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Chapter 14 - CP Violation and Weak Hadronic Interactions, Theory of Kaon…
Chapter 14 - CP Violation and Weak Hadronic Interactions
CP violation in early universe
Nowadays, there are
more (baryonic) matter than anti-matter
To generate the observed asymmetry, Sakharov proposed 3 conditions, including
CP violation
.
Today, CP violation is
only observed in quark sector during weak interaction
. However, this still
could not explain
the observed imbalance...
Weak interaction of quarks
Lepton universality
: \( G_F \) is same for all lepton flavors (at lepton-neutrino vertices).
But G is not the same for different quarks!
Cabbibo hypothesis
weak eigenstates
(d', s') = \(\begin{pmatrix} \cos\theta_c & \sin\theta_c \\ -\sin\theta_c & \cos\theta_c \\ \end{pmatrix}\)
mass eigenstates
(d, s)
Mixing only involves
down-type quarks
(by convention): \( (d', s') = U (d, s) \)
GIM mechanism
: explain suppression (\(BR\sim10^{-9}\)) of the "Flavor Changing Neutral Current", \(K_L \rightarrow \mu^+ \mu^- \)
Original box diagram only involves u quark
If add a diagram with a new "
charm
" quark, they cancel!
Cancellation is not exact since \( m_c \neq m_u \)
CKM Matrix
U = \( \begin{pmatrix} V_{ud} & V_{us} & V_{ub}\\ V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb}\\ \end{pmatrix}\)
Nearly diagonal
; largest suppression in
td
and
tb
\( \theta_c = \phi_{12} = 13^\circ \), \( \phi_{23} = 2.3^\circ\), and \(\phi_{13} = 0.2^\circ \)
$$\begin{pmatrix} 0.97 & 0.225 & 0.04 \\ 0.225 & 0.97 & 0.04 \\ 0.008 & 0.04 & 1 \end{pmatrix}$$
Number of
free params
of 3x3
Unitary
matrix: \( (N - 1)^2=4 \)
\( N = 1 \): Cabbibo angle
\( N = 2 \): 3 mixing angles, 1
complex phase
\(V_{ij} \):
(Relative) strength
of weak interaction between up-type quark i and down-type quark j
Weak CC
Vertex
: \( -\frac{ig_w}{\sqrt{2}}(\bar{u},\bar{c}, \bar{t})_i \gamma^\mu \frac{1}{2} (1-\gamma^5) CKM_{ij} (d, s, b)_j \)
If down-type quark enter vertex as adjoint spinor, use \(V^*_{ij} \)
Wolfenstein
Parametrization: \( A, \lambda, \rho, \eta \)
\(\begin{pmatrix} 1-\frac{\lambda^2}{2} & \lambda & A\lambda^3(\rho - i \eta)\\ -\lambda & 1-\frac{\lambda^2}{2} & A \lambda^2\\ A\lambda^3(1-\rho - i\eta) & -A\lambda^2 & 1\end{pmatrix}\) + \(O(\lambda^4)\)
CKM could be
complex
in \(V_{ub}, V_{td} \), corresponding to \(\eta \neq 0 \) (indicate
CP violation
)
Measuring \(|V_{ij}| \) (
in magnitudes
)
\( V_{ud} \): beta decay, \(n \rightarrow p + e^- + \bar{\nu}_e \)
\( V_{us} \): Kaon semi-leptonic decay, \(K^0(d\bar{s}) \rightarrow \pi^-(d\bar{u}) + e^+ + \nu_e \)
\( V_{ub} \): B semi-leptonic decay, \(B^0(d\bar{b}) \rightarrow \pi^-(d\bar{u}) + e^+ + \nu_e \)
\( V_{cd} \): neutrino-nucleon scattering, \( \nu_\mu d \rightarrow \mu^- c \)
\( V_{cb} \): B semi-leptonic decay, with charm at final state
\( V_{cs} \): \( D_s^+ \) leptonic decay, \( D_s^+ (c\bar{s}) \rightarrow \mu^+\nu_\mu \)
\( V_{td} \): \(B^0 \) meson oscillation
\( V_{ts} \): \(B^0_s \) meson oscillation
\( V_{tb} \): constrain from unitarity, or top quark decay, \( t \rightarrow bW \)
Neutral Kaon System
Flavor eigenstates
: \( K^0 (d\bar{s}), \overline{K}^0 (\bar{d}s) \)
Created
in interactions and appear in Feynman diagrams
Describes
semi-leptonic
decays (unique
tags
for kaon by detecting lepton charges): \(K^0(d\bar{s}) \rightarrow \pi^-e^+\nu_e\), \(\overline{K}^0 \rightarrow \pi^+ e^- \bar{\nu}_e\)
\( CP(K^0)=\overline{K}^0, CP(\overline{K}^0)=K^0\)
CP eigenstates
: \( K_1=\frac{1}{2}(K^0+\bar{K^0}) \), \( K_2 = \frac{1}{2}(K^0 - \bar{K^0})\)
\(CP(K_1)=1, CP(K_2)=-1\); \(CP(\pi\pi)=1, CP(\pi\pi\pi)=-1\)
Describes
hadronic
(\(\pi\pi/\pi\pi\pi\)) decays
Therefore, \(K_S \approx K_1, K_S \approx K_2\). Equal (physical state is CP eigenstate) if CP were exact.
