11institutetext: Department of Particle Physics and Astrophysics
Weizmann Institute of Science, Israel

# Flavour Physics and CP Violation

Y. Nir

###### Abstract

We explain the many reasons for the interest in flavor physics. We describe
flavor physics and the related CP violation within the Standard Model, and
explain how the B-factories proved that the Kobayashi-Maskawa mechanism
dominates the CP violation that is observed in meson decays. We explain the
implications of flavor physics for new physics, with emphasis on the “new
physics flavor puzzle”, and present the idea of minimal flavor violation as a
possible solution. We explain why the values flavor parameters of the Standard
Model are puzzling, present the Froggatt-Nielsen mechanism as a possible
solution, and describe how measurements of neutrino parameters are interpreted
in the context of this puzzle. We show that the recently discovered Higgs-like
boson may provide new opportunities for making progress on the various flavor
puzzles.

##  0.1 What is flavor?

The term “flavors” is used, in the jargon of particle physics, to describe
several copies of the same gauge representation, namely several fields that
are assigned the same quantum charges. Within the Standard Model, when
thinking of its unbroken $SU(3)_{\rm C}\times U(1)_{\rm EM}$ gauge group,
there are four different types of particles, each coming in three flavors:

  * •

Up-type quarks in the $(3)_{+2/3}$ representation: $u,c,t$;

  * •

Down-type quarks in the $(3)_{-1/3}$ representation: $d,s,b$;

  * •

Charged leptons in the $(1)_{-1}$ representation: $e,\mu,\tau$;

  * •

Neutrinos in the $(1)_{0}$ representation: $\nu_{1},\nu_{2},\nu_{3}$.

The term “flavor physics” refers to interactions that distinguish between
flavors. By definition, gauge interactions, namely interactions that are
related to unbroken symmetries and mediated therefore by massless gauge
bosons, do not distinguish among the flavors and do not constitute part of
flavor physics. Within the Standard Model, flavor-physics refers to the weak
and Yukawa interactions.

The term “flavor parameters” refers to parameters that carry flavor indices.
Within the Standard Model, these are the nine masses of the charged fermions
and the four “mixing parameters” (three angles and one phase) that describe
the interactions of the charged weak-force carriers ($W^{\pm}$) with quark-
antiquark pairs. If one augments the Standard Model with Majorana mass terms
for the neutrinos, one should add to the list three neutrino masses and six
mixing parameters (three angles and three phases) for the $W^{\pm}$
interactions with lepton-antilepton pairs.

The term “flavor universal” refers to interactions with couplings (or to
parameters) that are proportional to the unit matrix in flavor space. Thus,
the strong and electromagnetic interactions are flavor-universal. An
alternative term for “flavor-universal” is “flavor-blind”.

The term “flavor diagonal” refers to interactions with couplings (or to
parameters) that are diagonal, but not necessarily universal, in the flavor
space. Within the Standard Model, the Yukawa interactions of the Higgs
particle are flavor diagonal.

The term “flavor changing” refers to processes where the initial and final
flavor-numbers (that is, the number of particles of a certain flavor minus the
number of anti-particles of the same flavor) are different. In “flavor
changing charged current” processes, both up-type and down-type flavors,
and/or both charged lepton and neutrino flavors are involved. Examples are (i)
muon decay via $\mu\to e\bar{\nu}_{i}\nu_{j}$, and (ii)
$K^{-}\to\mu^{-}\bar{\nu}_{j}$ (which corresponds, at the quark level, to
$s\bar{u}\to\mu^{-}\bar{\nu}_{j}$). Within the Standard Model, these processes
are mediated by the $W$-bosons and occur at tree level. In “flavor changing
neutral current” (FCNC) processes, either up-type or down-type flavors but not
both, and/or either charged lepton or neutrino flavors but not both, are
involved. Example are (i) muon decay via $\mu\to e\gamma$ and (ii)
$K_{L}\to\mu^{+}\mu^{-}$ (which corresponds, at the quark level, to
$s\bar{d}\to\mu^{+}\mu^{-}$). Within the Standard Model, these processes do
not occur at tree level, and are often highly suppressed.

Another useful term is “flavor violation”. We explain it later in these
lectures.

##  0.2 Why is flavor physics interesting?

  * •

Flavor physics can discover new physics or probe it before it is directly
observed in experiments. Here are some examples from the past:

    * –

The smallness of
$\frac{\Gamma(K_{L}\to\mu^{+}\mu^{-})}{\Gamma(K^{+}\to\mu^{+}\nu)}$ led to
predicting a fourth (the charm) quark;

    * –

The size of $\Delta m_{K}$ led to a successful prediction of the charm mass;

    * –

The size of $\Delta m_{B}$ led to a successful prediction of the top mass;

    * –

The measurement of $\varepsilon_{K}$ led to predicting the third generation.

    * –

The measurement of neutrino flavor transitions led to the discovery of
neutrino masses.

