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Lectures 1 & 2
The Violation of Symmetry between
Matter and Antimatter
Andreas Höcker
CERN
CERN Summer Student Lectures, August 7-10, 2007
CERN Summer Student Lectures 2007
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
M.C. Escher
1
A definition, which we will understand later in this lecture:
The matter-antimatter symmetry violation
in physics reactions corresponds to the breaking of the
so-called CP symmetry  C∙P
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violation
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
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Matter-Antimatter Asymmetry
q
q
1
Early universe
?
Current
universe
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Sakharov Conditions
Is baryon asymmetry initial condition ? Possible ?
Dynamically generated ?
Sakharov conditions (1967) for Baryogenesis
1.
2.
Baryon number violation  new physics !
C and CP violation
 (probably) new physics !
3.
Departure from thermodynamic equilibrium (non-stationary system)
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Note: ring is not necessarily due to dark matter !
Something Else is Strange Out There …
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Hubble space telescope
picture of Cluster
ZwCl0024+1652
Image: NASA, ESA, M.J.
JEE AND H. FORD (Johns
Hopkins University)
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
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Something Else
is Strange
Dark
Matter Out There …
Dark matter does not emit or reflect
sufficient electromagnetic radiation to
be detected
Evidence for dark matter stems from:
gravitational lensing
kinetics of galaxies
Bullet cluster: Collision of galaxy clusters: baryonic matter,
stars – weakly affected by collisions – and
strongly
Dark
energy
affected gas (pink in picture),Image
and collisionless
dark matter
Dark matter
(blue)
Free H and He
Stars
Neutrinos
65%
Heavy elements
strongest gravitational lensing
Foreground
cluster
anisotropy of cosmic microwave
background (blackbody) radiation
30%
0.03%
0.03%
0.5%
4%
gravitational lensing
Mass density contours superimposed over photograph
taken with Hubble Space Telescope
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And: There is Much More Strangeness …
Empirical and Theoretical
Limitations of the
Standard Model
Dark matter (and, perhaps, dark energy)
Baryogenesis (CKM CPV too small)
Grand Unification of the gauge couplings
The gauge hierarchy Problem (Higgs sector, NP scale ~ 1 TeV)
The strong CP Problem (why is  ~ 0 ?)
Neutrino masses
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The understanding of matter-antimatter
symmetry violation
is crucial if we want to move closer into the heart of the Big
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Bang
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
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Lecture Themes
I.
Introduction
Antimatter
Discrete Symmetries
II.
The Phenomena of CP Violation
Electric and weak dipole moments
The strong CP problem
The discovery of CP violation in the kaon system
III.
CP Violation in the Standard Model
The CKM matrix and the Unitarity Triangle
B Factories
CP violation in the B-meson system and a global CKM fit
Penguins
IV.
CP Violation and the Genesis of a Matter World
Baryogenesis and CP violation
Models for Baryogenesis
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digression:
CP Violation, a Family History of Flavour
Discovery of strange particles
Neutral kaons can mix
(Rochester, Butler)
(Gell-Mann, Pais)
(1946-47)
(1952)
KL discovery (Lederman et al.)
(1956)
Parity (P) violation: possible explanation
(Lee, Yang) (1956)
P Violation found in  decay
(Wu et al.)
later: maximum P and C violation, but CP invariance
(1957)
Cabibbo-Theory
(1963)
CP violation (CPV) discovered
GIM-Mechanism
(Glashow, Illiopolous, Maiani)
J/ Resonance: c quarks
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(Cronin, Fitch et al.)
(Ting, Richter)
(1964)
(1970)
(1974)
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
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digression:
CP Violation, a Family History of Flavour
CPV Phase requires 3 families
Discovery of  lepton: 3rd family
(Kobayashi-Maskawa)
(Perl et al.)
(1973)
(1975)
 resonance: b quarks
(Lederman et al.)
(1977)
Neutral Bd mesons mix
(ARGUS)
(1987)
t-Quark discovery
(CDF)
(1995)
-Oscillation discovery (Super-K)
Direct CP violation in K system
(1998)
(NA31, NA48, KTeV)
(1999)
Start of Bd Factories: BABAR (PEP II), Belle (KEKB)
(1999)
CPV in Bd system : sin(2)  0
(2001)
Direct CPV in Bd system
(BABAR, Belle)
(BABAR, Belle)
(2004)
Bs mesons oscillate
(CDF)
(2006)
D0 mesons oscillate
(BABAR & Belle)
(2007)
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Through the
Looking
Glass
What’s the
Matter with
Antimatter ?
David Kirkby, APS, 2003
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Paul Dirac (1902 – 1984)
Combining quantum mechanics with special relativity,
and the wish to linearize /t, leads Dirac to the equation
i     x,t   m  x,t   0
(1928)
for which solutions with negative energy appear
Dirac, imagining holes
and seas in 1928
Energy
E 
me
s  1/ 2
Dirac identified holes in this sea as “antiparticles” with
opposite charge to particles … (however, he conjectured
that these holes were protons, despite their large difference in mass,
because he thought “positrons” would have been discovered already)
0
me
Vacuum represents a “sea” of such negative-energy
particles (fully filled according to Pauli’s principle)
E
s  1/ 2
This picture fails for bosons !
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An electron with energy E can fill this hole, emitting an
energy 2E and leaving the vacuum (hence, the hole
has effectively the charge +e and positive energy).
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
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Positron – Antiparticle
incoming of the Electron
Antineutron
Antiproton
incoming
antiproton
discovery 1955
antiproton
6 GeV Fixed target threshold
1956
Discovered
in cosmic rays by Carl Anderson in 1932 (Caltech)
energy required to produce
p + p  p + p + anti-p + p
Reproduction
Antiproton chargeof an
Has the same mass as the electron but positive charge
antiproton
exchangeannihilation
reaction into
star
asneutron-antineutron
seen in nuclear emulsion
pair in
63 MeV
(source:
O. Chamberlain,
Lecture)
propane
bubble Nobel
chamber
Anderson saw a track in a cloud
positron
chamber left by “something
(source: E.G. Segrè, Nobel Lecture)
track
positively charged, and with the
same mass as an electron”
6mm Pb plate

