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
• No lecture on Friday.
• Next lecture Tuesday 14th February!
• Tutorial next Monday: will post problem sheets and notes on the
website.
• Few typos found in posted lecture slides: see website.
2
using a set of simple rules
Basic Feynman Rules:
Basic Feynman Rules:
Propagator factor for each internal line
!" "
e+ +
Propagator(i.e.
factor
eachvirtual
internal
line
#
eachfor
internal
particle)
!
e
#
(i.e.
each
internal
virtual
particle)
Dirac Spinor for each external line
Dirac(i.e.
Spinor
external
line
each for
realeach
incoming
or outgoing
particle)
–
–
!–
e–
(i.e.
each
real
incoming
or
outgoing
particle)
Vertex
factor
for
each
vertex
e calculations! use spinors to define fermion current in Feynman
QED
Vertex factor for each vertex
From last Friday: Summary
• For full
diagrams.
µ
Michaelmas
2011
M.A. Thomson
ψ̄γ
ψ where
ψ̄ is the adjoint spinor:117
ψ̄ ≡ ψ † γ 0
QED,
fermion currents are Michaelmas
• ForProf.Prof.
2011
117
M.A. Thomson
• Spin-1 bosons are described by polarisation vectors, !µ (s)
for an unpolarised process need to average over
• To calculate the cross section
Basic
Rules
for QED
initial spins (helicities) and
sum Rules
over
allfor
possible
Basic
QEDfinal possibilities
External
Lines
energy scattering, E>>m, helicity is conserved.
• For highExternal
Lines
incoming particle
incoming
outgoingparticle
particle
outgoing particle
incoming antiparticle
incoming
outgoingantiparticle
antiparticle
outgoing antiparticle
incoming photon
incoming
outgoingphoton
photon
outgoing photon
spin 1/2
spin 1/2
spin 1
spin 1
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
Matrix Element
= product of all factors
3
Michaelmas 2011
Prof. M.A. Thomson
118
Higher Order QED
Michaelmas 2011
Prof. M.A. Thomson
118
Two photon “box” diagrams also contribute to QED scattering
*+!
/!
*,!
Two extra vertices
*-!
*+1!/!
/!
*,-!/!
!"!#$%&&'(!!
*+-!/!
0!-!/!
*.!
*-!
/!
*,!
/!
*--!/!
0!- /!
!)!#$%&&'(!!
*.!
contribution is suppress by a factor of # = 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)
4
Higher Order QED Diagrams - II
• A virtual photon can
be emitted and
reabsorbed across a
vertex.
• This effectively
changes the coupling
at the vertex.
• Fermion can emit and
reabsorb a virtual photon.
• In a virtual photon
propagator a fermionantifermion pair can be
produced and then
annihilate.
• Effectively modifies the
fermion wavefunction.
• When the photon is
emitted the fermion
becomes virtual and
changes four-momentum.
• This modifies the photon
propagator.
• Any charged fermion is
allowed
Each of these diagram adds two vertices suppressing M by factor of ! = 1/137
Four momentum flowing around the “loop” can be anything!
5
Renormalisation
• Impose a “cutoff” mass M, do not allow the loop four momentum to be larger
than M. Use M2 >> q2.
! 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 !(q2) are now functions of q2
• e.g. Renormalisation of electric charge (considering
only effects from one type of fermion):
�
� 2�
e2
M
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!
6
.-*##/
6)"($-7"89(.#):%5(*#$$($+$,"#)%(
,#$."/)%(.%5(.%%/'/+."/)%
)*(;/#":.+($+$,"#)%<7)4/"#)%(7./#4
Running Coupling Constant
correct for all possible fermion types in
• Renormalise !, and
0*122'&'(
�
�
the loop:
2
α(0)
−q
=.#$(,'.#&$
α(q 2 ) = α(0)
1 + .%5(-.44
zf ln( )*($+$,"#)%()%+8(;/4/>+$(
)
3π
M2
."(;$#8(4')#"(5/4".%,$4
all possible fermions+.#&$#(-)-$%":-("#.%4*$#
in the loop
/%,#$.4$4(?/"'(?/"'
• zf is the sum of charges!over
" At q2 ~ 1 MeV only electron, zf = 1
" At q2 ~ 100 GeV, f=e,µ,",u,d,s,c,b zf = 38/9
zf =
�
Q2f
f
• Instead of using M2 dependence, replace with a
reference value µ2:
e$e# % & $ & #
�
�−1
α(µ2 )
q2
α(q ) = α(µ ) 1 −
zf ln( 2 )
3π
µ
2
2
• Usual choices for µ are 1 MeV or mZ ~ 91 GeV.
" !(µ2=1 MeV2) = 1/137
" !(µ2=(91 GeV)2) = 1/128
@+.44/,.+(+/-/"
of µ where make a initial
• We choose a value
measurement of #, but
once we do the evolution
A1 0(B((((((((((((((!