  * •

CP violation is closely related to flavor physics. Within the Standard Model,
there is a single CP violating parameter, the Kobayashi-Maskawa phase
$\delta_{\rm KM}$ [1]. Baryogenesis tells us, however, that there must exist
new sources of CP violation. Measurements of CP violation in flavor changing
processes might provide evidence for such sources.

  * •

The fine-tuning problem of the Higgs mass, and the puzzle of the dark matter
imply that there exists new physics at, or below, the TeV scale. If such new
physics had a generic flavor structure, it would contribute to flavor changing
neutral current (FCNC) processes orders of magnitude above the observed rates.
The question of why this does not happen constitutes the new physics flavor
puzzle.

  * •

Most of the charged fermion flavor parameters are small and hierarchical. The
Standard Model does not provide any explanation of these features. This is the
Standard Model flavor puzzle. The puzzle became even deeper after neutrino
masses and mixings were measured because, so far, neither smallness nor
hierarchy in these parameters have been established.

##  0.3 Flavor in the Standard Model

A model of elementary particles and their interactions is defined by the
following ingredients: (i) The symmetries of the Lagrangian and the pattern of
spontaneous symmetry breaking; (ii) The representations of fermions and
scalars. The Standard Model (SM) is defined as follows:
(i) The gauge symmetry is

$G_{\rm SM}=SU(3)_{\rm C}\times SU(2)_{\rm L}\times U(1)_{\rm Y}.$ (1)

It is spontaneously broken by the VEV of a single Higgs scalar,
$\phi(1,2)_{1/2}$ $\left(\langle\phi^{0}\rangle=v/\sqrt{2}\right)$:

$G_{\rm SM}\to SU(3)_{\rm C}\times U(1)_{\rm EM}.$ (2)

(ii) There are three fermion generations, each consisting of five
representations of $G_{\rm SM}$:

$Q_{Li}(3,2)_{+1/6},\ \ U_{Ri}(3,1)_{+2/3},\ \ D_{Ri}(3,1)_{-1/3},\ \
L_{Li}(1,2)_{-1/2},\ \ E_{Ri}(1,1)_{-1}.$ (3)

###  0.3.1 The interaction basis

The Standard Model Lagrangian, ${\cal L}_{\rm SM}$, is the most general
renormalizable Lagrangian that is consistent with the gauge symmetry (1), the
particle content (3) and the pattern of spontaneous symmetry breaking (2). It
can be divided to three parts:

${\cal L}_{\rm SM}={\cal L}_{\rm kinetic}+{\cal L}_{\rm Higgs}+{\cal L}_{\rm
Yukawa}.$ (4)

As concerns the kinetic terms, to maintain gauge invariance, one has to
replace the derivative with a covariant derivative:

$D^{\mu}=\partial^{\mu}+ig_{s}G^{\mu}_{a}L_{a}+igW^{\mu}_{b}T_{b}+ig^{\prime}B^{\mu}Y.$
(5)

Here $G^{\mu}_{a}$ are the eight gluon fields, $W^{\mu}_{b}$ the three weak
interaction bosons and $B^{\mu}$ the single hypercharge boson. The $L_{a}$’s
are $SU(3)_{\rm C}$ generators (the $3\times 3$ Gell-Mann matrices ${1\over
2}\lambda_{a}$ for triplets, $0$ for singlets), the $T_{b}$’s are $SU(2)_{\rm
L}$ generators (the $2\times 2$ Pauli matrices ${1\over 2}\tau_{b}$ for
doublets, $0$ for singlets), and the $Y$’s are the $U(1)_{\rm Y}$ charges. For
example, for the quark doublets $Q_{L}$, we have

${\cal L}_{\rm
kinetic}(Q_{L})=i{\overline{Q_{Li}}}\gamma_{\mu}\left(\partial^{\mu}+{i\over
2}g_{s}G^{\mu}_{a}\lambda_{a}+{i\over 2}gW^{\mu}_{b}\tau_{b}+{i\over
6}g^{\prime}B^{\mu}\right)\delta_{ij}Q_{Lj},$ (6)

while for the lepton doublets $L_{L}^{I}$, we have

${\cal L}_{\rm
kinetic}(L_{L})=i{\overline{L_{Li}}}\gamma_{\mu}\left(\partial^{\mu}+{i\over
2}gW^{\mu}_{b}\tau_{b}-\frac{i}{2}g^{\prime}B^{\mu}\right)\delta_{ij}L_{Lj}.$
(7)

The unit matrix in flavor space, $\delta_{ij}$, signifies that these parts of
the interaction Lagrangian are flavor-universal. In addition, they conserve
CP.

The Higgs potential, which describes the scalar self interactions, is given
by:

${\cal L}_{\rm
Higgs}=\mu^{2}\phi^{\dagger}\phi-\lambda(\phi^{\dagger}\phi)^{2}.$ (8)

For the Standard Model scalar sector, where there is a single doublet, this
part of the Lagrangian is also CP conserving.