Physical (mass) eigenstates
: \(K_S\), \(K_L\)
Kaons
propagate
as physical states (time evolution)
Observed (dominant) decays: \( K_s \rightarrow \pi\pi \), \(K_L \rightarrow \pi\pi\pi \)
Lifetimes: \(\tau(K_S)\sim10^{-10}s, \tau(K_L)\sim10^{-7}s\); Mass: 497 MeV.
\(K^0\) and \(\bar{K^0}\) oscillate to each other through
box diagram
(u,c,t and W propagator). Consider as a whole when talking about a physical system!
Time evolution
Kaons are created in \(K^0\) or \(\bar{K^0}\), propagate as {\(K_S, K_L\)}, and then decay, described by flavor/CP eigenstates.
\(K(t)=\frac{1}{\sqrt{2}}(\theta_S(t)K_S+\theta_L(t)K_L) \)
\(\theta_i = \exp( -im_it - \frac{\Gamma_i}{2}t) \)
E.g., if produced as \(K^0\), At large t, \(K_S\) would
decay away
, leaving purely \(K_L\) (if CP is exact)
Produced in
strong interaction
: \(\pi^- p \rightarrow \Lambda K^0\), \(\pi^+ p \rightarrow p K^+ \bar{K^0} \)
Kaon (strangeness) Oscillation
Measuring Kaon mass difference
\(P(K^0_{t=0} \rightarrow K^0) \propto +2\cos(\Delta m t) \), \(P(K^0_{t=0} \rightarrow \overline{K}^0) \propto -2\cos(\Delta m t) \) (NOTE: CP violation neglected in derivation!)
\(T_{osc}=\frac{2\pi \hbar}{\Delta m} \), \(\Delta m = m_{K_L} - m_{K_S}\)
Measure
the
oscillation period
gives mass difference
Measured by
CPLEAR
(Low Energy Anti-proton Ring)
\(p\bar{p}\) collision to create equal number of \(K^0, \bar{K^0}\)
Identify Kaon type at production and at decay.
An "
asymmetry
" can be written as a ratio (cancel out systematics) involving \(P(K^0/\bar{K^0}_{t=0} \rightarrow K^0/\bar{K^0}) \), which is expressed in terms of \(\cos(\Delta m t)\)
Result: \(\Delta m = 10^{-15} GeV \)
\(\Delta m\) is
related to the magnitude of the matrix element
(\(M_{qq'} \propto V_{qd}V_{qs}^*V_{q's}^*V_{qd} \)) in Kaon mixing box diagram: \(\Delta m \propto \frac{(m_c^2-m_u^2)^2}{m_c^2} \)
Observe \(P(K^0 \rightarrow K^0/\overline{K}^0)\)
CP violation in Kaon (\(\epsilon\))
Observation of CP violation
Discovery
Create neutral Kaon beam & observe decay
Observed \(K_L \rightarrow \pi\pi\) event at large distance!
Finch-Cronin Experiment, 1964
Classification
of CP violation in Kaon
Direct
violation
\(K_L= K_2 \), but \(K_2 \rightarrow \pi\pi\)
\( \epsilon'=\frac{\Gamma(K_2\rightarrow\pi\pi)}{\Gamma(K_2\rightarrow \pi\pi\pi)} \), and \(Re\frac{\epsilon'}{\epsilon}\approx 10^{-3}\).