p + anti-p  n + anti-n
History of antiparticle discoveries:
23 MeV
“annihilation star”positron
(large energy release track
from antiproton destruction)
1955: antiproton (Chamberlain-Segrè, Berkeley)
“annihilation star”
antineutron (Cork et al., LBNL)
(large energy release
1965: antideuteron (Zichichi, CERN and Lederman,from
BNL)antineutron destruction)
1995: antihydrogen atom (CERN, by now millions produced !)
Every particle has an antiparticle
Some particles (e.g., the photon) are their own antiparticles !
outgoing
1956:
charged
particles

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Particles and Antiparticles Annihilate
Fermilab
What happens if we bring particles and antiparticles together ?
A modern
A particle
can annihilate with its
example:
antiparticle to form gamma rays
An example whereby matter is
converted into pure energy by
Einstein’s formula E = mc2
Conversely, gamma rays with
sufficiently high energy can turn
ALEPH
into a particle-antiparticle pair
Higgs candidate
ee  ZH (Z )  qqbb
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Particle-antiparticle tracks in a
bubble chamber
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Symmetries
A symmetry is a change of something that leaves
the physical description of the system unchanged.
1.
Physical symmetries:
People are approximately bilaterally symmetric
Spheres have rotational symmetries
2.
Laws of nature are symmetric with respect to mathematical operations,
that is: an observer cannot tell whether or not this operation has occurred
Pollen of the hollyhock exhibits spherical
symmetry (magnification x 100,000)
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Continuous Symmetries and Conservation Laws
In classical mechanics we have learned that to each continuous symmetry
transformation, which leaves the scalar Lagrange density invariant, can be
attributed a conservation law and a constant of movement (E. Noether, 1915)
Continuous symmetry transformations lead to additive conservation laws
Symmetry
Invariance under
movement in time
Homogeneity of
space
Isotropy of space
Transformation
Translation in time
Translation in
space
Rotation in space
Conserved
quantity
Energy
Linear momentum
Angular
momentum
No evidence for violation of these symmetries seen so far
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digression: Symmetry of Reference Systems
Another type of symmetry has to do with reference frames moving with respect
to one in which the laws of physics are valid (inertial reference frames):
Physical laws are unchanged when viewed in any reference frame
moving at constant velocity with respect to one in which the laws are valid
Note that while laws are unchanged between reference frames, quantities are not
The fact that the laws of motion are unchanged between frames, plus the fact that
the speed of light is always the same lead to the theory of special relativity with two
consequences
Two events that are simultaneous in one reference frame are not
necessarily simultaneous in a reference frame moving with respect to it
There are some quantities (called Lorentz scalars) that have values
independent of the reference frame in which their value is calculated
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Continuous Symmetries and Conservation Laws
In general, if U is a symmetry of the Hamiltonian H, one has: H,U   0  H  U †HU
f  H i   Uf H Ui  f U †HU i  f H i
Accordingly, the Standard Model Lagrangian satisfies local gauge symmetries
(the physics must not depend on local (and global) phases that cannot be observed):
U(1) gauge transformation