The quark Yukawa interactions are given by

$-{\cal L}_{\rm Y}^{q}=Y^{d}_{ij}{\overline{Q_{Li}}}\phi
D_{Rj}+Y^{u}_{ij}{\overline{Q_{Li}}}\tilde{\phi}U_{Rj}+{\rm h.c.},$ (9)

(where $\tilde{\phi}=i\tau_{2}\phi^{\dagger}$) while the lepton Yukawa
interactions are given by

$-{\cal L}_{\rm Y}^{\ell}=Y^{e}_{ij}{\overline{L_{Li}}}\phi E_{Rj}+{\rm
h.c.}.$ (10)

This part of the Lagrangian is, in general, flavor-dependent (that is,
$Y^{f}\not\propto{\bf 1}$) and CP violating.

###  0.3.2 Global symmetries

In the absence of the Yukawa matrices $Y^{d}$, $Y^{u}$ and $Y^{e}$, the SM has
a large $U(3)^{5}$ global symmetry:

$G_{\rm global}(Y^{u,d,e}=0)=SU(3)_{q}^{3}\times SU(3)_{\ell}^{2}\times
U(1)^{5},$ (11)

where

$\displaystyle SU(3)_{q}^{3}$ $\displaystyle=$ $\displaystyle SU(3)_{Q}\times
SU(3)_{U}\times SU(3)_{D},$ $\displaystyle SU(3)_{\ell}^{2}$ $\displaystyle=$
$\displaystyle SU(3)_{L}\times SU(3)_{E},$ $\displaystyle U(1)^{5}$
$\displaystyle=$ $\displaystyle U(1)_{B}\times U(1)_{L}\times U(1)_{Y}\times
U(1)_{\rm PQ}\times U(1)_{E}.$ (12)

Out of the five $U(1)$ charges, three can be identified with baryon number
($B$), lepton number ($L$) and hypercharge ($Y$), which are respected by the
Yukawa interactions. The two remaining $U(1)$ groups can be identified with
the PQ symmetry whereby the Higgs and $D_{R},E_{R}$ fields have opposite
charges, and with a global rotation of $E_{R}$ only.

The point that is important for our purposes is that ${\cal L}_{\rm
kinetic}+{\cal L}_{\rm Higgs}$ respect the non-Abelian flavor symmetry
$S(3)_{q}^{3}\times SU(3)_{\ell}^{2}$, under which

$Q_{L}\to V_{Q}Q_{L},\ \ \ U_{R}\to V_{U}U_{R},\ \ \ D_{R}\to V_{D}D_{R},\ \
L_{L}\to V_{L}L_{L},\ \ \ E_{R}\to V_{E}E_{R},$ (13)

where the $V_{i}$ are unitary matrices. The Yukawa interactions (9) and (10)
break the global symmetry,

$G_{\rm global}(Y^{u,d,e}\neq 0)=U(1)_{B}\times U(1)_{e}\times
U(1)_{\mu}\times U(1)_{\tau}.$ (14)

(Of course, the gauged $U(1)_{Y}$ also remains a good symmetry.) Thus, the
transformations of Eq. (13) are not a symmetry of ${\cal L}_{\rm SM}$.
Instead, they correspond to a change of the interaction basis. These
observations also offer an alternative way of defining flavor physics: it
refers to interactions that break the $SU(3)^{5}$ symmetry (13). Thus, the
term “flavor violation” is often used to describe processes or parameters that
break the symmetry.

One can think of the quark Yukawa couplings as spurions that break the global
$SU(3)_{q}^{3}$ symmetry (but are neutral under $U(1)_{B}$),

$Y^{u}\sim(3,\bar{3},1)_{SU(3)_{q}^{3}},\ \ \
Y^{d}\sim(3,1,\bar{3})_{SU(3)_{q}^{3}},$ (15)

and of the lepton Yukawa couplings as spurions that break the global
$SU(3)_{\ell}^{2}$ symmetry (but are neutral under $U(1)_{e}\times
U(1)_{\mu}\times U(1)_{\tau}$),

$Y^{e}\sim(3,\bar{3})_{SU(3)_{\ell}^{2}}.$ (16)

The spurion formalism is convenient for several purposes: parameter counting
(see below), identification of flavor suppression factors (see Section 0.5),
and the idea of minimal flavor violation (see Section 0.5.3).

###  0.3.3 Counting parameters

How many independent parameters are there in ${\cal L}_{\rm Y}^{q}$? The two
Yukawa matrices, $Y^{u}$ and $Y^{d}$, are $3\times 3$ and complex.
Consequently, there are 18 real and 18 imaginary parameters in these matrices.
Not all of them are, however, physical. The pattern of $G_{\rm global}$
breaking means that there is freedom to remove 9 real and 17 imaginary
parameters (the number of parameters in three $3\times 3$ unitary matrices
minus the phase related to $U(1)_{B}$). For example, we can use the unitary
transformations $Q_{L}\to V_{Q}Q_{L}$, $U_{R}\to V_{U}U_{R}$ and $D_{R}\to
V_{D}D_{R}$, to lead to the following interaction basis:

$Y^{d}=\lambda_{d},\ \ \ Y^{u}=V^{\dagger}\lambda_{u},$ (17)

where $\lambda_{d,u}$ are diagonal,

$\lambda_{d}={\rm diag}(y_{d},y_{s},y_{b}),\ \ \ \lambda_{u}={\rm
diag}(y_{u},y_{c},y_{t}),$ (18)

while $V$ is a unitary matrix that depends on three real angles and one
complex phase. We conclude that there are 10 quark flavor parameters: 9 real
ones and a single phase. In the mass basis, we will identify the nine real
parameters as six quark masses and three mixing angles, while the single phase
is $\delta_{\rm KM}$.

