III. Introduction to QED
4. Fermion pair production in e+e- annhihilation
1.
QED prediction
2.
Experimental methods
3.
e +e − → µ + µ −
4.
Bhabha scattering
5.
Discovery of the Tau-Lepton
6.
e +e − → hadrons
7.
Hadronic resonances
1
4. Fermion pair production in e+e- annhihilation
e +e − → f f
e
f
−
p1
p3
p2
p4
e
+
f f = e +e − , µ + µ − , τ +τ −
= uu , dd , ss , cc , bb
f
Observed as hadrons
In case of electron pair production there are two interfering amplitudes:
+
Nothing really new:
1/q4 term (Rutherford+spin)
2
J. Pawlowski / U. Uwer
1
III. Introduction to QED
4.1 QED prediction
+
−
+
e e →µ µ
e
−
dσ =
M fi
2
F
µ−
−
p1
e
F = 2E12E2
p4
p2
µ+
+
⋅
flux
p3
⋅ (2π )4 δ 4 ( p1 + p2 − p3 − p4 )
r
p1
d 3 p3
d 3 p4
⋅
3
2E3 (2π ) 2E4 (2π )3
E1
Tfi = µ + ( p4 )µ − ( p3 ) S e + ( p2 )e − ( p1) =
= −i ⋅ (2π ) δ 4 ( p1 + p2 − p3 − p4 ) Mfi
4
dσ
1
1
=
⋅ ⋅ M fi
dΩ 64π 2 s
2
r
dσ
1
1 p3
=
⋅ ⋅ r ⋅ M fi
dΩ 64π 2 s p1
2
≈1
Lorentz invariant transition amplitude Mfi
3
Reminder:
Jν = −ieψ γ ν ψ
e
electron current
Jeν = −ie v e γ ν ue
−
µ
u µ ( p3 )
u e ( p1 )
v e ( p2 )
e
+
1
q2
M fi = (ie)2 u µ ( p3 )γ ν v µ ( p4 )
ν
Jmuon
−
v µ ( p4 )
muon current
ν
Jmuon
= −ie u µ γ ν v µ
µ+
1
v e ( p3 )γ ν ue ( p4 )
q2
Je ν
See next lecture by J.Pawlowski.
4
J. Pawlowski / U. Uwer
2
III. Introduction to QED
Spin states
M fi = (ie)2 u µ ( p3 )γ ν v µ ( p4 )
1
v e ( p3 )γ ν ue ( p4 )
q2
This result is true only for one specific spin configuration.
For non-polarized ingoing particles and for non-observation of final
state spin one observes unpolarized cross sections ⇒ need to
average over possible initial spin states and sum over all final
spin states.
M fi
2
=
∑∑
1
⋅
M fi
(2se + 1)(2se′ + 1) se ,se′ rµ ,rµ′
2
See next lecture by J.Pawlowski.
5
See next lecture by J.Pawlowski.
Averaging and summing over spins of initial/final state:
2
=
1 e4
4 s2
=4
By using the
Mandelstam
variables in the
relativistic limit
neglect masses
e4
s2
Lorentz invariant!
s = ( p1 + p2 )2 = m 2 + m 2 + 2 p1p2 ≈ 2 p1p2 ≈ 2 p3 p4
t = ( p1 − p3 )2 = m 2 + M 2 − 2 p1p3 ≈ −2 p1p3 ≈ −2 p2 p4
u = ( p1 − p4 )2 = m 2 + M 2 − 2 p1p4 ≈ −2 p1p4 ≈ −2 p2 p3
Mfi
J. Pawlowski / U. Uwer
2
t 2 + u2
= 2e
s2
4
if masses
neglected
M fi
Remember: matrix
element squared can be
expressed in s, u, t!