Electromagnetic interaction
SU(2) gauge transformation

Weak interaction
SU(3)C gauge transformation

Strong interaction (QCD)
Conserved additive quantum numbers:
Electric charge (processes can move charge between quantum fields, but the sum of all charges is constant)
Similar: color charge of quarks and gluons, and the weak charge
Quark (baryon) and lepton numbers (however, no theory for these, therefore believed to be only
approximate asymmetries)  evidence for lepton flavor violation in “neutrino oscillation”
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Discrete Symmetries
Discrete symmetry transformations lead to multiplicative conservation laws
The following discrete transformations are fundamental in particle physics:
Parity P (“handedness”):
Reflection of space around an arbitrary center;
In particle physics:
P eL  eR
These are interesting because it is not obvious whether
 
the laws of nature should look the same for anyP of
these
Particle-antiparticle transformation C :
P n  n
transformations, and the answer was surprising
when
Change of all additive quantum numbers (for example the
Ce  e
these symmetries
were
firstconjugation”)
tested !
electrical
charge) in its opposite
(“charge
P invariance  cannot know whether we live in this world, or in its mirror world
0

L
0

L
Cu  u
Time reversal T :
Cd  d
The time arrow is reversed in the equations;
T invariance  if a movement is allowed by a the physics law, the movement in
the opposite direction is also allowed
C 0   0
Time reversal symmetry (invariance under change of time direction) does certainly not correspond to our daily experience. The
macroscopic violation of T symmetry follows from maximising thermodynamic entropy (leaving a parking spot has a larger solution space
than entering it). In the microscopic world of single particle reactions thermodynamic effects can be neglected, and T invariance is realised.
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C, P, T Transformations and the CPT Theorem
Quantity
P
C
T
–x
x
x
t
t
–t
–p
p
–p
s
s
–s
Electrical field
–E
–E
E
Magnetic field
B
–B
–B
Space vector
Time
Momentum
Spin
The CPT theorem (1954): “Any Lorentz-invariant local quantum field
theory is invariant under the successive application of C, P and T ”
proofs: G. Lüders, W. Pauli; J. Schwinger
Fundamental consequences:
Relation between spin and statistics: fields with integer spin (“bosons”) commute and
fields with half-numbered spin (“fermions”) anticommute  Pauli exclusion principle
Particles and antiparticles have equal mass and lifetime, equal magnetic moments
with opposite sign, and opposite quantum numbers
Best experimental test:
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m
K
0
 mK 0  / mK 0  1018
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If CPT is Conserved, how about P, C and T ?
Parity is often violated in the macroscopic world:
Strongly Left-sided
Strongly Right-sided
Mixed Sided
Handedness
5%
72%
22%
Footedness
4%
46%
50% (?)
Eyedness
5%
54%
41%
Earedness
15%
35%
60%
Porac C & Coren S. Lateral preferences and human behavior. New York: Springer-Verlag, 1981
About 25% of the population drives on the
left side: why ?
In ancient societies people walked (rode) on the left to have their
sword closer to the middle of the street (for a right-handed man) !?
The DNA is an oriented double helix
Two right-handed
polynucleotide
chains that are
coiled about the
same axis:
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Not so in the microscopic World ?
Electromagnetic and strong interactions are (so far) C, P and T invariant
Example: neutral pion decays via electromagnetic (EM) interaction : 0   but not 0  
0 
1
uu  dd 
L 0,S 0
2
C  B, E  B, E
, C  0    0
, C    
the initial (0) and final states () are C even: hence, C is conserved !
Generalization: P qq   1
L1
qq , C qq   1
LS
qq , G uu (d )   1
L S I
uu (d )
Experimental tests of P and C invariance of the EM interaction:


P invariance: BR   4   6.9  10
C invariance: BR  0  3  3.1 10 8
0
7
Experimental tests of C invariance of strong interaction: compare rates of positive and
negative particles in reactions like: pp     X , K K  X
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And … the Surprise in Weak Interaction !
T.D.
Lee
and C.N. Yang pointed out in 1956 (to explain the observation
Lee &
Yang:
of the decays K  2 and 3 - the cosmic-ray / puzzle) that P invariance
“Past
experiments
on theinteraction
weak interactions
hadperformed
actually noinbearing on
had not
been
tested in weak
 C.S. Wu
of parity
conservation.”
1957the
thequestion
experiment
they suggested
and observed parity violation
“In strong interactions, ... there were indeed many experiments that
Angular
distribution
of electron
intensity:
established
parity
conservation
to a high degree of accuracy...”
  Pe
v
unequivocally
whether
parity is conserved in weak interactions,
I (“To
)  1decide

 1   cos

E
c
e
one must perform
an experiment to determine whether weak interactions
differentiate
the right
from
where:
 - spin vector
of electron
helicity
the left.”
Pe - electron momentum
Ee - electron energy
Yang, C. N., The law of parity conservation and other symmetry
laws of physics, Nobel Lectures Physics: 1942-1962, 1964.
Lee, T. D., and C. N. Yang, Question of Parity Conservation in
Weak Interactions, The Physical Review, 104, Oct 1, 1956.
1 for electron
1 for positron
 
It was found that parity is even maximally
violated in weak interactions !
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TCO ~ 0.01 K
polarized in
magnetic field
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
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P and C Violation in Weak Interaction
Goldhaber et al. demonstrated in 1958 that in the  decay of the nucleus, the
neutrino (e–) is left-handed, while the antineutrino (e+) is right-handed:
Particle :
e
Helicity :  v / c
e
+v / c


1
1
(  C violation ! )
In the Dirac theory, fermions are described as 4-component spinor wave
functions upon which 44 Operators i apply, which are classified according
to their space reflection properties :
   † 0 , scalar (S )
 5 , pseudoscalar (P )
  4  4  current
Lorentz-covariant bilinear
  , vector (V )
  5 , axial vector (A)
          , tensor (T )
P -even
P -odd
P -even
P -odd
antisymmetric tensor
2i 
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P and C Violation in Weak Interaction
Let’s consider the  reaction: n    e   p
General ansatz for the current-current
some constants
matrix element :
G
M  F   p i n   e i Ci   5Ci  v 
2 i S,...
One
transparency
a bit
of math
Use the
helicity projection with
operators
for Dirac
spinors:

u u  uL  uR    uL…
 uRsorry
  uL u…
L  uR uR
so that one has for a V interaction:
while for a scalar interaction:
uL / R 
1
1  5  u
2
 at high energies, the EM interaction
conserves the helicity of the scattered fermions
won’t happen
again (almost) …
1
uL uR 
u L uR 
4
u  (1   52 )u  0
and similar for A
1
1
u (1   5 )u  0
2
and similar for P, T
Now, consider the weak neutrino-electron current in the relativistic limit:
ue V  A  u  ue 
1
1   5  u  ue,L  ue,R   u ,L  ue,L u ,L
2
It projects upon the left handed helicities, and hence violates P maximally, as required !
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P and C Violation in Weak Interaction
Weak interaction violates both C and P symmetries