How many independent parameters are there in ${\cal L}_{\rm Y}^{\ell}$? The
Yukawa matrix $Y^{e}$ is $3\times 3$ and complex. Consequently, there are 9
real and 9 imaginary parameters in this matrix. There is, however, freedom to
remove 6 real and 9 imaginary parameters (the number of parameters in two
$3\times 3$ unitary matrices minus the phases related to $U(1)^{3}$). For
example, we can use the unitary transformations $L_{L}\to V_{L}L_{L}$ and
$E_{R}\to V_{E}E_{R}$, to lead to the following interaction basis:

$Y^{e}=\lambda_{e}={\rm diag}(y_{e},y_{\mu},y_{\tau}).$ (19)

We conclude that there are 3 real lepton flavor parameters. In the mass basis,
we will identify these parameters as the three charged lepton masses. We must,
however, modify the model when we take into account the evidence for neutrino
masses.

###  0.3.4 The mass basis

Upon the replacement ${\cal R}e(\phi^{0})\to\frac{v+h^{0}}{\sqrt{2}}$, the
Yukawa interactions (9) give rise to the mass matrices

$M_{q}={v\over\sqrt{2}}Y^{q}.$ (20)

The mass basis corresponds, by definition, to diagonal mass matrices. We can
always find unitary matrices $V_{qL}$ and $V_{qR}$ such that

$V_{qL}M_{q}V_{qR}^{\dagger}=M_{q}^{\rm
diag}\equiv\frac{v}{\sqrt{2}}\lambda_{q}.$ (21)

The four matrices $V_{dL}$, $V_{dR}$, $V_{uL}$ and $V_{uR}$ are then the ones
required to transform to the mass basis. For example, if we start from the
special basis (17), we have $V_{dL}=V_{dR}=V_{uR}={\bf 1}$ and $V_{uL}=V$. The
combination $V_{uL}V_{dL}^{\dagger}$ is independent of the interaction basis
from which we start this procedure.

We denote the left-handed quark mass eigenstates as $U_{L}$ and $D_{L}$. The
charged current interactions for quarks [that is the interactions of the
charged $SU(2)_{\rm L}$ gauge bosons
$W^{\pm}_{\mu}={1\over\sqrt{2}}(W^{1}_{\mu}\mp iW_{\mu}^{2})$], which in the
interaction basis are described by (6), have a complicated form in the mass
basis:

$-{\cal
L}_{W^{\pm}}^{q}={g\over\sqrt{2}}{\overline{U_{Li}}}\gamma^{\mu}V_{ij}D_{Lj}W_{\mu}^{+}+{\rm
h.c.}.$ (22)

where $V$ is the $3\times 3$ unitary matrix ($VV^{\dagger}=V^{\dagger}V={\bf
1}$) that appeared in Eq. (17). For a general interaction basis,

$V=V_{uL}V_{dL}^{\dagger}.$ (23)

$V$ is the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix for quarks [2, 1]. As
a result of the fact that $V$ is not diagonal, the $W^{\pm}$ gauge bosons
couple to quark mass eigenstates of different generations. Within the Standard
Model, this is the only source of flavor changing quark interactions.

Exercise 1: Prove that, in the absence of neutrino masses, there is no mixing
in the lepton sector.

Exercise 2: Prove that there is no mixing in the $Z$ couplings. (In the
physics jargon, there are no flavor changing neutral currents at tree level.)

The detailed structure of the CKM matrix, its parametrization, and the
constraints on its elements are described in Appendix .9.

##  0.4 Testing CKM

Measurements of rates, mixing, and CP asymmetries in $B$ decays in the two B
factories, BaBar abd Belle, and in the two Tevatron detectors, CDF and D0,
signified a new era in our understanding of CP violation. The progress is both
qualitative and quantitative. Various basic questions concerning CP and flavor
violation have received, for the first time, answers based on experimental
information. These questions include, for example,

  * •

Is the Kobayashi-Maskawa mechanism at work (namely, is $\delta_{\rm KM}\neq
0$)?

  * •

Does the KM phase dominate the observed CP violation?

As a first step, one may assume the SM and test the overall consistency of the
various measurements. However, the richness of data from the B factories allow
us to go a step further and answer these questions model independently, namely
allowing new physics to contribute to the relevant processes. We here explain
the way in which this analysis proceeds.

###  0.4.1 $S_{\psi K_{S}}$

The CP asymmetry in $B\to\psi K_{S}$ decays plays a major role in testing the
KM mechanism. Before we explain the test itself, we should understand why the
theoretical interpretation of the asymmetry is exceptionally clean, and what
are the theoretical parameters on which it depends, within and beyond the
Standard Model.

