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Flavor Physics Theory
Lecture at the DESY Summer Student Programme 2015
Joachim Brod, JGU Mainz
August 21, 2015
Contents
1 Flavor in the SM
1.1 SM flavor symmetry and Yukawa interactions . . . . . . . . . . . . . . . . . .
1.2 The CKM matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
2 Phenomenology of meson mixing and CP Violation
2.1 Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Meson mixing . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 CP violation . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 CP violation in mixing (|q/p| =
6 1) . . . . . . . . .
2.3.2 CP violation in decay (|Āf¯/Af | =
6 1) . . . . . . . .
2.3.3 CP violation in the interference between decay and
6
6
7
8
8
9
9
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
mixing ( f 6= ±1)
3 Rare decays: Looking for New Physics
3.1 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 K + ! ⇡ + ⌫ ⌫¯ in the SM and beyond . . . . . . . . . . . . . . . . . . . . . . . .
10
10
11
Instead of giving you a broad overview, I will rather focus on a few selected details. My
discussion is partially based on Ref. [1, 2]; see also [3]. I recommend these references if you
want to continue reading. Other excellent sources are the PDG review articles.
Discussion 0.1 What is flavor physics? Why are we interested in it (there are many dedicated experiments: LHCb, Belle II, NA62, plus many specialized experiments . . . )
Well, I can tell you why I am interested in flavor physics. Several reasons: A beautiful
interplay of strong and weak interaction physics, you learn a lot about the SM. Flavor physics
allows us to study the properties of matter vs. antimatter (CP violation); this is of importance
also for cosmology and the early universe. And, rare processes allow us to look for new
phenomena (physics beyond the SM) “indirectly”. The goal of this short lectures is to explain
these ideas in a bit more detail.
1
Flavor in the SM
The first question we want to answer is: “What is flavor”?
1.1
SM flavor symmetry and Yukawa interactions
You have learned all basic facts about the standard model (SM) already in Jürgen Reuters
lecture, but here we will look in some more detail into the “flavor sector” of the SM. Let’s
look at the kinetic term for the quarks in the SM Lagrangian (the discussion for leptons is
very similar):
X
/ L i + URi iDU
/ Ri + DRi iDD
/ Ri
Lkin =
QL i iDQ
(1.1)
i=1,2,3
The quarks have SU (3) ⇥ SU (2) ⇥ U (1) quantum numbers
QL i (3, 2, 1/6) ,
URi (3, 1, 2/3) ,
DRi (3, 1, 1/3)
(1.2)
(where Q = I 3 + Y is the electric charge), and the covariant derivatives act accordingly.
Note that we have the freedom to rotate all LH and RH quark and lepton fields – we cannot
distinguish the di↵erent quarks and leptons! No reason to call one of them the top quark or
the bottom quark etc. Formally, we write
X Q
QL 0i =
VQ,ij QL j ,
(1.3)
j
where VQ is a 3 ⇥ 3 unitary matrix, and the Lagrangian (1.1) is invariant under this rotation.
We can perform similar rotations for UR and DR , so we see that the SM quark sector has a
global U (3)3 symmetry.
Of course, this symmetry is broken by the Yukawa interactions – we can actually distinguish the di↵erent quarks, because they have di↵erent masses and di↵erent couplings to the
Higgs boson. The Yukawa Lagrangian reads
X
LY =
Yijd QL i HDRj + Yiju QL i H c URj + h.c. ,
(1.4)
i,j=1,2,3
3
with the Higgs field H and its charge conjugate H c ⌘ i 2 H ⇤ . The Yukawa matrices Y d , Y u
are general 3 ⇥ 3 complex matrices.
How many parameters are independent (physical)? The two quark Yukawa matrices contain 2 ⇥ 9 = 18 real and 2 ⇥ 9 = 18 imaginary parameters. Recalling that Lkin is invariant
under unitary rotations, we see that the physics described by two Yukawa matrices (Y d , Y u )
and (Ye d = VQ† Y d Vd , Ye u = VQ† Y u Vu ) is the same. A 3 ⇥ 3 unitary matrix can be parameterized by three angles and six phases, so of the 18 + 18 we can remove at most 9 real and 18
imaginary parameters.
