Particle Physics
Dr Victoria Martin, Spring Semester 2012
Lecture 7: Renormalisation and Weak Force
★ Renormalisation in QED
★ Weak Charged Current
★ V−A structure of W-boson
interactions
★ Muon decay
★ Beta decay
★ Weak Neutral Current
★ Neutrino scattering
1
Reminders
• No lecture on Friday.
• Next tutorial: Monday 11th February.
• New tutorial sheet will be posted to webpage.
Paper copies
available at tutorial
• Next lecture: Tuesday 12th February!
2
Higher Order QED
Two photon “box” diagrams also contribute to QED scattering
*+!
*+1!/!
/!
*,!
Two extra vertices
*-!
/!
*,-!/!
!"!#$%&&'(!!
*+-!/!
0!-!/!
*.!
*-!
/!
*,!
/!
*--!/!
0!- /!
!)!#$%&&'(!!
*.!
contribution is suppress by a factor of α = 1/137
• The four momentum must be conserved at each vertex.
• However, four momentum k flowing round the loop can be anything!
• In calculating M integrate over all possible allowed momentum
configurations: ∫ f(k) d4k ~ ln(k) leads to a divergent integral!
• This is solved by renormalisation in which the infinities are “miraculously
swept up into redefinitions of mass and charge”
(Aitchison & Hey P.51)
3
Renormalisation
• Impose a “cutoff” mass M, do not allow the loop four momentum to be larger
than M. Use M2 >> q2, the momentum transferred between initial and final state.
➡ This can be interpreted as a limit on the shortest range of the interaction
➡ Or interpreted as possible substructure in pointlike fermions
➡ Physical amplitudes should not depend on choice of M
• Find that ln(M2) terms appear in the M
• Absorb ln(M2) into redefining fermion masses and vertex couplings
➡ Masses m(q2) and couplings α(q2) are now functions of q2
• e.g. Renormalisation of electric charge (considering
only effects from one type of fermion):
�
�
�
e2
M2
eR = e 1 −
ln
12π 2
q2
• Can be interpreted as a “screening” correction due to
the production of electron/positron pairs in a region
round the primary vertex
• eR is the effective charge we actually measure!
4
6)"($-7"89(.#):%5(*#$$($+$,"#)%(
,#$."/)%(.%5(.%%/'/+."/)%
)*(;/#":.+($+$,"#)%<7)4/"#)%(7./#4
Running Coupling Constant
correct for all possible fermion types in
• Renormalise α, and
0*122'&'(
�
�
α(0)
=.#$(,'.#&$
.%5(-.44 −q
)*($+$,"#)%()%+8(;/4/>+$(
2
α(q ) = α(0) 1 +
zf ln( 2 )
3π
M
."(;$#8(4')#"(5/4".%,$4
the loop:
2
/%,#$.4$4(?/"'(?/"'
all possible fermions+.#&$#(-)-$%":-("#.%4*$#
in the loop
• zf is the sum of charges! over
➡ At
q2 ~
➡ At
q2
1 MeV only electron, zf = 1
~ 100 GeV, f=e,µ,τ,u,d,s,c,b zf = 38/9
zf =
�
f
• Instead of using M2 dependence, replace with a
reference value µ2:
�
2
Qf
e$e# % & $ & #
�−1
α(µ )
q
2
2
α(q ) = α(µ ) 1 −
zf ln( 2 )
3π
µ
2
2
• Usual choices for µ are 1 MeV or mZ ~ 91 GeV.
➡ α(µ2=1 MeV2) = 1/137
➡ α(µ2=(91 GeV)2) = 1/128
@+.44/,.+(+/-/"
of µ where make a initial
• We choose a value
measurement of α, but
once we do the evolution
A1 0(B((((((((((((((!
! 0(C2CDE
of the values of α are determined by the above eqn.
