδ(p2 − m2)
(2π)4θ(p0)
This is the Lorentz-invariant phase space measure .
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Introduction Free particles Cross section 2 → 2 Decay width
Back to cross section
Cross section is Lorentz-invariant, but we will derive it in TRF.
σ =Ns
∆tΦaNb,
Nb = nbV = 2ETRFb = 2mb, number of target particles
Φa = na|vTRFa | = 2ETRF
a |vTRFa |/V flux of particle a (flux = density × velocity)
2ETRFa |vTRF
a |/V = 2|pTRFa |/V =
q
λ(s, m2a, m2
b)/(mbV )
Thus (ΦaNb)TRF = 2q
λ(s, m2a, m2
b)/V =⇒ Back to Lorentz-invariant form,
up to volume factor V .
Ns∆t =scattering events per time = probability to go to state |f |; summedover states, divided by ∆t
Ns
∆t=
N(ab → 1 . . . n)
∆t=
|Tfi |2∆t
× (number of possible final states),
=
Z nY
m=1
V2Ep.(2π)3
d3pm × |Tfi |2∆t
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Introduction Free particles Cross section 2 → 2 Decay width
T -matrix elementThe last, and most important, thing we need
Number of scatterings, T -matrix
The number of scatterings is given by the T -matrix element between finaland initial states. We separate out some factors:
Tfi = −iNaNbN1 . . .Nn(2π)4δ(4)
pa + pb −n
X
i=1
pi
!
Mfi ,
Why the factors?
energy, momentum always conserved in amplitude : useful to extract δ(4)
to explicitly treat (δ(4))2 in |Tfi |2 plane wave normalizations: conventions for N vary; for Mfi not.
(Justification: Tfi is matrix element between our box-normalized plane waves
Tfi ∼ 〈f |T |i〉 ∼ (Nae−ipa·xNbe−ipb·x )∗T (N1e−ip1·x . . .Nne−ipn·x )
The “invariant amplitude” Mfi , o.t.o.h. is defined using unnormalized plane waves
Mfi ∼ (e−ipa·x e−ipb·x )∗T (e−ip1·x . . . e−ipn·x )
Why this complication: needed the box to understand delta functions, count particles. . . )
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Introduction Free particles Cross section 2 → 2 Decay width
Putting things together
What is then (δ(k))2? Integral representation of δ:Z
dx eikx = (2π)δ(k), thus δ(k = 0) =1
2π
Z
dx =L
2πand (2π)4δ(4)(0) = ∆tV
Now we have
σ =1
ΦaNb
Ns
∆t=
V
2q
λ(s, m2a, m2
b)
Z nY
m=1
»
V2Ep.(2π)3
d3pm
–
|−iNaNbN1 . . .Nn|2h
(2π)4δ(4)(pa + pb − Pni=1 pi)
i2|Mfi |2
∆t
=V
2q
λ(s, m2a, m2
b)
Z nY
m=1
»
V2Ep.(2π)3
d3pm
–
V−n−2(2π)4δ(4)(pa + pb − Pni=1 pi) [∆tV ]|Mfi |2
∆t
Now volume cancels — time interval cancels
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Introduction Free particles Cross section 2 → 2 Decay width
Final master formula for cross section
The final formulae
Total σ(ab → 1 . . . n) =1
2q
λ(s, m2a, m2
b)
"
Z nY
m=1
d3pm
(2π)32Em
×(2π)4δ(4)“
pa + pb −Xn
i=1pi
”
#
|Mab→1...n|2 ,
Differentialdσab→1...n
d3p1 . . . d3pn=
1(2(2π)3E1) · · · (2(2π)3En)
1
2q
λ(s, m2a, m2
b)
×(2π)4δ(4)“
pa + pb −Xn
i=1pi
”
|Mab→1...n|2
Everything is Lorentz-invariant (Derived in TRF, but now works in any frame.)