! 0(C2CDE
of the values of # are determined by the above eqn.
F"(4')#"(5/4".%,$4
A1 0(GHB(I$JK1 ! ! 0(C2C1L
Nuclear and Particle Physics
Franz Muheim
7
12
Review: Weak Charged Current
• Exchange of heavy W+ or W# bosons (mW = 80.40±0.03 GeV)
• The exchange of W± bosons changes the flavour of the interacting fermion.
• Lepton couplings are between e#$ %e µ# $ %µ "# $ %" and antiparticle equivalents
• No lepton flavour changes allowed: lepton numbers Le, Lµ, L! observed to be
conserved.
" Le = N(e#) " N(e+) + N(%e) " N(%!e)
" Lµ = N(µ#) " N(µ+) + N(%µ) " N(%!µ)
" L! = N("#) " N("+) + N(%") " N(%!")
• All lepton couplings are equal, ! gW
• Quark couplings between dj $ ui (ui=u,c,t, dj=d,s,b)
• All possible quark flavour changes allowed
• Baryon number, B = #(N(q) " N(q! )) conserved.
gW VCKM(ui dj)
• Coupling strength depends on quark flavours involved,
√ 2
2gW
• gW is measured experimentally
GF =
from muon decay (see later today)
8m2W
8
incoming antiparticle
ming particle
outgoing antiparticle
oing particle
incoming photon
spin
ming1 antiparticle
outgoing photon
Feynman Rules for Charged Current
oing antiparticle
ernal
Lines (propagators)
spin 1photon
photon
ming
oing photon
spin 1/2
fermion
propagator
opagators)
spin 1/2
interaction
vertex
fermion (charge -|e|)
on
rix Element
mson
$
!
tex Factors
on
$
!
gµν
= product
of all factors
1
√
g γ µ (1 − γ 5 )
W-boson
2
2
2 2 W
q − mW
Michaelmas 2011
gµν
photon, $
q2
= product of all factors
• W-boson has:
Michaelmas 2011
on (charge -|e|)
• Left-handed interactions known as
V!A theory
" $µ gives a vector current (V)
118
e γµ
" $µ$5 gives an axial vector current (A)
•Photon interactions are purely vector
118
" q2 # mW2 as denominator of propagator
" Factor of gW(1"&5)/$8 in vertex term
" Quark couplings have extra factor of Vuidj
•
•
•
•
•
The (1"&5) term is observed experimentally.
Overall factor of 1/'8 is conventional
Recall PL=(1"&5)/2 is the Left Handed projection operator
W-boson interactions only act on left-handed components of fermions
For low energy interactions q << mW: effective propagator is gµ%/mW2
9
“Inverse Muon Decay”
• Start with a calculation of the process %µ e! & µ! %e
• Not an easy process to measure experimentally,
%µ
µ!
but easy to calculate!
!
e
g2
gµν
ν
5
u(ν
)γ
(1
−
γ
)u(µ)
M = W ū(νe )γ µ (1 − γ 5 )u(e− ) 2
µ
8�
q − m2W
�2
|M| =
2
2
gW
8m2W
%e
�
�2 �
�2
ū(νe )γ µ (1 − γ 5 )u(e− )
ū(µ)γ µ (1 − γ 5 )u(νµ )
• Usually we would average over initial spin and sum over final spin states:
• However the massless neutrino only has one possible spin state (↓): left handed.
• The equation can�be solved
�2 as (see Griffiths section 9.1):
|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
gW
m2W
�2
�2 �
2
mµ
1−
2E 2
10
“Inverse Muon Decay” Cross Section
scattering:
• From lecture 4,�for 2→2
�2
p ∗f |
S|M|2 |�
dσ
1
=
dΩ
8π
(E1 + E2 )2 |�
p ∗i |
%µ
µ!
e!
%e
• Substitute:
! centre of mass energy, (E1+E2)2=4E2
! For elastic scattering particle |p*f|=|p*i|
! S=1 as no identical particles in final state
! Fermi coupling constant GF = '2gW2/8mW2�
� 2 �2
dσ
E2
gW
=
1
dΩ
32π 2 m2W
−
m2µ
2E 2
�2
! Unlike electromagnetic interaction, no angular dependence
! Integral over 4( solid angle
�
=
σ
dσ
4
dΩ = E 2 G2F
dΩ
π
�
m2µ
1−
2E 2
�2
11
Muon Decay
%!e
muon decay µ! & e! %!e %µ
• Examine
�
�
|M| =
2
=2
�
2
gW
8m2W
2
gW
m2W
�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 ))
• We haven’t done 1 → 3 space space, quote result from Griffiths:
dΓ
1
=
dEe
4π 3
�√
2
2gW
2
8MW
�2
m2µ Ee2
• Integrate over allowed values of Ee:
Γ=
�
0
mµ /2
G2F m2µ
dΓ
dEe =
dEe
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! %!e %µ) " 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
12
! p (MeV/c)
Muon Decay Measurements
0.2
Data
0.1
Monte Carlo
2
2000 "10
(a)
• TWIST experiment at TRIMF in Canada measures µ
& e+ %e !%µ decay spectrum.