The CP asymmetry in neutral meson decays into final CP eigenstates $f_{CP\ }$
is defined as follows:

${\cal A}_{f_{CP\ }}(t)\equiv\frac{d\Gamma/dt[\overline{B}{}^{0}_{\rm
phys}(t)\to f_{CP\ }]-d\Gamma/dt[{B^{0}}_{\rm phys}(t)\to f_{CP\
}]}{d\Gamma/dt[\overline{B}{}^{0}_{\rm phys}(t)\to f_{CP\
}]+d\Gamma/dt[{B^{0}}_{\rm phys}(t)\to f_{CP\ }]}\;.$ (24)

A detailed evaluation of this asymmetry is given in Appendix .10. It leads to
the following form:

$\displaystyle{\cal A}_{f_{CP\ }}(t)$ $\displaystyle=$ $\displaystyle
S_{f_{CP\ }}\sin(\Delta mt)-C_{f_{CP\ }}\cos(\Delta mt),$ $\displaystyle
S_{f_{CP\ }}$ $\displaystyle\equiv$ $\displaystyle\frac{2\,{\cal
I}m(\lambda_{f_{CP\ }})}{1+|\lambda_{f_{CP\ }}|^{2}},\ \ \ C_{f_{CP\
}}\equiv\frac{1-|\lambda_{f_{CP\ }}|^{2}}{1+|\lambda_{f_{CP\ }}|^{2}}\;,$ (25)

where

$\lambda_{f_{CP\ }}=e^{-i\phi_{B}}(\overline{A}_{f_{CP\ }}/A_{f_{CP\ }})\;.$
(26)

Here $\phi_{B}$ refers to the phase of $M_{12}$ [see Eq. (129)]. Within the
Standard Model, the corresponding phase factor is given by

$e^{-i\phi_{B}}=(V_{tb}^{*}V_{td})/(V_{tb}V_{td}^{*})\;.$ (27)

The decay amplitudes $A_{f}$ and $\overline{A}_{f}$ are defined in Eq. (107).

Figure 1: Feynman diagrams for (a) tree and (b) penguin amplitudes
contributing to $B^{0}\to f$ or $B_{s}\to f$ via a
$\bar{b}\to\bar{q}q\bar{q}^{\prime}$ quark-level process.

The $B^{0}\to J/\psi K^{0}$ decay [3, 4] proceeds via the quark transition
$\bar{b}\to\bar{c}c\bar{s}$. There are contributions from both tree ($t$) and
penguin ($p^{q_{u}}$, where $q_{u}=u,c,t$ is the quark in the loop) diagrams
(see Fig. 1) which carry different weak phases:

$A_{f}=\left(V^{\ast}_{cb}V_{cs}\right)t_{f}+\sum_{q_{u}=u,c,t}\left(V^{\ast}_{q_{u}b}V_{q_{u}s}\right)p^{q_{u}}_{f}\;.$
(28)

(The distinction between tree and penguin contributions is a heuristic one,
the separation by the operator that enters is more precise. For a detailed
discussion of the more complete operator product approach, which also includes
higher order QCD corrections, see, for example, ref. [5].) Using CKM
unitarity, these decay amplitudes can always be written in terms of just two
CKM combinations:

$A_{\psi K}=\left(V^{\ast}_{cb}V_{cs}\right)T_{\psi
K}+\left(V^{\ast}_{ub}V_{us}\right)P^{u}_{\psi K},$ (29)

where $T_{\psi K}=t_{\psi K}+p^{c}_{\psi K}-p^{t}_{\psi K}$ and $P^{u}_{\psi
K}=p^{u}_{\psi K}-p^{t}_{\psi K}$. A subtlety arises in this decay that is
related to the fact that ${B}^{0}\to J/\psi K^{0}$ and $\overline{B}^{0}\to
J/\psi\overline{K}{}^{0}$. A common final state, e.g. $J/\psi K_{S}$, can be
reached via $K^{0}-\overline{K}{}^{0}$ mixing. Consequently, the phase factor
corresponding to neutral $K$ mixing,
$e^{-i\phi_{K}}=(V^{*}_{cd}V_{cs})/(V_{cd}V^{*}_{cs})$, plays a role:

$\frac{\overline{A}_{\psi K_{S}}}{A_{\psi
K_{S}}}=-\frac{\left(V_{cb}V^{\ast}_{cs}\right)T_{\psi
K}+\left(V_{ub}V^{\ast}_{us}\right)P^{u}_{\psi
K}}{\left(V^{\ast}_{cb}V_{cs}\right)T_{\psi
K}+\left(V^{\ast}_{ub}V_{us}\right)P^{u}_{\psi
K}}\times\frac{V_{cd}^{\ast}V_{cs}}{V_{cd}V_{cs}^{\ast}}.$ (30)

The crucial point is that, for $B\to J/\psi K_{S}$ and other
$\bar{b}\to\bar{c}c\bar{s}$ processes, we can neglect the $P^{u}$ contribution
to $A_{\psi K}$, in the SM, to an approximation that is better than one
percent:

$|P^{u}_{\psi K}/T_{\psi
K}|\times|V_{ub}/V_{cb}|\times|V_{us}/V_{cs}|\sim({\rm loop\ factor})\times
0.1\times 0.23\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.005.$ (31)

Thus, to an accuracy better than one percent,

$\lambda_{\psi
K_{S}}=\left(\frac{V_{tb}^{*}V_{td}}{V_{tb}V_{td}^{*}}\right)\left(\frac{V_{cb}V_{cd}^{*}}{V_{cb}^{*}V_{cd}}\right)=-e^{-2i\beta},$
(32)

where $\beta$ is defined in Eq. (105), and consequently

$S_{\psi K_{S}}=\sin 2\beta,\ \ \ C_{\psi K_{S}}=0\;.$ (33)

(Below the percent level, several effects modify this equation [6, 7, 8, 9].)

Exercise 3: Show that, if the $B\to\pi\pi$ decays were dominated by tree
diagrams, then $S_{\pi\pi}=\sin 2\alpha$.

Exercise 4: Estimate the accuracy of the predictions $S_{\phi K_{S}}=\sin
2\beta$ and $C_{\phi K_{S}}=0$.

When we consider extensions of the SM, we still do not expect any significant
new contribution to the tree level decay, $b\to c\bar{c}s$, beyond the SM
$W$-mediated diagram. Thus, the expression $\bar{A}_{\psi K_{S}}/A_{\psi
K_{S}}=(V_{cb}V_{cd}^{*})/(V_{cb}^{*}V_{cd})$ remains valid, though the
approximation of neglecting sub-dominant phases can be somewhat less accurate
than Eq. (31). On the other hand, $M_{12}$, the $B^{0}-\overline{B}^{0}$
mixing amplitude, can in principle get large and even dominant contributions
from new physics. We can parametrize the modification to the SM in terms of
two parameters, $r_{d}^{2}$ signifying the change in magnitude, and
$2\theta_{d}$ signifying the change in phase:

$M_{12}=r_{d}^{2}\ e^{2i\theta_{d}}\ M_{12}^{\rm SM}(\rho,\eta).$ (34)

This leads to the following generalization of Eq. (33):

$S_{\psi K_{S}}=\sin(2\beta+2\theta_{d}),\ \ \ C_{\psi K_{S}}=0\;.$ (35)

The experimental measurements give the following ranges [10]:

$S_{\psi K_{S}}=+0.68\pm 0.02,\ \ \ C_{\psi K_{S}}=+0.005\pm 0.017\;.$ (36)

###  0.4.2 Self-consistency of the CKM assumption

The three generation standard model has room for CP violation, through the KM
phase in the quark mixing matrix. Yet, one would like to make sure that indeed
CP is violated by the SM interactions, namely that $\sin\delta_{\rm KM}\neq
0$. If we establish that this is the case, we would further like to know
whether the SM contributions to CP violating observables are dominant. More
quantitatively, we would like to put an upper bound on the ratio between the
new physics and the SM contributions.

As a first step, one can assume that flavor changing processes are fully
described by the SM, and check the consistency of the various measurements
with this assumption. There are four relevant mixing parameters, which can be
taken to be the Wolfenstein parameters $\lambda$, $A$, $\rho$ and $\eta$
defined in Eq. (100). The values of $\lambda$ and $A$ are known rather
accurately [11] from, respectively, $K\to\pi\ell\nu$ and $b\to c\ell\nu$
decays:

$\lambda=0.2254\pm 0.0007,\ \ \ A=0.811^{+0.022}_{-0.012}.$ (37)

Then, one can express all the relevant observables as a function of the two
remaining parameters, $\rho$ and $\eta$, and check whether there is a range in
the $\rho-\eta$ plane that is consistent with all measurements. The list of
observables includes the following:

  * •

The rates of inclusive and exclusive charmless semileptonic $B$ decays depend
on $|V_{ub}|^{2}\propto\rho^{2}+\eta^{2}$;

  * •

The CP asymmetry in $B\to\psi K_{S}$, $S_{\psi K_{S}}=\sin
2\beta=\frac{2\eta(1-\rho)}{(1-\rho)^{2}+\eta^{2}}$;

  * •

The rates of various $B\to DK$ decays depend on the phase $\gamma$, where
$e^{i\gamma}=\frac{\rho+i\eta}{\sqrt{\rho^{2}+\eta^{2}}}$;

  * •

The rates of various $B\to\pi\pi,\rho\pi,\rho\rho$ decays depend on the phase
$\alpha=\pi-\beta-\gamma$;

  * •

The ratio between the mass splittings in the neutral $B$ and $B_{s}$ systems
is sensitive to $|V_{td}/V_{ts}|^{2}=\lambda^{2}[(1-\rho)^{2}+\eta^{2}]$;

  * •

The CP violation in $K\to\pi\pi$ decays, $\epsilon_{K}$, depends in a
complicated way on $\rho$ and $\eta$.