 e      e
Consider the collinear decay of a polarized muon: polarized
P
The preferred emission
direction of the light
left-handed electron is
opposite to the muon
polarization.
handedness of the electron:
C
suppressed
P transformation (i.e.
reversing all three
directions in space)
yields constellation that
is suppressed in nature.
Similar situation for C
transformation (i.e.
replace all particles with
their antiparticles).
suppressed
handedness of the positron:
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B. Cahn, LBL
Applying CP, the resulting
reaction—in which an
antimuon preferentially
emits a positron in the
same direction as the
polarization—is observed.
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
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… and tomorrow, we will
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see
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
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Lecture Themes
I.
Introduction
Antimatter
Discrete Symmetries
II.
The Phenomena of CP Violation
Electric and weak dipole moments
The strong CP problem
The discovery of CP violation in the kaon system
III.
CP Violation in the Standard Model
The CKM matrix and the Unitarity Triangle
B Factories
CP violation in the B-meson system and a global CKM fit
Penguins
IV.
CP Violation and the Genesis of a Matter World
Baryogenesis and CP violation
Models for Baryogenesis
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CP Violation
CP Symmetry requires that processes and
their anti-processes have the same rates
1.
Due to the CPT theorem, CP symmetry also requires T symmetry
2.
CP violation would enable us to distinguish between particles and
antiparticles, and between past and future in an absolute way(*) !
two worlds that are far away want to gather together: if one were made of matter,
and the other one of antimatter, the spaceship of the one that landed on the planet of the other
would lead to a disastrous annihilation. This could be prevented by measuring a CP violation
parameter (if exists) by both worlds and comparing the results.
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© CPEP
(*)Imagine
31
Dipole moments
Can there be CP violation in the electromagnetic or neutral weak current ?
Let’s modify the Standard Model Lagrangian to allow for CP violation through
electromagnetic and weak dipole moments:
LCP  
i
   5
2
d
EM
F  d weak Z

where F and Z are electric and weak field strength tensors.
In the nonrelativistic limit one obtains the Pauli equation with the additional terms:
LCP  d EM E  d weak Z
But…. why do these dipole moments violate CP symmetry ?
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Dipole Moments and CP Violation
Spin is the only explicit “direction” of an elementary particle. Hence the dipole moment
must be proportional to it
d s
The electric dipole moment is the average of a charge density distribution  polar vector
d   d 3 x   (x)  x
spin
The spin has the form of angular momentum  axial vector
s r p
Parity transformation gives:
Pd  d , Ps  s

d 0
+
–
P
dipole
moment
–
+
P invariance
Time reversal transformation gives Td  d , Ts  s

d 0
T invariance
T
+
–
+
–
Non-vanishing electric or weak dipole moments require the
presence of a P- and T-violating (=CP-violating) interaction
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Experimental Limit on dEM (e.cm)
d(muon)  7×10–19
neutron:
electron:
10-20
10-20
10-22
10-22
10-24
10-24
10-26
Multi
Higgs
Electromagnetic
d(proton)  6×10–23
d(neutron)  6×10–
SUSY
26
d(electron)  1.6×10–
27
10-28
Left-Right
present experimental limits
10-30
10-30
none of this seen yet, why ?
1960 1970 1980 1990 2000
10-32
The Measurement of EDMs:
History of the experimental
progress
10-34
Standard Model
10-36
10-38
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34
digression: CP Violation in the QCD Lagrangian
It was found in 1976 that the perturbative QCD Lagrangian was missing a term L
LQCD 

LpQCD
perturbative QCD
L ,
where:
L  
P ,T -violating
s
8
a
G
G  ,a
, and G  ,a 
Gluon field tensors
1
  G ,a
2
dual field tensor
that breaks through an axial triangle anomaly diagram the U(1)A symmetry of LpQCD , which
is not observed in nature
when classical symmetries are broken on
the quantum level, it is denoted an anomaly
a
 ,a
a
The term G
is P-and T-odd, since:
G ,a contained in LpQCD is CP-even, while G G