Discussion 1.1 Can you see why we can remove only 17 out of the 18 phases? So, how
many physical real and imaginary parameters do we have in the quark sector? What are their
physical interpretations?
The remaining single phase is related to CP violation. Before we show this in some more
detail, let us introduce. . .
1.2
The CKM matrix
You all know that after electroweak symmetry breaking,
✓
◆
0 p
H!
(h + v)/ 2
(1.5)
and the Yukawa Lagrangian leads to terms bilinear in the quark fields – the quark mass terms:
Lmass =
vYijd
vYiju
p dLi dRj + p uLi uRj + h.c. .
2
2
i,j=1,2,3
X
(1.6)
In general, we can always find two unitary matrices to transform a complex matrix into
diagonal and positive-definite form: M ! D = V † M U . However, to transform the two
Yukawa matrices, we have only three rotations available that leave the kinetic term invariant;
uL and dL have to be rotated by the same matrix due to SU (2) gauge invariance.
In other words, when we go to the mass basis where we diagonalize the mass matrices
using four di↵erent matrices,
v
Mddiag = p Ld Y d Rd† ,
2
v
Mudiag = p Lu Y u Ru† ,
2
(1.7)
using the rotations to the mass eigenstates (denoted by a prime)
dR ! d0R = Rd dR ,
uR ! u0R = Ru uR ,
dL ! d0L = Ld dL , uL ! u0L = Lu uL ,
(1.8)
we will not leave all the kinetic terms invariant.
Discussion 1.2 Why are we allowed to use four matrices?
Now, which terms do actually change? The couplings of the neutral gauge boson will
remain invariant. Let’s consider the Z couplings to RH up quarks as an example:
2esw X k
/ kR .
LZ
uR Zu
(1.9)
3cw
k=1,2,3
4
Since all RH up quark couple to the Z in the same way, this term is invariant (Ru† Ru = 1).
What about the charged-current (W ) interactions?
✓
◆
e
k
/ + dkL + dL W
/ ukL .
p
LW
ukL W
(1.10)
2sw
We see that, for instance,
+
k
+
k
+
0 / V km d0m .
/ dkL = u0 L Lkl
/ (L†d )lm d0m
ukL W
uW
L ⌘ u LW
L
(1.11)
In the last step we introduced the Cabibbo-Kobayashi-Maskawa (CKM) matrix V ⌘ Lu L†d .
We will drop the primes in the following, always working in the mass basis.
There are several useful representations of the CKM matrix; you find lots of information
(including a general, but standard, parameterization) in the PDG [4]. A form useful for
phenomenology is the Wolfenstein parameterization
0
1 0
1
Vud Vus Vub
1 12 2
A 3 (⇢ i⌘)
A + O( 4 ) . (1.12)
1 12 2
A 2
V = @ Vcd Vcs Vcb A = @
Vtd Vts Vtb
A 3 (1 ⇢ i⌘)
A 2
1
Here, ⌘ |Vus | ⇡ 0.23, and the other parameters are of order unity. This representation
makes it manifest that the CKM matrix is very hierarchical.
We have already argued that the CKM matrix contains three mixing angles and one phase.
This phase is the only source of CP violation in the SM. To motivate this a bit further, we
perform a CP transformation on, let’s say, the W bu coupling in the SM. The corresponding
part of the Lagrangian reads (neglecting the primes from now on)
✓
◆
e
+
⇤
/ bL + Vub bL W
/ uL .
p
LW
Vub uL W
(1.13)
2sw
Under a CP transformation (see, for instance, Ref. [1]), this goes into
✓
◆
e
+
1
⇤
/ uL + Vub uL W
/ bL .
p
CP LW (CP )
Vub bL W
2sw
(1.14)
⇤ =V .
The two expressions are only the same if Vub
ub
The CKM matrix is, as a product of two unitary matrices, itself unitary. This has some
interesting consequences.
We have already seen that the Z couplings remain diagonal in the SM. We say that there
are no flavor-changing neutral currents (FCNC) at tree level. The unitarity of the CKM
matrix ensures that this property is approximately preserved also at loop level. We best see
this by an example. Let us look at the decay K + ! ⇡ + ⌫ ⌫¯.
Discussion 1.3 Can you draw the Feynman diagrams for this process? Write down the
contributing CKM factors.