Very small
effect!
Observe \(K_{L=2}\rightarrow \pi\pi\)
Violation in
mixing
{\(K_S, K_L\)} are not CP eigenstates {\(K_1, K_2\)}; violation parametrized by \(\epsilon\)
How? Write \(K^0, \overline{K}^0\) in terms of \(K_S, K_L\), apply time evolution, then write in terms of \(K_1, K_2\). Ignore direct CP violation.
Observe \(K^0/\overline{K}^0 \rightarrow \pi\pi\)
\(P(K^0_{t=0}\rightarrow \pi\pi) \approx (K^0\rightarrow K_S \rightarrow\pi\pi)\) + \(O(|\epsilon|^2)(K^0\rightarrow K_L \rightarrow \pi\pi)\) + \(O(|\epsilon|)\) interference
CPLEAR
: Measure asymmetry \(A_{+-}\propto\Gamma(\overline{K}^0\rightarrow\pi^+\pi^-) - \Gamma(K^0\rightarrow\pi^+\pi^-) \)
Result: \(|\epsilon|\approx 2.2 \times 10^{-3}, \phi=43^\circ\)
(
Observe
) violation in
semileptonic decay
(another observable effect)
Since \(K_L\) is a combination of \(K^0\) and \(\overline{K}^0\), look for asymmetry in semileptonic decay product from \(K_L\) at large distance.
Charge asymmetry
\(\delta \propto\) \(\frac{\Gamma(K_L\rightarrow \pi^-e^+\nu_e) - \Gamma(K_L\rightarrow \pi^+e^-\bar{\nu}_e)}{\Gamma(K_L\rightarrow \pi^-e^+\nu_e) + \Gamma(K_L\rightarrow \pi^+e^-\bar{\nu}_e)}\) = \(2Re(\epsilon)=2|\epsilon|cos(\phi)\)
Result: \(\delta=0.32\%\)
Observe \(K_L \rightarrow \pi^-e^+\nu/\pi^+e^-\bar{\nu} \)
How? Write \(K_L\) in terms of \(K^0, \overline{K}^0\). Calculate probabilities directly in terms of \(\epsilon\).
B meson
Neutral B meson systems
Oscillation: \(B^0(d\bar{b})\leftrightarrow \overline{B}^0(\bar{d}b)\). Box diagram only has
t-quark contribution
.
Interference in decay is small
(few common decay channels)
Physical states
: \(B_L = \frac{1}{\sqrt{2}}(B^0 + e^{-i2\beta}\overline{B}^0)\) \(B_H = \frac{1}{\sqrt{2}}(B^0 - e^{-i2\beta}\overline{B}^0)\) (similar \(\tau\), different mass)
B oscillation is also used to measure \(\Delta m_d = m_{B_H} - m_{B_L}\) = \(2 |M_{12}| \propto |(V_{td}V^*_{tb})^2|\)
CP violation is
very small in mixing
for B meson! Need huge datasets to observe.
Measure oscillation -> measure \(m_d\) -> measure \(V_{td}\)
Measure \(\Delta m, V\)
Experiments: BaBar (SLAC), Belle (KEK); \(B^0(d\bar{b}) \leftrightarrow \overline{B}^0(\bar{d}b)\)
\(e^+ e^- \rightarrow \Upsilon (4S) \rightarrow B^+B^-/B^0B^0 \). Asymmetric collision to ensure B has large momentum. Distance from two B decay vertices \(\Delta z \sim 200\mu m\)
Lepton flavor asymmetry \(A = \frac{N_{OF} - N_{SF}}{N_{OF} + N_{SF}} = \cos(\Delta m_d t)\)
\(\Delta_m\) gives \(|V_{td}|\). Result: \(|V_{td}| = 0.008\)
Experiments: CDF (Tevatron), LHCb; \(B_s^0 (s\bar{b})\leftrightarrow \overline{B}_s^0(\bar{s}b)\) (Asymmetry related to \(\Delta m_s\))
Result: \(|V_{ts}|=0.043\)
SF: probe B that oscillated to a different flavor. OF: probe B having same flavor when decays.