GG   Ea  Ba
a

2
GG   Ea  Ba
a

2

 E
P,T


2
a
a

P,T

   Ea  Ba
a
 Ba
2


color electric and magnetic fields
Relativistic invariants,
similar to electric field
tensors: F F  , F F 
  F   j  ,   F   0
Maxwell equations
This CP-violating term contributes to the EDM of the neutron:
dn
  5  1016 ecm, so that  tiny or zero
"Strong CP (finetuning) Problem"
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35
The Strong CP Problem
Remarks:
If at least one quark were massless, L could be made to vanish; if all quarks are
massive, one has uncorrelated contributions, which have no reason to disappear
Peccei-Quinn suggested a new global, chiral UPQ(1) symmetry that is broken, with the
“axion” as pseudoscalar Goldstone boson; the axion field, a ,compensates the contribution
from L :

   a  ,a
axion coupling to SM particles is
L     a  s G
G
suppressed by symmetry-breaking
fa  8

scale (= decay constant)
QCD nonperturbative effects (“instantons”) induce a potential for a with minimum at a =  fa
The axion mass depends on the UPQ(1) symmetry-breaking scale fa
 107 GeV 
ma  
  0.62 eV ,
f
(GeV)
 a

and axion coupling strength: ga  ma
If fa of the order of the EW scale (v), ma~250 keV  excluded by collider experiments
CERN Summer Student Lectures 2007
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36
The Search for Axions (the axion is a dark matter candidate)
The axion can be made “invisible” by leaving scale and coupling free, so that one has:
ma ~ 10–12 eV up to 1 MeV  18 orders of magnitude !
Axion scale and mass, together
with the exclusion ranges from
experimental non-observation

Axion decays to 2, just as for the
0,
or in a static magnetic field:
a
f

Schematic view of
CAST experiment
at CERN:
Axion source
CERN Summer Student Lectures 2007
Axion detection (LHC magnet)
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
37
The Discovery of CP Violation
CERN Summer Student Lectures 2007
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
38
Ingredients … Strange Particles
Strange mesons have an “s” valence quark
Non-strange particles: ( ,  , )I 1 : ud , (uu  dd ) / 2
(,, )I 0 : (uu  dd ) / 2 
neutral particles are
eigenstates of C operator
(K ,K , )I 1/ 2 : K   us , K   us, K 0  ds , K 0  ds
Strange particles:
neutral strange particles are
not eigenstate of C operator
Production of strange particles via strong or electromagnetic interaction has to respect
conservation of the S (“strangeness”) quantum number (they are “eigenstates” of these interactions)





S  0 S  0 S 0
p

S  0 S  0 S 0
p
   S 1KS01 
S 0
  pS 0KS1KS01 
S 0
 pS 0 pS 0 S 0   S0KS1KS01 S 0 ,  S0KS1KS01 S 0
e



S 0 S 0 S 0
CERN Summer Student Lectures 2007
e
 S 0   KS01KS01 
S 0
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
39
The Discovery of CP Violation
Empirically (in the experiment) one does however not observe the neutral “flavor
eigenstates” K 0 and K 0 but rather long- and short-lived neutral states: KL and KS
Their observed pionic decays are: KS   
and it was believed that: CP KS   KS
0
and K L   
and CP KL   KL
0
Larger phase space of
2 decay:
  KL /  KS
580
However, Cronin, Fitch et al. discovered in 1964 the CP-violating decay: K L    
Measurement of opening angle of pion
tracks and their invariant mass:
+ –
Jim Cronin
KL    events
Today’s most precise
measurement of amplitude ratio:
 
A  K L     
A  KS     
  2.282  0.017   10 3
Val Fitch
CERN Summer Student Lectures 2007
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40
Observing CP Violation at the  Factory
The KLOE experiment at the  Factory DANE (Frascati, Italy) can detect single CPviolating decays:
  K 0K 0
and equivalently:
  KS K L
KS    
K L    
Note that the
quantum coherence
is broken after the
decay of one of the
two K0’s
CERN Summer Student Lectures 2007
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41
The Discovery of CP Violation in
the Charged Weak Current
To understand the observed CP violation from the flavor perspective, let us construct CP
eigenstates with CP eigenvalues ±1:
K1 