The invariant amplitude M(ab → 1 . . . n) from Feynman rules. Part in [·] is Lorentz-invariant n-particle phase space .
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Introduction Free particles Cross section 2 → 2 Decay width
Special case: 2 → 2 scattering
First work in general frame, start from the general formula
σab→cd =1
2q
λ(s, m2a, m2
b)
Z
d3pc
2Ec(2π)3
d3pd
2Ed(2π)3(2π)4δ(4)(pa+pb−pc−pd )|M|2.
Now δ(4)(p) = δ(E)δ(3)(p) =⇒ integrate d3pd . Other variable d3pc = |pc |2 d|pc | dΩc
σab→cd =(2π)−2
8q
λ(s, m2a, m2
b)
Z
dΩc
Z |pc |2 d|pc |EcEd
δ(Ea + Eb − Ec − Ed)|M|2
Get rid of δ(Ea + Eb − Ec − Ed), usingR
d|pc |.Remember, now Ec =
p
|p2c | + m2
c , Ed =q
|pa + pb − pc |2 + m2d .
=⇒ δ-function has |pc |-dependence in both Ec and Ed .Complicated in general frame, specialize to CMS.
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Introduction Free particles Cross section 2 → 2 Decay width
Special case: 2 → 2 scattering in CMS
In CMS |pc | = |pd | ≡ p∗f , E∗
a + E∗b =
√s
Get rid of δ(Ec + Ed −√s), using
R
d|pc |.
pa
pb
pc
pd
θ∗
θ∗
Remember δ(f (x)) =δ(x − x0)
|f ′(x0)|, where f (x0) = 0
Nowd
dpE(p) =
ddp
p
p2 + m2 =p
p
p2 + m2=
pE(p)
So δ(E∗c (p∗
f ) + E∗d (p∗
f ) −√
s) =δ(p∗
f − . . . )p∗fE∗
c+
p∗fE∗
d
=E∗
c E∗d
p∗f
√s
δ(p∗f − . . . )
and σab→cd =(2π)−2
8q
λ(s, m2a, m2
b)
Z
dΩc
Z
(p∗f )2 dp∗
f
E∗c E∗
d
E∗c E∗
d
p∗f
√s
δ(p∗f − . . . )|M|2
=(2π)−2
8q
λ(s, m2a, m2
b)
p∗f√s
Z
dΩ∗c |M|2.
p∗f =
q
λ(s,m2c ,m2
d )
(2√
s)(Remember:
q
λ(s, m2a, m2
b) came from p∗
i .)
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Introduction Free particles Cross section 2 → 2 Decay width
2 → 2 scattering in CMS, final result I
Not integrating over the angle gives differential σ
dσab→cd
dΩ∗c
=|Mab→cd |2
64π2s
s
λ(s, m2c , m2
d )
λ(s, m2a, m2
b)=
|Mab→cd |264π2s
p∗f
p∗i
.
(Differential: scattering events with final particle in angle Ω∗
c , divided by angle dΩ∗
c )
Special case (ma = mc and mb = md ) or (ma = md and mb = mc)
Leads to p∗i = p∗
f =⇒
dσab→cd
dΩ∗c
=|Mab→cd |2
64π2s
Remarks Energy-momentum conservation constraints now all implicitly in |M|
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Introduction Free particles Cross section 2 → 2 Decay width
2 → 2 scattering, invariant dσ/ dt
Convenient to also express dΩ∗c in a Lorentz-invariant way, via t .