Measure CP violation
CP violation for B meson is small in 1. direct violation and 2. violation in mixing. But
violation in interference is large
.
Measure interference between \(B^0 \rightarrow \psi K_S\) and \(B^0 \rightarrow \overline{B}^0 \rightarrow \psi K_S\)
Asymmetry \(A = \Gamma(\overline{B}^0\rightarrow \psi K_S) -\Gamma(B^0\rightarrow \psi K_S) \)= \(\sin(\Delta m_d t)\sin(2\beta)\)
Result: \(sin(2\beta) = 0.685\)
Unitary triangle
A triangle parametrized by \(\rho, \eta\) can be drawn from the Unitary relation of CKM matrix.
Measured \(\Delta m_d, sin(2\beta), |\epsilon|\) with associated uncertainties constrain the vertex of the triangle. If all converge at same point -> CKM is unitary.
constrain 1: \(\Delta m_d = |V_{td}V_{tb}^*|^2 \propto (1-\rho)^2 + \eta^2 \)
constrain 2: \(|\epsilon| = \eta(1 - \rho + const)\)
constrain 3: \(tan(\beta) = \eta/(1-\rho)\)
Theory
of Kaon Mixing (time evolution of \(K^0\leftrightarrow \overline{K}^0\) system)
Effective Hamiltonian: \(H = M - \frac{i}{2}\Gamma\)
M = \(\begin{pmatrix} M & M_{12} \\ M_{12}^* & M\\ \end{pmatrix} \)
Off-diagonal terms come from \(K^0 \leftrightarrow \overline{K}^0 \)
mixing
box diagrams
\(\Gamma=\begin{pmatrix} \Gamma & \Gamma_{12}\\ \Gamma_{12}^* & \Gamma\\ \end{pmatrix}\)
Off-diagonal terms come from
interference
between \(K^0\) and \(\overline{K}^0\) decays.
Diag terms: real.
Off-diag terms
are non-zero: \(K^0, \bar{K}^0\) are
not physical states
; their linear combinations are.
CKM matrix is
complex
-> \(M_{12}^* \neq M_{12}\) ->
CP violation in mixing
-> \(K_S \neq K_1, K_L\neq K_2\), but instead altered by \(\epsilon\)
Mass (physical) eigenstates {\( K_S, K_L\)} are found by
diagonalizing
the Hamiltonian
Solutions (physical states): \(K_+(t) = \frac{1}{\sqrt{1+|\xi|^2}} (K^0 + \xi\overline{K}^0) e^{-i\lambda_+t} \), \(K_-(t) = \frac{1}{\sqrt{1+|\xi|^2}} (K^0 - \xi\overline{K}^0) e^{-i\lambda_-t} \)
Since CP violation is small, write \(\xi = \frac{1-\epsilon}{1+\epsilon}\)
Therefore, written in
flavor
states
, \(K_S(t) = \frac{1}{\sqrt{2(1+|\epsilon|^2)}}\Big((1+\epsilon)K^0 + (1-\epsilon)\overline{K}^0\Big) e^{-i\lambda_S t} \), \(K_L(t) = \frac{1}{\sqrt{2(1+|\epsilon|^2)}}\Big((1+\epsilon)K^0 - (1-\epsilon)\overline{K}^0\Big) e^{-i\lambda_S t} \)
Written in
CP eigstates
, \(K_S(t) = \frac{1}{\sqrt{1+|\epsilon|^2}}(K_1 + \epsilon K_2) e^{-i\lambda_St}\), \(K_L(t) = \frac{1}{\sqrt{1+|\epsilon|^2}}(K_2 + \epsilon K_1)e^{-i\lambda_L t}\)
If \(\epsilon=0\), then physical states {\(K_S, K_L\)} = CP eigenstates {\(K_1, K_2\)}. In reality, \(\epsilon \neq 0\) but is very small,
allowing CP violation through mixing
.
Interpreting \(\epsilon\)
CP violation occurs
in Kaon mixing
because the matrix element of \(K^0 \rightarrow \overline{K}^0 \neq \overline{K}^0 \rightarrow K^0\) (CKM is complex; see above)
In Wolfenstein parametrization, \(|\epsilon| \propto \eta(1 - \rho + const) \)
Therefore, measure \(\epsilon\) constrains \(\eta, \rho\)