This decay arises first at the one-loop level. From the diagrams we see that the amplitude
is proportional to
⇤
⇤
Vus Vud
F (mu ) + Vcs Vcd
F (mc ) + Vts Vtd⇤ F (mt ) .
5
(1.15)
(ρ,η)
α
V*
Vtd Vtb*
Vcd Vcb*
Vud ub
Vcd Vcb*
γ
β
(1,0)
(0,0)
Figure 1: Unitarity triangle (image credit: Ref. [1]).
⇤ +V V ⇤ +V V ⇤ = 0, so the
Now you can convince yourself that CKM unitarity implies Vus Vud
cs cd
ts td
amplitude would vanish, were it not for the di↵erent quark masses. This is called the GlashowIliopoulos-Maiani (GIM) mechanism. It suppresses FCNC processes (like K + ! ⇡ + ⌫ ⌫¯) in
the SM and need not hold in models of new physics. This is one of the reasons that FCNC
processes are sensitive probes of new physics.
⇤ + V V ⇤ + V V ⇤ = 0 also have a nice geometrical interpretaIn fact, relations like Vud Vub
cd cb
td tb
tion; they define a triangle in the complex plane. This one has lengths of similar magnitude
and you have probably seen the unitarity triangle many times.
Discussion 1.4 How can we determine the elements of the CKM matrix? (Hint: Read the
PDG review article on the CKM matrix.)
2
Phenomenology of meson mixing and CP Violation
The main “problem” that we encounter is that we never observe quark flavor transitions in
nature – quarks are confined into hadrons. So we need to study transitions between di↵erent
hadrons (mesons, baryons, . . . ). The connection between the quark picture and the picture
of hadrons is non-trivial and there is no time to discuss this very interesting subject within
this lecture.
2.1
Mesons
Consider the neutral mesons
K ⇠ s̄d ,
D ⇠ cū ,
Bd ⇠ b̄d ,
Bs ⇠ b̄s
(2.1)
B̄d ⇠ bd¯,
B̄s ⇠ bs̄
(2.2)
and their antiparticles
K̄ ⇠ sd¯,
D̄ ⇠ c̄u ,
These quantum numbers are “good” quantum numbers from the viewpoint of QCD –
they are conserved by the strong interactions. Consider, for instance, the pair production
⌥(4S) ! B 0 B̄ 0 or ⌥(4S) ! B + B . Once the two B mesons are separated, they are stable
in the limit of vanishing weak interactions. However, the weak interaction induces their
mixing and decay.
6
2.2
Meson mixing
We want to understand the three di↵erent types of CP violation:
1. CP violation in mixing
2. CP violation in decay
3. CP violation in the interference of decays with and without mixing
We start with introducing some formalism. Here we will take B mesons as an example, but
most of the discussion applies also to Bs , D, and K mesons. Let us denote by |B 0 (t)i the state
vector of a B meson that is tagged as B 0 at time t = 0, in other words, |B 0 (t = 0)i = |B 0 i,
and similarly for a B̄ 0 . (Flavor tagging will be explained in the next lecture, by Torben
Ferber.)
The weak interaction induces mixing between neutral mesons via the famous box diagrams.
The time evolution of these states is given by
! ✓
!
◆
|B 0 (t)i
d |B 0 (t)i
i
= M i
,
(2.3)
dt |B 0 (t)i
2
|B 0 (t)i
where M and are Hermitian matrices. M12 and 12 are the dispersive and absorptive part
of the transition amplitudes from B 0 to B̄ 0 . We diagonalize the Hamiltonian and call the
mass eigenstates
|BL,H i = p|B 0 i ± q|B 0 i
(2.4)
with |p|2 + |q|2 = 1. The two eigenstates will, in general, have di↵erent mass and life time,
m ⌘ MH
⌘
ML ,
L
H
.
(2.5)
The connection between these “phenomenological” parameters and the “theoretical” parameters obtained by solving the system of eigenvalue equations. One obtains
( m)2
1
4(
)2 = 4|M12 |2
m
=
4Re(M12
q
=
p
Introducing
m
⌘ arg( M12 /
= 4|M12 ||
|
12 |
2
,
⇤
12 ) ,
(2.6)
⇤
2M12
m + i /2
=
2M12 i 12
12 ),
12 | cos
m+i
i
⇤
12
/2
.
we can also write
.
