Download Weak interaction Weak interaction, Spontaneous symmetry Breaking

Weak interaction,
interaction
Spontaneous symmetry Breaking,
and
Electro-weak unification
Particle Physics Phenomenology 1
guidance:
g
particle p
p
physics
y
p
phenomenology1
gy
• Text : 前半はスライド、後半は
L Ryder,
L.
Ryder Quantum Field Theory (2nd.ed),
ed) Chap8
• Goals
★弱い相互作用の基本事項を理解する。
★弱い相互作用の基本事項を理解する
★V-A理論を用いて崩壊率などが計算できる。
★Goldstoneの定理、およびHiggs機構について説明できる。
定
機構
★電弱統一理論の構造を理解する
• 前提となる知識
場 理論と 準量 化
場の理論と正準量子化
S行列と摂動展開、ファインマン規則
相対論的量子力学
guidance (cont.)
(cont )
• 使用言語：英語
• 成績評価
レポート 合計4回課し成績判定を行う。
最低でも３回提出しないと不合格になります。
S h d l
Schedule
１．Weak interaction & Standard Model: Brief review
２．Phenomenology of Weak Interaction: Fermi’s theorem
３．Spontaneous
p
Symmetry
y
y Breakingg and Goldstone theorem
４．SSB with gauge field 1
５．SSB with gage field2
６．Electro-weak theory 1
７．Electro-weak theory2
chapter 1
Weak Interaction: Brief review
History:
y Puzzle of beta decayy
β-decay of the atom or neutron: puzzling energy spectrum
A(N , Z ) → A(N , Z ± 1) + e ∓
N.
N Bohr
Bohr’ss comment
‘At the present stage of atomic theory, however, we may say
that
h we h
have no argument, either
i h empirical
i i l or theoretical,
h
i l ffor
upholding the energy principle in the case of β-ray
di i
disintegrations.’
i
’
Pauli’s Discoveryy ((1930))
Dear Radioactive Ladies and Gentlemen,
As the bearer of these lines, to whom I graciously ask you to listen,
will explain to you in more detail, how because of the "wrong"
statistics of the N and Li6 nuclei and the continuous beta spectrum, I
have hit upon a desperate remedy to save the "exchange
exchange theorem
theorem" of
statistics and the law of conservation of energy. Namely, the
possibility that there could exist in the nuclei electrically neutral
particles, that I wish to call neutrons, which have spin 1/2 and obey
the exclusion principle and which further differ from light quanta in
Neutrino
at theofpresent
that they do not travel with
the velocity
light. The mass of the
neutrons should be of the same order of magnitude as the electron
mass and in any event not larger than 0.01
0 01 proton masses.
masses The
continuous beta spectrum would then become understandable by
the assumption that in beta decay a neutron is emitted in addition
to the electron such that the sum of the energies of the neutron and
the electron is constant...
‘Weak interaction’ is weak!
Decay of particles induced by strong, EM, and weak forces.
n
(life time = 10n [s ])
2
1/(life time) ≈(coupling constant) ×(phase space)
Fermi’s theoryy ((1932))
construct effective 4-point interaction
universal for any
y kind of hadrons
e.g.
Calculated neutrino cross section (Bethe)
(
)
N.B.
Need 50 light-years of water to stop one 1 MeV neutrino.
linear in Energy E ⇒ indicate Fermi’s theory is an ‘effective
theory’.
Introduction to the Weak interaction in SM
Interaction between charged leptons, neutrino, and quarks in the
Standard Model
1★Charged Current Interaction by W± exchange (Fermi)
2★Neutral Current Interaction by
y Z0 exchange
g ((1973))
Gauge boson propagator
Weak vertex: V-A (left handed)
Physical Processes induced by the Weak interaction
1-A: Leptonic process
1-B:Semi-leptonic process
1-C: Non-leptonic process
only tree diagrams are shown for the strong interaction
→ Need knowledge of QCD and quark structure of hadron to
calculate them.
2 Neutral interaction (discovered in 1973 @CERN)
Vertex for Z0
depends on the Weinberg angle
(see Higgs mechanism)
Effective Interaction at the low energy
Introduce the effective Fermi coupling constant for the low energy
weak interaction.
yields the weak coupling constant
in SM.
Coupling constant itself is not so week!
Lepton
p
Familyy
lepton number is conserved in SM
Quark Family
quark WFs are eigenstates of the strong interaction, but not
for the weak and EM interactions
interactions.
Cabibbo angle :
Experimental
p
evidence of the Cabibbo angle
g
s→u process is suppressed largely compared with d→s process
Leptonic
epto c decays of
o Kaon
ao aandd pion
po
→
More on the q
quark familyy
Puzzle of K0 decay into muon pairs.
calculation is smaller than the data
GIM mechanism (1970)
add the charm quark contribution,
which
w
c be
belong
o g to the
t e generation
ge e at o of
o s-quark.
s qua .