,
CP  1
K2

,
CP  1
1
K0  K0
2
1

K0  K0
2
While the flavor eigenstates are distinguished by their production mechanism, the CP
eigenstates are distinguished by their decay into an even and odd number of pions.
Since there is CP violation, the physical states (“mass eigenstates”) are not exactly the
same as the CP eigenstates:
 KS 
1


K 
1 
 L 
where: q / p  1    / 1   
CERN Summer Student Lectures 2007
2
 K1   K 2

  K  K
1
2

0

1  p q   K


  0

2
p

q

 K





0.995  1 (!)
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
42
CP Violation and Neutral Kaon Mixing
CPLEAR (CERN) measured the rates of K 0 ,K 0 (t )     (using initial state strangeness
tagging) as a function of the decay time, t, and finds quite a surprise:
after acceptance correction
and background subtraction
K 0    
rates are at
different
for the
s-tagging
production:
two flavor states
  0
K  K 
pp     0  charge of K 
K  K 
K 0    
there is a non-exponential
component (?)
CERN Summer Student Lectures 2007
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43
Neutral Kaon Mixing
Neutral kaons can “mix” through the charged weak current, which does not conserve
strangeness, and neither P nor C. Weak interaction cannot distinguish K 0 from K 0
Simple picture: they mix through common virtual states:
 
0
K0
K0
0
Neutral kaons with fixed
  strangeness quantum
number do not exist in nature !
Note: A priori, mixing has nothing to do with
Because Δm(K) = m(KL) – m(KS) = 3.5 10–12 MeV > 0, a K 0 will change with time into
CP violation !
a
and
K 0 vice versa
These oscillations are described in QCD by ΔS = 2 Feynman “box” diagrams:
[S=2]
s
K
CERN Summer Student Lectures 2007
0
t,c
d
W
d

W
t ,c
s
K0
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
44
Neutral Kaon Mixing
An initially pure K0 state, will evolve into a superposition of states:
K (t )  g(t ) K 0  h(t ) K 0
The time dependence is obtained by solving the time-dependent Schrödinger equation:
 K 0 (t )
d
i  0
dt  K (t )


 K 0 (t )
 M i  
 0
2
 

 K (t )







with 22 matrices M, , of which the offdiagonals  Δm, Δ govern the mixing
The respective time-dependent intensities
are found to be (neglecting CP violation):
I (T ) / I (0)
K0
IK 0 t   e Lt  2e Lt / 2 cos  m  t 
IK 0 t   e
Lt
 2e
Lt / 2
K L0
cos  m  t 
K0
After several KS lifetimes, only KL are left
T  t /S
CERN Summer Student Lectures 2007
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45
Neutral Kaon Mixing and CP Violation
Since KS and KL are not CP eigenstates, the time dependence has to be slightly modified
by the size of , giving rise to an additional sine term.


 
 


 K 0+ 
     K 0    
–
Let’s get back
rates:
Asymmetry:
Ato the 0  decay
  cos  m  t   
 
 K      K 0    
Neglecting other sources of CP
violation & assuming arg() = /4.
CPLEAR 1999
dominated by KS+–
K 0    
A
N(KS+–) ~ N(KL+–amplitude
)
 ||
 Large interference with opposite sign
K 0    
CERN Summer Student Lectures 2007
dominated by KL+–
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
46
There are in Fact Four Meson Systems with Mixing
Pairs of self-conjugate mesons that can be transformed to each other via flavor changing
weak interaction transitions are:
K 0  sd
Bd0  bd
D0  cu
Bs0  bs
They have very different oscillation properties that can be understood from the “CKM
couplings” (see later in this lecture) occurring in the box diagrams
N(T ) / N0
for the plot
xsD == 15
0.02
ysD == 0.10
0
sD == 00
Bs0  Bs0
00
0
0 B 00
B
0
K
s 
s0
D
KD
B
d  Bd
mixing probability:
 ~~
=210
18%–6
50%
K L0
Bd0  Bd0 0
K  K0
T
CERN Summer Student Lectures 2007
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47
Three Types of CP Violation
The CP violation discovered by Cronin, Fitch et al. involves two types of CPV:
CP Violation in mixing:
Prob(K 0  K 0 )  Prob(K 0  K 0 )
CP Violation in interference between decays with and without mixing :
also called:
“indirect CPV”
Prob(K 0 (t )     )  Prob(K 0 (t )     )
However, there is yet another, conceptually “simpler” type of CP violation yet to discover:
CP Violation in the decay:
Prob(K  f )  Prob(K  f )
CERN Summer Student Lectures 2007
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also called:
“direct CPV”
48
“Direct” CP Violation = CP Violation in Decay
General signature: rate differences between CP-conjugated processes:
 i  f