(2.7)
Homework 2.1 Derive these equations.
The time evolution of the mass eigenstates is given by
|BL,H (t)i = e
(iMH,L +
H,L /2)t
|BL,H i
(2.8)
and we could transform this back into the flavor basis if we wanted (see, e.g., Ref. [1]).
These formulas are valid generally; the di↵erent meson systems exhibit very di↵erent
behaviour regarding mass di↵erences and life-time di↵erences. For instance, for neutral kaons,
KS ⇡ 500 KL , whereas B mesons have similar life times.
7
2.3
CP violation
We now want to study properties (. . . deacy rates!) of particles vs. antiparticles. For convenience, we introduce the decay amplitudes
Af = hf |Bi ,
Āf = hf |B̄i ,
(2.9)
etc.
Side remark: Phase conventions. Note that the phases of
M12 ,
12 ,
Af
Āf
q
,
p
(2.10)
depend on the phase conventions of the CP transformation (e.g., CP |B 0 (P ⇢ )i = ⌘CP |B 0 (P⇢ )i).
Physical observables cannot depend on them. On the other hand, the following expressions
are independent of phase conventions:
Af
,
Āf
q
,
p
,
m,
,
f
.
(2.11)
Here we defined
f
=
q Āf
.
p Af
(2.12)
Its phase is a physical observable.
Now, let’s employ this abstract formalism to see how we can learn something about CP
symmetry and its breaking by experiments.
2.3.1
CP violation in mixing (|q/p| =
6 1)
This type of CP violation occurs if the mass eigenstates are di↵erent from the CP eigenstates.
We want to look the di↵erence of decay rates of a neutral B meson and its antiparticle. Let
us consider the so-called semi-leptonic asymmetry
asl (t) =
(B 0 (t) ! `+ ⌫X)
(B 0 (t) ! ` ⌫¯X)
(B 0 (t) ! `+ ⌫X) + (B 0 (t) ! ` ⌫¯X)
=
1 |q/p|4
.
1 + |q/p|4
(2.13)
This expression holds also for the other neutral mesons. If it is non-zero, it means that the
probability of a B mixing into B̄ is di↵erent from the probability of a B̄ mixing into B – CP is
violated. From the semileptonic asymmetry in the B system, we find adsl = (0.7 ± 2.7) ⇥ 10 3
and hence CP violation in mixing is a small e↵ect: |q/p| = 0.9997 ± 0.0013.
Discussion 2.1 Can you explain why the charge of the lepton emitted in the decay of the
neutral B meson tags its flavor? Draw the corresponding quark-level Feynman diagram! How
do we know the flavor of the B meson “before it mixes”?
8
2.3.2
CP violation in decay (|Āf¯/Af | =
6 1)
Here, the decay rate of a meson into a final state f di↵ers from the rate of the CP-conjugated
mode. This type of CP violation is also called direct CP violation. A simple example is given
by charged B decays:
(B ! f )
(B + ! f¯)
=
(B ! f ) + (B + ! f¯)
|Āf¯/Af |2
1
1 + |Āf¯/Af |2
.
(2.14)
A necessary condition is that two amplitudes that di↵er in their weak and strong phases
contribute to the decay rate. Strong phases are CP conserving and arise from the absorptive
part of the decay amplitude (final-state rescattering e↵ects). We can, thus, write
X
X
Af =
Ak ei( k + k ) ,
Af¯ =
Ak ei( k k ) .
(2.15)
k
k
Discussion 2.2 Considering the case k = 2 (two di↵erent amplitudes), can you show that
|Āf¯/Af | = 1 if either 1 = 2 or 1 = 2 ?
This type of CP violation can also occur in neutral meson decays. The contributions can
be difficult to disentangle. As it involves strong phases that a hard to calculate, it is typically
a source of theoretical uncertainty.
An example were all hadronic parameters can be extracted from experiment is direct CP
violation in B ! DK decays. It allows to determine the CKM angle .