-0.1
Yield (Events)
0
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 = ρ +6∆ρ and η = η + ∆η, then the angle! # = (2.19703±0.00002)$10
s
integrated muon decay spectrum can be written as:
H
H
! m = 105.658367 ± 0.000004 MeV
N (x) = NS (x, ρH , ηH ) + ∆ρN∆ρ (x) + ∆ηN∆η (x). (3)
This expansion is exact. It can also be generalized to
include the angular dependence [2]. This is the basis for
the blind analysis. The measured momentum-angle spectrum is fitted to the sum of a MC ‘standard’ spectrum
NS produced with unknown Michel parameters ρH , ηH ,
ξH , δH and additional ‘derivative’ MC distributions N∆ρ ,
N∆ξ , and N∆ξδ , with ∆ρ, ∆ξ, and ∆ξδ as the fitting pa!5The hidden
!2Michel parameters associated with
rameters.
NS are revealed only after all data analysis has been completed. The fiducial region adopted for this analysis requires p < 50 MeV/c, |pz | > 13.7 MeV/c, pT < 38.5
MeV/c, and 0.50 < | cos θ| < 0.84.
2
Fits over
W
W this momentum range including both ρ and η
contain very strong correlations [3]. To optimize the precision for ρ, η was fixed at the hidden value ηH throughout the blind analysis, then a refit was performed to shift
η to the accepted value. We find that ρ depends linearly
on the assumed value of η, with dρ/dη = 0.018.
J.R. set
Musser
Figure 3(a) shows the momentum spectrum from
B
in the angular range 0.70 < | cos θ| < 0.84. The probability for reconstructing muon decays is very high, as shown
in Figs. 3(b) and (d). Thus, higher momentum decays
that undergo hard interactions and are reconstructed at
lower momenta can lead to an apparent reconstruction
probability above unity. Figures 3(c) and (e) show the
residuals of the fit of the decay spectrum from set B.
Similar fits have been performed to the other data sets,
yielding the results shown in Table I. The fit results for δ
are not adopted due to a problem with the polarizationdependent radiative corrections in the event generator
[2], but are consistent with the separate analysis reported
in [2]. Fits to the angle-integrated spectra, which are independent of δ, give nearly identical results for ρ.
The 11 additional data sets have been combined with
further MC simulations to estimate the systematic uncertainties shown in Tables I and II. The largest effects
• Applying small corrections for finite electron
mass and second order effects
! GF = 1.166364(5) $ 10 GeV
•
•
Implies gW = 0.653, ! =g
/4% = 1/29.5
#W >> #EM, the weak force not intrinsically weak,
just appears so due to mass of W-boson
Summary
Renormalisation
• QED: Photons and fermions can appear in loops.
(effective charge) and masses of fermions.
0.70 < |cos #| < 0.84
1500
All events
In fiducial
1000
500
00
0
Probability
prediction!
10
20
30
40
50
(b)
0.7 < |cos #| < 0.84
(c)
0.7 < |cos #| < 0.84
(d)
0.5 < |cos #| < 0.7
(e)
0.5 < |cos #| < 0.7
1.005
1
0.995
0.99
(Data - Fit)/ $
-0.2
-0.3
Probability
• Excellent agreement between data and
2
0
-2
1.005
1
0.995
0.99
(Data - Fit)/$
+
3
2
0
-2
0
10
20
30
40
Momentum (MeV/c)
50
et al., Physical Review Letters 94, 101805 (2005)
FIG. 3: (color online) Panel (a) shows the muon decay spec13
trum (solid curve) from surface muon set B vs. momentum,
for events within 0.70 < | cos θ| < 0.84, as well as the events
within this angular region that pass the fiducial constraints
(dot-dashed curve). The spectrum within 0.5 < | cos θ| < 0.7
is similar. Panels (b) and (d) show the probability for reconstructing decays for the two angular ranges, as calculated by
the Monte Carlo. Panels (c) and (e) show the residuals for
the same angular ranges from the fit of set B to the Monte
Carlo ‘standard’ spectrum plus derivatives.
arise from time-variations of the cathode foil locations [9]
and the density of the DME gas, which change the drift
velocities and influence the DC efficiencies far from the
sense wires. These parameters were monitored throughout the data taking, but only average values were used in
the analysis. Special data sets and MC simulations that
Effectively modifying coupling
• This leads to running coupling # 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
14