  i  f

It necessarily involves interference of amplitudes contributing to the processes.
To obtain interference, we need phases that change sign under CP
Example: if the decay amplitudes are given by: a1,2 ,1,2 
A i  f

A i  f



 a1e i1e i1  a2e i2 e i2

a e
1
i1
e  i1  a2e i2 e  i2

e 2i (i f )
j
alters sign under CP
(“weak phase”)
j
CP invariant
(“strong phase”)
unphysical phase
where:
 i  f

 A i  f

2
and

 i  f


 A i  f

2
We can define the following CP asymmetry ACP:
ACP


 i
 i  f
CERN Summer Student Lectures 2007
 f
   i
   i
 f
 f


 a
2a1a2 sin 1   2  sin 1  2 
2
2
1  a2  2a1a2 cos 1   2  cos 1  2 
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
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CP Violation in the Kaon Decay
We have seen that at least two amplitudes with different CP-violating (weak) and conserving (strong) phases have to contribute to the decay for direct CPV. This suppresses this
type of CPV, so that the observable effect should be small compared to .
To allow for (small) direct CPV, we need to slightly modify our previous definitions:
   
2
  K L     
  KS   



and use also:
  2  
2
  KL   0 0 
  KS   0 0 
“Clebsch-Gordon isospin” factor when
passing from charged to neutral pions
If the observed CP violation is different in the two decay modes, we have a prove for a
contribution from direct CP violation. From the measurement of the ratio of these decayrate ratios we can determine  ’
The observable
  KL   0 0 
  KS   0 0 
  KL     
  KS     
  2 

  
2
 
 
1  6  Re  
 
First order Taylor expansion
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50
The Discovery of CP Violation in the Decay
Due to the smallness of the effect, it took several experiments and over 30 years of
effort to establish the existence of direct CPV
Feynman graphs:
K
0
“Tree”
(born-level)
amplitudes
K
0
s
d
s
d
u
d
u
d
W
W
u
u
d
d


Experimental average
Indeed, a very small CPV effect !

0
0
(16.7 ± 2.3)10–4
Interference
W
“Penguin”
(loop-level)
amplitude
s
K
0
d
CERN Summer Student Lectures 2007
t, c,u
g
d
u
u
d


A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
51
And the Theory ?
Direct CP violation is in general very hard to calculate due to its sensitivity to the
relative size and phase of different amplitudes of similar size…
Many theoretical groups have put serious efforts into this. All agree that the effect is
much smaller than the indirect CPV (which is a success for the Standard Model !),
but the theory uncertainties are much larger than the measurement errors:
Theoretical
pre(post)dictions
plot not updated !
...the ball is on
the theory side
courtesy:
G. Hamel de Monchenault
e n d
CERN Summer Student Lectures 2007
o f
L e c t u r e
1&2
A. Höcker: The Violation of Symmetry between Matter and Antimatter (1 & 2)
52
Conclusions of the first two Lectures
No CP violation without antimatter !
P, C, T are good symmetries of electromagnetic and strong interactions
P, C are maximally violated in weak interaction
CP, T are not good symmetries of weak interaction
no other source of CP violation has been found so far (EDM’s)
CP violation has been first discovered in the kaon system, and both,
Cartoon
shownCP
by N.violation
Cabibbo in 1966…
then,
there was tremendous
direct and
indirect
havesince
been
observed
progress in the understanding (better: description) of CP violation  next lecture !
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