N.B. It was proposed before the discovery
the charm quark!
q
3-generations:
g
Cabibbo-Kobayashi-Maskawa
y
matrix
which
hi h is
i expressedd with
i h 3-real
3
l andd 1-imaginary
1i
i
parameters
The imaginary phase produces CP-violating processes!
q
quiz
• Suppose CKM matrix is unitary, show that the CKM
matrix is parametrized by 3 real parameters and 1
imaginary phase.
chapter 2
C
Consequences
off V
V-A
A interaction
i
i
・V-A structure & left-handed current
・parity violation
・ΔI = ½ rule
Current-current Interaction
Feynman and Gell-Mann, Sudarshan and Marshak (1958)
where
‘Left-handed’ structure of the weak current
Define the left- and right-handed spinors as
1±γ5
ψ L ≡ PLψ
ψ R ≡ PRψ
PR / L =
2
with the gamma5 matrix
⎛0 1⎞
γ 5 ≡ iγ 0γ 1γ 2γ 3 = ⎜ 1 0 ⎟
{γ μ , γ 5 } = 0
⎝
⎠
We introduce ‘chirality’ ±1 for right/handed spinors as the
eigenvalues
i
l
off th
the gamma55 matrix;
ti
γ 5ψ R/L = ±1 ⋅ ψ R/L
・For the mass-less fermion,
chirality coincides with the helicity h
h=
p ⋅s
|p|
spin projection along
the momentum
Proof: chiralityy = helicityy in the mass-less limit
begin with the Dirac eq.
(α i pi + β m )ψ = Eψ
⎛ 0
α =⎜ i
⎝σ
i
⎛1
σi ⎞
β = ⎜0
0 ⎟⎠
⎝
0 ⎞
− 1 ⎟⎠
⎛φ ⎞
ψ
=
Rewrite with two component spinor
⎜ χ ⎟ as,
⎝ ⎠
σ ⋅ p χ + mφ = E φ
σ ⋅ pφ − m χ = E χ
Thus, in the mass-less case, one easily finds eigenvalue
equation
equat
o for
o tthee helicity.
e c ty.
σ ⋅p
(φ ± χ ) = ± 1 ⋅ (φ ± χ )
E
On the other hand, the chirality projection provides
1±γ5
1 ⎛φ ± χ ⎞
PR/Lψ =
ψ = ⎜
⎟
2
2 ⎝⎜ φ ± χ ⎟⎠
Q.E.D
V-A
V
A interaction plays a role of ‘chirality
chirality projection
projection’
Weak current-current interaction
ue γ μ (1 − γ 5 )uν e u p γ μ (1 − γ 5 )u n
only acts on only the left-handed spinors, which is peculiar
f t
feature
off th
the weak
k iinteraction,
t
ti
becasuse
γ μ (1 − γ 5 )u n = γ μ (1 − γ 5 )(u nR + u nL )
1 + γ5
1 − γ5
= γ μ (1 − γ 5 )(
un +
un )
2
=γμ
2
1 − γ5
1 − γ5 L
un = γ μ
un
2
2
ue L γ μ uν e L u pL γ μ u nL
Parityy operation
p
on Dirac spinor
p
Parity transformation
for the Dirac spinor
p
The free Dirac eq.
eq is invariant under this operation
★Decay rate includes the parity violating term due to the
V-A structure of the interaction.
Experiment : Parity non-conservation
non conservation
θ-τ puzzle (1956~) : kaon decays in S-wave final state
1958 Lee
Lee-Yang
Yang
postulate observation of the
parity violation
parity-violation
1961 Co60 experiment (Wu)
Example1:
p
electron-neutrino elastic scattering
g
With the neutral current exchange, elastic scattering
amplitude
p
is ggiven byy
Ignoring the electron mass m and taking the CM frame,
one finds
and also for the total cross section
Example2:
p
Neutron beta decayy
With the charged current interaction
Straightforward calculation yields
In the neutron rest frame,
Resulting energy spectrum is
consistent
i
to data
d
ΔI=1/2 rule
In strangness changing weak meson and baryon decays,
there should exist both ΔI=1/2 and ΔI=3/2 processes.
p
e.g. Λ baryon (I=0) can decay into Λ 0 → pπ − or Λ 0 → nπ 0
nucleon: II=1/2,
1/2, pion: II=1
1 ⇒ final state has II=3/2
3/2 or 1/2.
for total I=3/2,
for I=1/2,
2 1 1
1
, − 1,0 +
3 2 2
3
nπ 0
1 1 1
2
, − 1,0 −
3 2 2
3
1 1
, 1, −1
2 2
pπ −
1 1
, 1, −1
2 2
ΔI=3/2
ΔI=1/2
Exp: nπ0 15.6, pπ- 22.4
★Data indicate complete dominance of ΔI
ΔI=1/2
1/2 amplitudes over
ΔI=3/2.
Amp (ΔI = 1 / 2) ~ 20 ⋅ Amp (ΔI = 3 / 2)
Origin of the ΔI=1/2 rule
For the hyperon decay, we calculate a weak matrix element
< N | d γ μ ((1 − γ 5 )u u γ μ (1
( − γ 5 )s | Λ >
between nucleon and Λ states.
In the non
non-relativisitc
relativisitc limit, we can rewrite
< N | d †u † (1 − σ u ⋅ σ d )u s | Λ >
spin
i projection
j ti operator
t
This operator forces ud-pair to be spin=0 and isospin=0 due to the
Pauli principle. Hence, this weak transition must be ΔI=1/2.
(Pati-Woo theorem)
NB. However,,
understanding of ΔI=1/2 rule for kaon decay is still incomplete
quiz 1
q
• calculate neutrino-electron elastic scattering
amplitude and show that the amplitude square with
the electron mass being 0 becomes
i the
in
th CM ((center
t off mass fframe).
)
chapter 3
Spontaneous Symmetry Breaking and
Nambu Goldstone boson
Nambu-Goldstone