F"(4')#"(5/4".%,$4
5
Fortunately
isn’t necessary
– can just write down matrix element
using a setthis
of simple
rules
using a set of simple rules
QED Summary
•
Basic Feynman Rules:
Basic Feynman Rules:
"
Propagator factor for each internal line
+
!
e
#
Propagator(i.e.
factor
eachvirtual
internal
line
eachfor
internal
particle)
+
!"
e
For full QED calculations
use spinors
define
fermion
in Feynman
#
(i.e.
each
internal
virtualcurrent
particle)
Dirac to
Spinor
for each
external
line
Dirac(i.e.
Spinor
external
line
each for
realeach
incoming
or outgoing
particle)
diagrams. e–
!–
(i.e. each
real
incoming
or outgoing particle)
Vertex
factor
for
each vertex
–
µ!–
e
ψ̄ isfactor
Fermion current are ψ̄γ ψ where
the for
adjoint
spinor: ψ̄ ≡ ψ † γ 0
Vertex
each vertex
•
• Spin-1 bosons are described by polarisation vectors, εµ (s)
• To calculate the cross section for an unpolarised process need to average over initial
Michaelmas 2011
Prof. M.A. Thomson
117
Michaelmas 2011
Prof. M.A. Thomson
117
helicities and sum over all possible final states.
the Rules
couplingfor
constant
• Fermion masses, charges and
Basic
QED α evolve as a function of
Basic Rules for QED
momentum
transfer.
External Lines
External Lines incoming particle
spin 1/2
spin 1/2
spin 1
spin 1
incoming
outgoingparticle
particle
outgoing particle
incoming antiparticle
incoming
outgoingantiparticle
antiparticle
outgoing antiparticle
incoming photon
incoming
outgoingphoton
photon
outgoing photon
Internal Lines (propagators)
Internal Lines (propagators)
spin 1
spin 1
spin 1/2
spin 1/2
photon
photon
fermion
fermion
!
!
Vertex Factors
Vertex
Factors
spin
1/2
fermion (charge -|e|)
spin 1/2
fermion (charge -|e|)
Matrix Element
= product of all factors
$
$
6
The Weak Force
• Exchange of massive W and Z bosons.
• mW = 80.385 ± 0.015 GeV
• mZ = 91.1876 ± 0.0021 GeV
p → n + e+ + νe
• Responsible for:
• Beta decay
• Fusion
• Neutrino interactions
ν̅µ e− → ν̅µ e−
7
Review: Baryon and Lepton Number; Chirality
• Lepton number is the number of leptons minus the number of
anti-leptons:
➡ Electron number:
Le = N(e−) − N(e+) + N(νe) − N(ν̅e)
➡ Muon number:
Lµ = N(µ−) − N(µ+) + N(νµ) − N(ν̅µ)
➡ Tau-lepton number: Lτ = N(τ−) − N(τ+) + N(ντ) − N(ν̅τ)
• Baryon number is a measure of the net number of quarks:
B = ⅓(N(q) − N(q̅ ))
• LH projection operator PL = (1 − γ5)/2 projects out left-handed chiral state
• RH projection operator PR = (1 + γ5)/2 projects out right-handed chiral state
• Massive fermions have both left-handed and right-handed chiral components
• Massless fermions would have only left-handed components
• Massless anti-fermions would have only right-handed components
• To date, only left-handed neutrinos and right-handed anti-neutrinos have been
observed
8
W and Z boson interactions
• Any fermion (quark, lepton) may emit or absorb a
Z-boson.
➡ That fermion will remain the same flavour.
➡ Very similar to QED, but neutrinos can interact
with a Z boson too.
• Any fermion (quark, lepton) may emit or absorb a
W-boson.
➡ To conserve electric charge that fermion
must change flavour!
➡ To conserve lepton number e↔νe, µ↔νµ, τ↔ντ
➡ To conserve baryon number (d, s, b) ↔ (u, c, t)
down-type quark
up-type quark
9
incoming photon
rticle
outgoing
photon
Feynman Rules for Charged Current
rticle
(propagators)
photon
n
nermion
s)
propagator
!
s
$
!