In general dΩc = dφc d(cos θc). If scattering is unpolarized (summed, averaged
over spins) , cross sections do not depend on φc =⇒dσ
d(cos θc )= 2π dσ
dΩc.
t = (pa − pc)2 = m2
a + m2c − 2E∗
a E∗c + 2|p∗
a ||p∗c | cos θ∗
c
=⇒ dt = 2p∗i p∗
f d(cos θ∗c )
=⇒dσ
dt=
2π
2p∗i p∗
f
dσ
dΩ∗c
pa
pb
pc
pd
θ∗
θ∗
Using the expressions from the previous slide:dσab→cd
dΩ∗
c= |Mab→cd |2
64π2s
p∗fp∗i
and p∗i =
p
λ(s, m2a, m2
a)/(2√
s), this leads to
Differential cross section in manifestly Lorentz-invaria nt form
dσab→cd
dt=
|Mab→cd |216πλ(s, m2
a, m2b)
If s ≫ m2a, m2
b: λ(s, m2a, m2
b) = s2 + m4a + m4
b − 2sm2a − 2m2
am2b − 2m2
bs ≈ s2
=⇒ this is the easiest version to remember.
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Introduction Free particles Cross section 2 → 2 Decay width
Example: unpolarized e+e−→ µ+µ− scattering
A little flavor of things to come
e− µ−, q
e+ µ+, q
γ, Z
Assume√
s small enough so we can neglectZ0-boson =⇒ only one Feynman diagram at LO(=leading order)
Use Feynman rules for QuantumElectroDynamics= QED =⇒ get expression for|Me+e−→µ+µ− |2.
unpolarized = summed over spins of µ±;averaged over the spins of e±, denote |M|2
Assuming√
s ≫ me, mµ
|M|2 ≡ 12
12
X
spins
|M|2 = 2e4 t2 + u2
s2.
Origin of Mandelstams: M contains dot productsof pa, pb, pc and pd =⇒ written in terms of s, tand u.
29
Introduction Free particles Cross section 2 → 2 Decay width
Example: e+e−→ µ+µ− continued
The differential cross section for the above process becomes(p∗
i ≈ p∗f in
√s ≫ me, mµ limit)
dσ
dΩ∗c
=α2
4s(1 + cos2 θ∗
µ).
From this, the total cross section is
σ(e+e− → µ+µ−) =4πα2
3s.
e− µ−, q
e+ µ+, q
γ, Z Two interactions vertices ∼ e, so
M(e+e− → µ+µ−) ∝ e2, and |M|2 ∝ e4 ∝ α2;α = e2/(4π) ≈ 1/137.
Check dimensions: [s] = GeV 2
30
Introduction Free particles Cross section 2 → 2 Decay width
e+e−→ µ+µ−, comments
Angular distribution depends on spin structure
Angular distribution 1 + cos2 θ∗ characteristic of spin-1 particle (virtualphoton γ∗) decaying into two spin 1/2 particles.
For spin interpretation, there are two spin states with amplitudes:∼ (1 ± cos θ). These are different final states =⇒ they do not interfere,each amplitude is squared separately, then added.
Cross section for γ∗ → 2 spin-0 particles would be ∼ sin2 θ ∼ |Y 11 (θ, φ)|2;
spin has to go into m = 1, ℓ = 1 angular momentum of the pair.
See Halzen, Martin, sec 6.6.
Squaring and averaging Sum of diagrams=amplitude with one initial and final state (incl. spin) Then square each amplitude separately Only then sum/average |M|2’s from different spin states =⇒ this is
denoted by |M|2
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Introduction Free particles Cross section 2 → 2 Decay width
Particle decay
Initial state |i〉 = unstable particle a, decaying to n particle final state |f 〉. Only natural frame is CMS = rest frame of a. Momentum conservation p2
a = (p1 + · · · + pn)2.
In CMS: m2a = (E1 + · · · + En)
2 ≥ (m1 + . . . mn)2
=⇒ threshold ma ≥ m1 + · · · + mn.(Thus a has ma > 0, there is always a rest frame.)
Terminology In general different possible final states |f 〉: decay channels . Define the decay width Γf for channel f as
Γf ≡Nf
Na∆Tin a rest frame.
Nf is the number of decays to state fNa is the number of decaying particles a
∆T is the time interval required for Nf decays. All
expe
rimen
tally
acce
ssib
le,
like
def.
ofσ
.