The resulting constraints are shown in Fig. 2.

Figure 2: Allowed region in the $\rho,\eta$ plane. Superimposed are the
individual constraints from charmless semileptonic $B$ decays
($|V_{ub}/V_{cb}|$), mass differences in the $B^{0}$ ($\Delta m_{d}$) and
$B_{s}$ ($\Delta m_{s}$) neutral meson systems, and CP violation in
$K\to\pi\pi$ ($\varepsilon_{K}$), $B\to\psi K$ ($\sin 2\beta$),
$B\to\pi\pi,\rho\pi,\rho\rho$ ($\alpha$), and $B\to DK$ ($\gamma$). Taken from
[12].

The consistency of the various constraints is impressive. In particular, the
following ranges for $\rho$ and $\eta$ can account for all the measurements
[11]:

$\rho=+0.131^{+0.026}_{-0.013},\ \ \ \eta=+0.345\pm 0.014.$ (38)

One can make then the following statement [13]:
Very likely, CP violation in flavor changing processes is dominated by the
Kobayashi-Maskawa phase.

In the next two subsections, we explain how we can remove the phrase “very
likely” from this statement, and how we can quantify the KM-dominance.

###  0.4.3 Is the KM mechanism at work?

In proving that the KM mechanism is at work, we assume that charged-current
tree-level processes are dominated by the $W$-mediated SM diagrams (see, for
example, [14]). This is a very plausible assumption. I am not aware of any
viable well-motivated model where this assumption is not valid. Thus we can
use all tree level processes and fit them to $\rho$ and $\eta$, as we did
before. The list of such processes includes the following:

  1. 1.

Charmless semileptonic $B$-decays, $b\to u\ell\nu$, measure $R_{u}$ [see Eq.
(104)].

  2. 2.

$B\to DK$ decays, which go through the quark transitions $b\to c\bar{u}s$ and
$b\to u\bar{c}s$, measure the angle $\gamma$ [see Eq. (105)].

  3. 3.

$B\to\rho\rho$ decays (and, similarly, $B\to\pi\pi$ and $B\to\rho\pi$ decays)
go through the quark transition $b\to u\bar{u}d$. With an isospin analysis,
one can determine the relative phase between the tree decay amplitude and the
mixing amplitude. By incorporating the measurement of $S_{\psi K_{S}}$, one
can subtract the phase from the mixing amplitude, finally providing a
measurement of the angle $\gamma$ [see Eq. (105)].

In addition, we can use loop processes, but then we must allow for new physics
contributions, in addition to the $(\rho,\eta)$-dependent SM contributions. Of
course, if each such measurement adds a separate mode-dependent parameter,
then we do not gain anything by using this information. However, there is a
number of observables where the only relevant loop process is
$B^{0}-\overline{B}{}^{0}$ mixing. The list includes $S_{\psi K_{S}}$, $\Delta
m_{B}$ and the CP asymmetry in semileptonic $B$ decays:

$\displaystyle S_{\psi K_{S}}$ $\displaystyle=$
$\displaystyle\sin(2\beta+2\theta_{d}),$ $\displaystyle\Delta m_{B}$
$\displaystyle=$ $\displaystyle r_{d}^{2}(\Delta m_{B})^{\rm SM},$
$\displaystyle{\cal A}_{\rm SL}$ $\displaystyle=$ $\displaystyle-{\cal
R}e\left(\frac{\Gamma_{12}}{M_{12}}\right)^{\rm SM}\frac{\sin
2\theta_{d}}{r_{d}^{2}}+{\cal I}m\left(\frac{\Gamma_{12}}{M_{12}}\right)^{\rm
SM}\frac{\cos 2\theta_{d}}{r_{d}^{2}}.$ (39)

As explained above, such processes involve two new parameters [see Eq. (34)].
Since there are three relevant observables, we can further tighten the
constraints in the $(\rho,\eta)$-plane. Similarly, one can use measurements
related to $B_{s}-\overline{B}_{s}$ mixing. One gains three new observables at
the cost of two new parameters (see, for example, [15]).

The results of such fit, projected on the $\rho-\eta$ plane, can be seen in
Fig. 3. It gives [12]

$\eta=0.44^{+0.05}_{-0.23}\ \ (3\sigma).$ (40)

[A similar analysis in Ref. [16] obtains the $3\sigma$ range $(0.31-0.46)$.]
It is clear that $\eta\neq 0$ is well established:
The Kobayashi-Maskawa mechanism of CP violation is at work.

Figure 3: The allowed region in the $\rho-\eta$ plane, assuming that tree
diagrams are dominated by the Standard Model [12].

Another way to establish that CP is violated by the CKM matrix is to find,
within the same procedure, the allowed range for $\sin 2\beta$ [16]:

$\sin 2\beta^{\rm tree}=0.80\pm 0.03.$ (41)

Thus, $\beta\neq 0$ is well established.

The consistency of the experimental results (36) with the SM predictions
(33,41) means that the KM mechanism of CP violation dominates the observed CP
violation. In the next subsection, we make this statement more quantitative.