$
interaction
vertex
ermion (charge -|e|)
nt
gµν
= product
of all factors
1
µ
5
√
g
γ
(1
−
γ
)
W
W-boson
2
2
2
2
q − mW
Michaelmas 2011
gµν
photon, γ
2
q
product of all factors
-|e|)
• Key differences w.r.t QED.
haelmas 2011
• Left-handed interactions are also
known as V−A theory
118
e γµ
➡ γµ gives a vector current (V)
➡ γµγ5 gives an axial vector current (A)
•Photon interactions are purely vector
118
➡ q2 − mW2 as denominator of propagator
➡ The ½(1−γ5) term: this is observed experimentally.
•
•
The overall factor of 1/√8 is conventional
Recall PL=(1−γ5)/2 is the Left Handed projection operator
➡ W-boson interactions only act on left-handed chiral components of fermions
•
For low energy interactions with q << mW: effective propagator is gµν/mW2
10
“Inverse Muon Decay”
• Start with a calculation of the process νµ →
• Not an easy process to measure experimentally,
e−
µ−
νe
νµ
µ−
e−
νe
but easy to calculate!
2
gW
gµν
ν
5
u(ν
)γ
(1
−
γ
)u(µ)
M=
ū(νe )γ (1 − γ )u(e ) 2
µ
2
8�
q − mW
�2
|M| =
2
µ
2
gW
8m2W
5
−
�
� �
�2
µ
5
− 2
µ
5
ū(νe )γ (1 − γ )u(e )
ū(µ)γ (1 − γ )u(νµ )
• Usually we would average over initial spin and sum over final spin states:
• However the neutrinos are only left handed
as (see Griffiths section 9.1):
• The equation can�be solved
�2
|M|2 = 2
2
gW
m2W
(pµ (e) · pµ (νµ )) (pµ (µ) · pµ (νe ))
• In the CM frame, where E is energy of initial electron or neutrino, and me
neglected as me << E:
|M|2 = 8E 4
�
�
�
2
�
2
2
2
mµ
gW
1−
2
mW
2E 2
11
“Inverse Muon Decay” Cross Section
• Cross section = |M|2 ρ, substituting for ρ (see problem sheet 1):
dσ
dΩ
=
• Substitute:
�
1
8π
�2
➡ centre of mass energy,
∗
|�
p
S|M|
f|
(E1 + E2 )2 |�
p ∗i |
2
(E1+E2)2=4E2
νµ
µ−
e−
νe
➡ For elastic scattering particle |p*f|=|p*i|
➡ S=1 as no identical particles in final state
dσ
dΩ
2
E
32π 2
=
�
�
�
2
�
2
2
2
m
gW
µ
1−
2
mW
2E 2
➡ Fermi coupling constant GF = √2gW2/8mW2
➡ Unlike electromagnetic interaction, no angular dependence
➡ Integral over 4π solid angle
σ
=
�
dσ
4 2 2
dΩ = E GF
dΩ
π
�
m2µ
1−
2E 2
�2
12
Muon Decay
µ− → e− ν̅e νµ
• Muon decay:
�
�
|M|2 =
=2
�
2
gW
8m2W
2
gW
m2W
�2
ν̅e
(Griffiths 9.2):
2
[ū(νµ )γ µ (1 − γ 5 )u(µ)]2 [ū(e)γ µ (1 − γ 5 )v(ν̄e )]2
(pµ (e) · pµ (νµ )) (pµ (µ) · pµ (νe ))
• The phase space, ρ, for a 1 → 3 decay is, (Griffiths equation 6.21):
dΓ
1
=
dEe
4π 3
�√
2
2gW
2
8MW
�2
m2µ Ee2
• Integrate over allowed values of Ee:
Γ=
�
0
mµ /2
dΓ
dEe =
dEe
G2F m2µ
4π 3
�
0
�
4Ee
1−
3m2µ
mµ /2
Ee2
�
�
4Ee
1−
3m2µ
�
G2F m5µ
dEe =
192π 3
• Only muon decay mode for muons BR(µ− → e− ν̅e νµ) ≈ 100%, only one decay
mode contributes to lifetime
1
192π 3
192π 3 �7
τ≡ = 2 5 = 2 5 4
Γ
GF mµ
GF mµ c
13
! p (MeV/c)
Muon Decay Measurements
0.2
Data
0.1
Monte Carlo
2
2000 "10
(a)
-0.2
15
20
25
30
35
40
45
50
Momentum (MeV/c)
FIG. 2: (color online) Average momentum change of decay
positrons through the target and detector materials as a function of momentum for data (closed circles) and Monte Carlo
(open circles), measured as described in the text.