Now want to tie this definition with theoretically calculable quantities, skippingdetails that are similar to the derivation with cross sections.
32
Introduction Free particles Cross section 2 → 2 Decay width
Decay width and amplitude
Recall: for cross section we had
Ns
∆t=
N(ab → 1 . . . n)
∆t=
|Tfi |2∆t
× (number of possible final states),
Now define alsoNf
∆t=
|Tfi |2∆t
× (number of possible final states).
We again have the transition amplitude Tfi and the invariant amplitude Mfi :
Tfi = −iNaN1 . . .Nn(2π)4δ(4)“
pa −Xn
j=1pj
”
Mfi(a → 1 . . . n).
|Tfi |2 has a δ(4) squared =⇒ treated similarly as before.Again the number of final states per particle is 1
2EV
(2π)3 d3p
Result: decay width
Γf =1
2Ea
Z nY
m=1
d3pm
(2π)32Em(2π)4δ(4)
“
pa −Xn
i=1p i
”
|M(a → 1 . . . n)|2.
Γa and Ea depend on frame; in CMS Ea = ma. Rest is Lorentz-invariant.
33
Introduction Free particles Cross section 2 → 2 Decay width
Branching ratios
One unstable particle; many possible finals tates |f 〉, each has width Γf .Define total decay width
Γ ≡X
f
Γf ,
Fraction of each final state is branching ratio
Bf ≡Γf
Γ; often in %.
mean lifetimeτ ≡ 1
Γ.
34
Introduction Free particles Cross section 2 → 2 Decay width
Branching ratio example
Example
For the Z0 boson we have the following branching ratios:Z0 → e+e− 3.363%
µ+µ− 3.366%τ+τ− 3.370%
In addition: Z0 can decay ”invisibly” to a pair of any of the three neutrino species:
total branching ratio 20.00%. Hadronic decay modes: remaining 69.91% of the total width
Γ(Z0) = 2.4952 ± 0.0023GeV τ ≃ 0.079fm ≈ 3 · 10−25 s.
Hence the Z0 boson is very short-lived.In particle physics “long-lived particles” have lifetime ∼ 10−16s or more=⇒ can be actually seen directly in experiment.
35
Introduction Free particles Cross section 2 → 2 Decay width
Some decay times
Out of gauge bosons γ is absolutely stable, W± and Z 0 extremelyunbstable, Γ ∼ 2GeV and τ ≈ 3 · 10−25s.
gluon is confined, stability/unstability not well defined Neutrinos, e± stable τ±, µ± decay via weak interaction: τµ ≈ 2 · 10−6s and ττ ≈ 3 · 10−13 s. Quarks u, d , c, s and b form first a hadron which then decays. Out of all
hadrons only the proton is absolutely stable. Heaviest t quark decays before it has formed a hadron, τt ≈ 0.02fm ; It
would take a time of the order of 1 fm to make a hadron.
36
Introduction Free particles Cross section 2 → 2 Decay width
1 → 2 decay
Analogous to σ(2 → 2) =⇒ we’ll be brief. Frame is CMS, omit ∗.
Γa→cd =1
2ma
Z
d3pc
(2π)32Ec
d3pd
(2π)32Ed(2π)4δ(ma − Ec − Ed)δ(3)(pc + pd)|M|2.
Use δ(3)(pc + pd) to doR
d3pd ; sets |pc | = |pd | ≡ pf
Write d3pc = dΩp2f dpf , want to use δ(ma − Ec − Ed) to do
R
dpf
Energy conservation
δ(ma − Ec − Ed) =EcEd
p0ma|pf =p0δ(pf − p0),
where p0 is obtained by solving ma = Ec + Ed , i.e. explicitly
p0 =
q
λ(s, m2c , m2
d)
2√
s=
q
λ(m2a, m2
c , m2d)
2ma.