2.3.3
CP violation in the interference between decay and mixing (
f
6= ±1)
For decays of neutral mesons into a CP eigenstate a third type of CP violation is possible. It
results from the interference of, e.g., B 0 ! fCP and B 0 ! B 0 ! fCP .
af (t) =
=
(B 0 (t) ! f )
(B 0 (t) ! f )
(B 0 (t) ! f ) + (B 0 (t) ! f )
(1
(1 |
| f |2 ) cos( mt)
2
t/2)
f | ) cosh(
(2.16)
2Im
2Re
sin( mt)
+ ... .
t/2)
f sinh(
f
In both the Bd and Bs meson systems |q/p| 1 . O(10 2 ). If only one amplitude contributes
to the decay, we have |Āf¯/Af | = 1 automatically (no direct CP violation). These modes are
“clean”, because Af drops out and we have (note that now | f | ⇡ 1)
af (t) =
cosh(
Im f sin( mt)
t/2) Re f sinh(
t/2)
+ ... .
(2.17)
Example: Measuring sin(2 )
The best-known example is the asymmetry in B ! J/ KS . It is dominated by the tree-level
b ! cc̄s transition and its CP conjugate. The mixing parameter is
J/ KS
q AJ/
=
p AJ/
KS
KS
=
✓
Vtd Vtb⇤
Vtd⇤ Vtb
◆✓
Vcb Vcs⇤
Vcb⇤ Vcs
9
◆✓
⇤
Vcs Vcd
Vcs⇤ Vcd
◆
.
(2.18)
The first factor comes from B mixing. In both B-meson systems, we have empirically |
|M12 |, and from Eq. (2.6) we obtain using this approximation
q
=
p
⇤
M12
.
|M12 |
12 |
⌧
(2.19)
The second factor is the ratio of CKM elements for the b ! cc̄s transition and its CP conjugate. The last factor comes from K K̄ mixing which is crucial in order to get interference
between B 0 ! J/ K 0 and B 0 ! J/ K 0 . In the kaon system, the relative phase between
⇤ /|M |. The phase of
M12 12 is small and we find q/p ⇡ M12
12
J/ KS is exactly given by
(twice) the CKM angle :
✓
◆✓
◆✓
✓
◆✓ ⇤
◆
⇤◆
Vtd Vtb⇤
Vcs Vcd
Vtd Vtb⇤
Vcd Vcb
Vcb Vcs⇤
=
⌘ e 2i .
(2.20)
Vtd⇤ Vtb
Vcb⇤ Vcs
Vcs⇤ Vcd
Vcd Vcb⇤
Vtd⇤ Vtb
Therefore, this asymmetry allows to measure sin 2 precisely. Penguin amplitudes come in
at the O(%) level, which actually is an issue given the precision of experimental data.
bc
In general, all three types of CP violation occur in the decays of neutral mesons. An
example is KL ! ⇡⇡. Historically, CP violation was first seen by observing this decay in
1964.
3
Rare decays: Looking for New Physics
In this last part of the lectures we will have a closer look at the rare decay K + ! ⇡ + ⌫ ⌫¯. We
will discuss why it is one of the “cleanest” decays in flavor physics and how we can use it to
look for new physics. I will try to give you a flavor of how an actual calculation is done.
3.1
Factorization
Recall that QCD behaves very di↵erently at di↵erent energy scales.
• At high energies (weak scale), the strong coupling is small and we can calculate observables involving quarks perturbatively
• At low energies (hadronic scale), quarks are confined into hadrons, and a perturbative
expansion in ↵s is not possible.
We can use e↵ective field theories to separate these scales, and apply appropriate techniques
in each regime. To this end, we write a general transition amplitude for the process i ! f
schematically as
X
A=
Ck hf |Qk |ii .
(3.1)
k
The Wilson coefficient Ck contains the high-energy information (weak interactions, possible
new physics etc.). The matrix element of the e↵ective interaction (“operator”) contains the
“nonperturbative” QCD e↵ects.
10
3.2
K + ! ⇡ + ⌫ ⌫¯ in the SM and beyond
The Wilson coefficients for K + ! ⇡ + ⌫ ⌫¯ are, in the SM, induced by the so-called Z-penguin
diagrams. They have been calculated up to three loops in QCD and two loops in the electroweak interactions and are thus known at the percent level. New-physics contributions
would enter as additional diagrams and have been calculated in several models (SUSY etc.).