• Measurements of muon lifetime and mass used to
define a value for GF (values from PDG 2010)
If ρ = ρH +6∆ρ and η = ηH + ∆η, then the angleintegrated muon decay spectrum can be written as:
➡ τ = (2.19703±0.00002)×10 s
N (x) = N (x, ρ MeV
, η ) + ∆ρN
➡ m = 105.658367 ± 0.000004
S
H
H
∆ρ (x)
+ ∆ηN∆η (x). (3)
All events
In fiducial
1000
500
10
20
30
40
50
(b)
0.7 < |cos #| < 0.84
(c)
0.7 < |cos #| < 0.84
1.005
1
0.995
0.99
2
mass and second order effects
➡ GF = 1.166364(5) × 10 GeV
Implies gW = 0.653, α =g
/4π = 1/29.5
αW >> αEM, the weak force not intrinsically weak,
just appears so due to mass of W-boson
Probability
This expansion is exact. It can also be generalized to
0
include the angular dependence [2]. This is the basis for
-2
the blind analysis. The measured momentum-angle spectrum is fitted to the sum of a MC ‘standard’ spectrum
(d)
0.5 < |cos #| < 0.7
NS produced with unknown Michel parameters ρH , ηH ,
1.005
ξH , δH and additional ‘derivative’ MC distributions N∆ρ ,
N∆ξ , and N∆ξδ , with ∆ρ, ∆ξ, and ∆ξδ as the fitting pa1
−5The hidden
−2Michel parameters associated with
rameters.
0.995
NS are revealed only after all data analysis has been completed. The fiducial region adopted for this analysis re0.99
quires p < 50 MeV/c, |pz | > 13.7 MeV/c, pT < 38.5
(e)
0.5 < |cos #| < 0.7
MeV/c, and 0.50 < | cos θ| < 0.84.
2
2
Fits over
W
W this momentum range including both ρ and η
0
contain very strong correlations [3]. To optimize the precision for ρ, η was fixed at the hidden value ηH through-2
out the blind analysis, then a refit was performed to shift
0
10
20
30
40
50
η to the accepted value. We find that ρ depends linearly
Momentum (MeV/c)
on the assumed value of η, with dρ/dη = 0.018.
J.R. set
Musser
et al., Physical Review Letters 94, 101805 (2005)
Figure 3(a) shows the momentum spectrum from
B
FIG. 3: (color online) Panel (a) shows the muon decay spec-14
in the angular range 0.70 < | cos θ| < 0.84. The probabil-
• Applying small corrections for finite electron
•
•
0.70 < |cos #| < 0.84
1500
0
00
(Data - Fit)/$
prediction!
-0.3
Probability
• Excellent agreement between data and
(Data - Fit)/ $
0
→ e+ νe ̅νµ decay spectrum.
-0.1
Yield (Events)
• TWIST experiment at TRIMF in Canada measures µ
+
3
Summary
Renormalisation
• QED: Photons and fermions can appear in loops.
(effective charge) and masses of fermions.
Effectively modifying coupling
• This leads to running coupling α as a function of energy of scattering q.
• This happens in QED, Weak and QCD
Weak Charged Current
gµν
• Carried by the massive W-boson: acts on all quarks and leptons. q2 − m2
W
• A W-boson interaction changes the flavour of the fermion.
• Acts only on the left-handed components of the fermions: V−A structure.
ū(νe )γ µ (1 − γ 5 )u(e− )
• At low energy, responsible for muon & tau decay, beta decay…
Weak Neutral Current
gµν
• Carried by the massive Z-boson: acts on all quarks and leptons. q2 − m2
Z
• No flavour changes observed.
ū(e)γ µ (ceV − ceA γ 5 )u(e)
• At low energies, responsible for neutrino scattering
15