Final result Γ(a → cd) =
q
λ(m2a, m2
c , m2d )
64π2m3a
Z
dΩ|Mfi(a → cd)|2.
37
Introduction Free particles Cross section 2 → 2 Decay width
Example: π− decay
Two possible final states, µ−νµ and e−νe (threshold,conservation laws).
Experiments:Γ(π− → e−νe)
Γ(π− → µ−νµ)≈ 1.23 · 10−4 very small, why?
π−
u
dW−
νℓ
ℓ
Annihilate du =⇒ charged weak current One W -boson, 2 weak vertices:
M ∼ g2W
m2W
∼ GF (“Fermi constant”)
π− ↔ du: “pion decay constant” M ∼ fπ.
Evaluation of the Feynman diagram gives:(Actually Feynman rule for Wℓνℓ vertex + information of pion wavefunction in fπ)
|M(π− → ℓ−νℓ)|2 = 4G2F f 2
πm2ℓ (pℓ · pν) = 2G2
F f 2πm2
ℓm2π
„
1 − m2ℓ
m2π
«
,
Then (exercise) the decay width is Γ(π− → ℓ−νℓ) =1
8πG2
F f 2πmπm2
ℓ
„
1 − m2ℓ
m2π
«2
,
which implies the ratioΓ(π− → e−νe)
Γ(π− → µ−νµ)=
„
me
mµ
«2 „
m2π − m2
e
m2π − m2
µ
«2
≈ 1.28·10−4.
38
Introduction Free particles Cross section 2 → 2 Decay width
Example 2: Muon decay µ−→ e−
+ νe + νµ, estimates I1 → 3 decay
µ−
νµ
e−
νe
W−
1. Conservation laws determine finalparticles:1.1 Charge conservation: e− (only negative
particle with m < mµ).1.2 µ number conservation: Lµ = 1 =⇒ νµ
1.3 e number: Le = 0: νe to compensate e−
2. Only e− detected. 2 others=⇒ continuous distribution of Ee.
As before, we can estimate the decay width as One W propagator, 2 W -fermion vertices: M ∼ g2
W /m2W ∼ GF
mµ ≫ me, mνℓ =⇒ dimensionally must have Γ ∼ G2F m5
µ
39
Introduction Free particles Cross section 2 → 2 Decay width
Example 2: Muon decay µ−→ e−
+ νe + νµ, calculation
The actual invariant amplitude, from S.M. Feynman rules, is
|M(µ → e−νeνµ)|2 = 64G2F (k · p′)(k ′ · p).
µ−, p
νµ, k
e−, p′
νe, k′
W−
Denote p = (mµ, 0)
p′ = (E ′, p′)
k = (ω, k)
k ′ = (ω′, k′)
Differential width is dΓ =1
2mµ
M(µ → e−νeνµ)|2 dQ,
with dQ =d3p′
(2π)32E ′d3k′
(2π)32ω′d3k
(2π)32ω(2π)4δ(4) `
p − p′ − k − k ′´ .
This can be simplified usingZ
d3k2ω
=
Z
Eats δ(4)
z|
d4k θ(ω)
1dz |
δ(k2) .
40
Introduction Free particles Cross section 2 → 2 Decay width
Example 2: Muon decay µ−→ e−
+ νe + νµ, result
We get dQ =1
(2π)5
d3p′
(2π)32E ′d3k′
(2π)32ω′ θ(mµ − E ′ − ω′)δ((p − p′ − k ′)2)
After intermediate steps (exercise) we get
dΓ =G2
F
2π3dE ′ dω′mµω′(mµ − 2ω′),
with restrictions12
mµ − E ′ ≤ ω′ ≤ 12
mµ
0 ≤ E ′ ≤ 12
mµ
Integrating, final result is (exercise) Γ =G2
F m5µ
192π3
Estimate ∼ G2F m5
µ true But constant of proportionality is very small!
Details: Halzen-Martin, sec. 12.5. Also in KJE’s notes, after page 90.