To obtain the hadronic matrix element of the operator
Q⌫ = (s̄L
⌫L
µ dL )(¯
µ
⌫L )
(3.2)
we use a trick: Isospin symmetry. It relies on the fact that QCD is invariant under the
exchange of up and down quarks. (This is, of course, an approximation, but a very good one.
Consider the example of proton and neutron. They behave the same from the viewpoint of
QCD; isospin symmetry is broken by their slightly di↵erent masses and their di↵erent electric
charge.)
p
K + ! ⇡ + ⌫ ⌫¯ , h⇡ + |s̄L µ dL |K + i = 2h⇡ 0 |s̄L µ uL |K + i , K + ! ⇡ 0 `+ ⌫ . (3.3)
p
Here, the 2 is a group-theory factor (Clebsch-Gordon coefficient). The decays K + !
⇡ 0 `+ ⌫ are precisely measured and allow us to extract the hadronic matrix element from data
(recall that the matrix element contains the QCD e↵ects and is not a↵ected by new-physics
contributions).
Putting together all pieces, we obtain the SM prediction
BrSM (K + ! ⇡ + ⌫ ⌫¯) = 7.81(75)(29) ⇥ 10
11
.
(3.4)
Discussion 3.1 Why is this number so small?
• This decay is loop induced in the SM
• It is CKM suppressed
We see that the decay rate is strongly suppressed in the SM. The suppression might not
hold in extensions of the SM. So, by comparing the SM prediction with the measurement, we
might get a hint of new physics. The current bound is
Brexp. (K + ! ⇡ + ⌫ ⌫¯) = 17.3+11.5
10.5 ⇥ 10
11
.
(3.5)
The NA62 experiment at CERN started taking data this summer; they plan to perform
a measurement with an error of 10% within two years.
As an example of a model that modifies the K + ! ⇡ + ⌫ ⌫¯ decay rate, we add a new
vector-like down quark to the SM.
Homework 3.1 Write down an explicit mass term and a generalized Yukawa interaction
for a vector-like down quark D that is a singlet under SU (2)L ; i.e. it has the SM quantum
numbers (3, 1, 1/3) for both LH and RH components. Show that, after electroweak symmetry
breaking and rotation to the mass eigenbasis, there will be, in general, flavor o↵-diagonal Zboson couplings between all LH quarks. Why are the RH couplings still diagonal?
11
Solution 3.1 Here are some hints to obtain the solution.
First, note that we can write down a mass term for the new quark (why does it not break
SM gauge invariance?)
Lm
k
DL mD DR
+ h.c. .
(3.6)
(You could also write down general mixed mass terms between the new quark and the SM
quarks, but you can neglect them for simlicity.) Write the formal rotation of all LH and RH
down quarks to the mass basis.
Second, write down the couplings of all down-type quarks to the Z boson in the flavor basis
(k = 1, 2, 3 denotes the SM generations):

✓
◆
e
1 2 k
1 2
1 1 2 k k
1
k
/
/
/ L + s2w DL ZD
/ L ,
LZ =
sw dR ZdR + sw DR ZDR
sw dL Zd
(3.7)
sw cw 3
3
2 3
3
Can you see already why, after rotation to the mass basis, the RH couplings remain diagonal, whereas the LH couplings will have o↵-diagonal components? If not, perform the
rotations explicitly.
References
[1] K. Anikeev et al. “B physics at the Tevatron: Run II and beyond”. In “Workshop on B
Physics at the Tevatron: Run II and Beyond Batavia, Illinois, September 23-25, 1999”,
2001. e-print:hep-ph/0201071. (Pages 3, 5, 6, and 7.)
[2] Y. Nir. “CP violation in and beyond the standard model”. In “CP violation: In and beyond
the standard model: Proceedings, 27th SLAC Summer Institute on Particle Physics (SSI
99): Stanford, USA, 7-16 Jul 1999”, 1999. e-print:hep-ph/9911321. (Page 3.)
[3] U. Nierste. “Three Lectures on Meson Mixing and CKM phenomenology”. In “Heavy
quark physics. Proceedings, Helmholtz International School, HQP08, Dubna, Russia, August 11-21, 2008”, 1–38. 2009. e-print:0904.1869. (Page 3.)
[4] K. Olive et al.
“Review of Particle Physics”.
doi:10.1088/1674-1137/38/9/090001. (Page 5.)
12
Chin.Phys. C38:090001, 2014.