Download The Higgs Boson and Fermion Masses

H
Quarks – “the building
blocks of the Universe”
The number of
quarks increased
with discoveries of
new particles and
Charm came as
surprise but
completed the
picture
have reached
For unknown
reasons Nature
created 3 copies
(generations) of
quarks and leptons
6
Discovery History
u
d
ne
nm
nt
1995
1956
1963
2000
s
b
e
m
t
1947
1977
1895
1936
1975
c
t
1974
six quarks
six leptons
Now we have a beautiful pattern of three pairs of quarks and
three pairs of leptons. They are shown here with their year of
discovery.
Matter and Antimatter
The first
generation is
what we are
made of
Antimatter was
created together
with matter
during the “Big
bang”
Antiparticles are created at accelerators in ensemble with
particles but the visible Universe does not contain antimatter
Quark’s Colour
Baryons are “made” of quarks

 ( d  d  d )
  ( s  s  s )
?
  (u  u  u )
To avoid Pauli principle veto one can
antisymmetrize the wave function
introducing a new quantum number
- “colour”, so that
   (di  d j  dk )

ijk
The Number of Colours

The x-section of
electron-positron
annihilation into
hadrons is proportional
to the number of quark
colours. The fit to
experimental data at
various colliders at
different energies gives
Nc = 3.06  0.10
The Number of Generations
 Z-line
Ng = 2.982  0.013
shape
obtained at LEP
depends on the
number of
flavours and
gives the
number of (light)
neutrinos or
(generations) of
the Standard
Model
Quantum Numbers of Matter

SU(3)c
Quarks

UY(1)
doublets
triplets
 up 
QL  

 down  L
U R  upR
DR 
SU(2)L
V-A
downR currents in
weak
interactions
3
3
3
2
1
1
1/ 3
4/3
2 / 3
Leptons
n 
1
2
LL    1
 e L - 2
NR n R ?
0
ER  eR
0
singlets
T3
T3
1
1
1
2
1
1
1
0
2
Electric charge
Q  T3  Y / 2
The group structure of the SM
Casimir Operators
For SU(N)
QCD analysis
definitely singles out
the SU(3) group as
the symmetry group of
strong interactions
Electro-weak sector of the SM
SU(2) x U(1) versus O(3)
3 gauge bosons
1 gauge boson
3 gauge bosons
After spontaneous symmetry breaking one has
2 massive gauge bosons
(W+ , W- ) and 1 massless (γ)
3 massive gauge bosons
(W+ , W- , Z0) and 1 massless (γ)
 Discovery of neutral currents was a
crucial test of the gauge model of
weak interactions at CERN in 1973

The heavy photon gives the
neutral current without flavour
violation
Gauge Invariance
 i ( x)  U ij ( x) j  exp[i a ( x)Tij a ] j
Gauge transformation

ji
 i ( x)   j U ( x)
matrix
a  1, 2,..., N

parameter matrix


m
i ( x)  m ( x)  i ( x)U ( x) m  m U ( x) ( x)
Fermion Kinetic term

U U 1

 i ( x)  m ( x)  ( x) U ( x) m U ( x) ( x)
m
m
 m  Dm   m I  gAmaT a   m I  g Am
Covariant derivative


Dm ( x)  U ( x) Dm ( x)
Am ( x)  U ( x) Am ( x)U ( x)   m U ( x)U ( x)
Gauge invariant kinetic term
1
g
i ( x) m Dm ( x)
[ Dm , Dn ]  G mn   m An  n Am  g[ Am , An ]
Gauge field kinetic term
Gauge field
 Tr G mn G
1
4

G mn ( x)  U ( x)G mn ( x)U ( x)
mn
Field strength tensor
Lagrangian of the SM
SU c (3)  SU L (2) UY (1)
L  Lgauge  LYukawa  LHiggs
a
a
i
i
Lgauge   14 Gmn
Gmn
 14 Wmn
Wmn
 14 Bmn Bmn
iL  Dm L  iQ  Dm Q  iE   Dm E
m
m
m
iU   m DmU  iD  m Dm D  ( Dm H )† ( Dm H )
LYukawa  y L E H  y Q D H  y QU  H
L
D
U
LHiggs  V  m H H  ( H H )
2
†

2
†
2
H  it 2 H †
α,β=1,2,3 - generation index
Fermion Masses in the SM
Direct mass terms are forbidden due to SU(2)L invariance !
Dirac Spinors
left
right
Dirac conjugated
Charge conjugated
1  5
1  5
, L 
, R 
 ,     0 ,  c  C 0  i 2 *
2
2
Lorenz invariant Mass terms
 L R
SUL(2)
  R L
 L L   R R  0
Unless Q=0, Y=0
SU(2) doublet SU(2) singlet
SUL(2) & UY(1)
c
 L L  L Lc
UY(1)
c
 R R  R Rc
c
R
n nR
Majorana mass term
Spontaneous Symmetry Breaking
SU c (3)  SU L (2) UY (1)  SU c (3) U EM (1)
H 
Introduce a scalar field with quantum numbers: (1,2,1) H  
0
H