6
3
III. Introduction to QED
M
2
+ −
+
e e →µ µ
−
(s, t , u ) = 2e 4
t 2 + u2
s2
dσ
1
1
=
⋅ ⋅ M fi
dΩ 64π 2 s
dσ
e
1 t +u
=
⋅ ⋅
2
2
dΩ 32π s
s
4
e
2
Kinematics for high-relativistic particles
CMS
2
e
µ
−
k
r
pi
p′
r
pf
θ
θ
∗
µ
k′
+
−
e
+
r
r
pi = pf
p
2
s = (k + k ′)2 ≈ 4Ei
2
t = (k − p)2 ≈ −2kp ≈ −2Ei (1 − cos θ ∗ )
2
4
1
2
=
⋅ ⋅ (1 + cos θ )
2
64π s
s
≈ − (1 + cos θ )
2
2
2
u = (k − p′) ≈ −2kp′ ≈ −2Ei (1 − cos θ )
e = 4πα
2
s
≈ − (1 − cos θ )
2
dσ
α
2
=
⋅ (1 + cos θ )
dΩ CMS 4s
2
1/s dependence from flux factor
7
dσ
α
2
=
⋅ (1 + cos θ )
dΩ CMS 4s
2
4πα
86.86 nb GeV
=
3s
s
2
σ tot
=
2
8
J. Pawlowski / U. Uwer
4
III. Introduction to QED
4.2 Experimental methods
e+e- accelerators (personal selection)
Accelerator
Lab
s
Lint / Exper.
SPEAR
PEP
PETRA
TRISTAN
SLAC
SLAC
DESY
KEK
2 – 8 GeV
→29 GeV
12 - 47 GeV
50 – 60 GeV
220 - 300 pb-1
~20 pb-1
~20 pb-1
LEP
CERN
90 GeV
~200 pb-1
today
Z physics
Cross section (experimental definition)
• Nff number of detected e+e-→ ff events
N (1 − b)
σ (e e → f f ) = ff
ε Lint
+
−
• b background fraction
• ε acceptance / efficiency
• Lint integrated luminosity of collider
9
PETRA, 1978 - 1986
Circumference 2.3 km
Emax = 23.4 GeV / beam
Experiments:
Pluto, Tasso,
Jade, Mark J,
Cello
10
J. Pawlowski / U. Uwer
5
III. Introduction to QED
Particle detectors
11
Japan – Deutschland – England
12
J. Pawlowski / U. Uwer
6
III. Introduction to QED
13
Experimental Signatures:
e−
f
e+
f
OPAL / LEP
e −e +
f f = e −e +
+
µ −µ +
µ µ
−
+
τ −τ +
q q mit
q = u, d, s, c,b, (t)
Hadron jets
qq
14
J. Pawlowski / U. Uwer
7
III. Introduction to QED
Determination of integrated luminosity
σ (e +e − → f f ) =
∫
Lint = Lee (t )dt
Nff (1 − b)
ε Lint
small angle Bhabha scattering
(low momentum transfer, QED works !!):
e +e − → e +e − (γ )
e − e−
1.7o
θ
e−
−z
3.2o
Effect of flange
in beam pipe
+z
+
L3
Small angle Bhabha scattering is
t channel dominated: theoretical
cross section σtheo well known.
4.3
Lint =
Nee
σ theoε
At LEP:
typ. errors < 0.5%
15
e +e − → µ + µ −
Acollinearity
+
−
e e → µ + µ − (γ )
Good agreement with QED!
Quantitative limit for new physics ?
Effect of bremsstrahlung:
There will always be
additional photons
16
J. Pawlowski / U. Uwer
8
III. Introduction to QED
Possible deviation from QED:
Rµµ =
• additonal heavy photon
σ µµ
σ QED
Modifies
propagator
1
1
1
→ 2− 2
q2
q
q − Λ2
=
q2
1
(1 − 2
)
2
q
q − Λ2
≈
Λ corresponds to the
mass of new photon
1
q2
(1 + 2 )
2
q
Λ
γ +Z
σ µµ
σ QED
To also account for possible lower
cross sections:
γ/Z
Additional heavy photon:
σ µµ
q2
1
1
→ 2 (1 m 2
)
2
q
q
q − Λ2±
Meaning of Λ- not clear
4πα 2
=
s
2
⎛
s ⎞
⎜⎜ 1 m
⎟
s − Λ ± ⎟⎠
⎝
Λ ± > 200 GeV
Confirms “Coulomb law” down to 10-18 m
17
Effect of Z boson exchange
„heavy photon w/ different couplings“
dσ α 2
=
⋅ (1 + cos 2 θ )
dΩ 4s
QED
dσ α 2
=
⋅ (1 + cos 2 θ + A cos θ )
dΩ 4s
γ +Z
Clear deviation from QED:
The effect of the “heavy” Z boson
is already seen at low energies!
⇒Effect of electro-weak γ/Z interference
18
J. Pawlowski / U. Uwer
9