With potential
V  m 2 H † H  2 ( H † H ) 2
Unstable maximum
At the minimum
v.e.v.
scalar
Stable minimum



H
H  


H  0
S  iP   exp(i
2
H  v

2 

0

)
S

v
2






pseudoscalar
Gauge transformation
Higgs boson


(   )
h 
H  H   exp(i
) H 
H  
S


2
v

2


0
The Higgs Mechanism
Q: What happens with missing d.o.f. (massless goldstone bosons P,H+ or ξ ) ?
A: They become longitudinal d.o.f. of the gauge bosons Wμi, i=1,2,3
Gauge transformation
 a   a
i a a
Wm e
Wm e
i a a
e
i a a
Longitudinal components
2
Higgs field kinetic term
Dm H   m H  W m H 
 gWm3  g ' Bm
 14 (0 v) 
+

2gW
m

 gWm3  g ' Bm

3
+
-gWm  g ' Bm 
2gW
m

g
2
2gWm-
g2 2   1 2

v Wm Wm  v ( gWm3  g ' Bm )2
2
4
tan W  g  / g
M W2  12 g 2 v 2
M Z2  12 ( g 2 +g'2 )v 2

i a a
 m e
1
g
M  0
g'
2
2
Bm H
0
H  
 v
 0
 
3
-gWm  g ' Bm   v 
Wm 
2gWm-
Wm1 Wm2
2
Zm   sin W Bm  cos W Wm3
 m  cosW Bm  sin W Wm3
The Higgs Boson and Fermion
Masses
 0 

H 
h
 v 

2

V  m 2 H † H  2 ( H † H ) 2
V 
 v4
2
 v h 
2 2
v
2
h 
3

8
h4
v2  m2 / 
mh  2m  2 v
LYukawa  y L E H  y Q D H  y QU  H
E
D
U
α, β =1,2,3 - generation index
Dirac fermion mass
u
d
l
M iu  Diag ( y
)v, M id  Diag ( y
)v, M il  Diag ( y
)v
N
N
y
L N  H  M in  Diag ( y
)v
Dirac neutrino mass
The Running Couplings
~α (log Λ2/p2 +fin.part)
Radiative Corrections
Renormalization operation
 Bare ()  Z ( / m )  R (m )
UV cutoff
 R (m )
 Log (2 / p 2 )   Log (2 / m 2 )   Log ( m 2 / p 2 )
d
m
  ( ),
2
dm
2
Renormalization scale
Z ( / m )  1  b Log ( 2 / m 2 )  ...
Renormalization constant
Subtraction of UV div
UV divergence
 ( )   m
Running coupling
2
d
dm
2
Log Z ( / m )
Finite
Renormalization Group
Observable
RPT (Q 2 m 2 ,  ( m ))   (1  b Log (Q 2 m 2 )  O( 2 ))
d 
m
R  (m
m
)R  0
2
2
2
dm
m
d m 
2
RG Eq.
d
2

2
 ( )
Solution to RG eq.
RRG (Q 2 m 2 ,  ( m ))  RPT (1,  (Q 2 m 2 ,  ))
Q2
d
  ( )
2
dQ
Effective coupling
Solution to RG eq. sums up an infinite series of the leading
Logs coming from Feynman diagrams
RPT (1,  )   ,  

1  b Log (Q 2 m 2 )
  (1  b Log (Q 2 m 2 )  ...)
Asymptotic Freedom and
Infrared Slavery
One-loop order
 ( )  b
4/3 nf
QED

2
-11+2/3 nf
QCD
_
α
_
α
QED
QCD
UV Pole
IR Pole

1  b Log (Q 2 m 2 )
Comparison with Experiment
Global Fit to Data
Remarkable agreement of ALL the data
with the SM predictions - precision tests
of radiative corrections and the SM
Higgs Mass Constraint
Though the values of sin w extracted
from different experiments are in good
agreement, two most precise
measurements from hadron and lepton
asymmetries disagree by 3
The SM and Beyond
The problems of the SM:
• Inconsistency at high energies due to Landau poles
• Large number of free parameters
• Still unclear mechanism of EW symmetry breaking
• CP-violation is not understood
• The origin of the mass spectrum in unclear
• Flavour mixing and the number of generations is arbitrary
• Formal unification of strong and electroweak interactions
The way beyond the SM:
• The SAME fields with NEW
interactions and NEW fields
• NEW fields with NEW
interactions
GUT, SUSY, String, ED
Compositeness, Technicolour,
preons
We like elegant solutions