###  0.4.4 How much can new physics contribute to $B^{0}-\overline{B}{}^{0}$
mixing?

All that we need to do in order to establish whether the SM dominates the
observed CP violation, and to put an upper bound on the new physics
contribution to $B^{0}-\overline{B}{}^{0}$ mixing, is to project the results
of the fit performed in the previous subsection on the $r_{d}^{2}-2\theta_{d}$
plane. If we find that $\theta_{d}\ll\beta$, then the SM dominance in the
observed CP violation will be established. The constraints are shown in Fig.
4(a). Indeed, $\theta_{d}\ll\beta$.

Figure 4: Constraints in the (a) $r_{d}^{2}-2\theta_{d}$ plane, and (b)
$h_{d}-\sigma_{d}$ plane, assuming that NP contributions to tree level
processes are negligible [12].

An alternative way to present the data is to use the $h_{d},\sigma_{d}$
parametrization,

$r_{d}^{2}e^{2i\theta_{d}}=1+h_{d}e^{2i\sigma_{d}}.$ (42)

While the $r_{d},\theta_{d}$ parameters give the relation between the full
mixing amplitude and the SM one, and are convenient to apply to the
measurements, the $h_{d},\sigma_{d}$ parameters give the relation between the
new physics and SM contributions, and are more convenient in testing
theoretical models:

$h_{d}e^{2i\sigma_{d}}=\frac{M_{12}^{\rm NP}}{M_{12}^{\rm SM}}.$ (43)

The constraints in the $h_{d}-\sigma_{d}$ plane are shown in Fig. 4(b). We can
make the following two statements:

  1. 1.

A new physics contribution to $B^{0}-\overline{B}^{0}$ mixing amplitude that
carries a phase that is significantly different from the KM phase is
constrained to lie below the 20-30% level.

  2. 2.

A new physics contribution to the $B^{0}-\overline{B}^{0}$ mixing amplitude
which is aligned with the KM phase is constrained to be at most comparable to
the CKM contribution.

One can reformulate these statements as follows:

  1. 1.

The KM mechanism dominates CP violation in $B^{0}-\overline{B}^{0}$ mixing.

  2. 2.

The CKM mechanism is a major player in $B^{0}-\overline{B}^{0}$ mixing.

##  0.5 The new physics flavor puzzle

###  0.5.1 A model independent discussion

It is clear that the Standard Model is not a complete theory of Nature:

  1. 1.

It does not include gravity, and therefore it cannot be valid at energy scales
above $m_{\rm Planck}\sim 10^{19}$ GeV:

  2. 2.

It does not allow for neutrino masses, and therefore it cannot be valid at
energy scales above $m_{\rm seesaw}\sim 10^{15}$ GeV;

  3. 3.

The fine-tuning problem of the Higgs mass suggests that the scale where the SM
is replaced with a more fundamental theory is actually much lower, $m_{\rm
top-partners}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$ a few TeV.

  4. 4.

If the dark matter is made of weakly interacting massive particles (WIMPs)
then, again, a low scale of new physics is likely, $m_{\rm wimp}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$ a
few TeV.

Given that the SM is only an effective low energy theory, non-renormalizable
terms must be added to ${\cal L}_{\rm SM}$ of Eq. (4). These are terms of
dimension higher than four in the fields which, therefore, have couplings that
are inversely proportional to the scale of new physics $\Lambda_{\rm NP}$. For
example, the lowest dimension non-renormalizable terms are dimension five:

$-{\cal L}_{\rm Yukawa}^{\rm dim-5}={Z_{ij}^{\nu}\over\Lambda_{\rm
NP}}L_{Li}^{I}L_{Lj}^{I}\phi\phi+{\rm h.c.}.$ (44)

These are the seesaw terms, leading to neutrino masses.

Exercise 5: How does the global symmetry breaking pattern (14) change when
(44) is taken into account?

Exercise 6: What is the number of physical lepton flavor parameters in this
case? Identify these parameters in the mass basis.

As concerns quark flavor physics, consider, for example, the following
dimension-six, four-fermion, flavor changing operators:

${\cal L}_{\Delta F=2}=\frac{z_{sd}}{\Lambda_{\rm
NP}^{2}}(\overline{d_{L}}\gamma_{\mu}s_{L})^{2}+\frac{z_{cu}}{\Lambda_{\rm
NP}^{2}}(\overline{c_{L}}\gamma_{\mu}u_{L})^{2}+\frac{z_{bd}}{\Lambda_{\rm
NP}^{2}}(\overline{d_{L}}\gamma_{\mu}b_{L})^{2}+\frac{z_{bs}}{\Lambda_{\rm
NP}^{2}}(\overline{s_{L}}\gamma_{\mu}b_{L})^{2}.$ (45)

Each of these terms contributes to the mass splitting between the
corresponding two neutral mesons. For example, the term ${\cal L}_{\Delta
B=2}\propto(\overline{d_{L}}\gamma_{\mu}b_{L})^{2}$ contributes to $\Delta
m_{B}$, the mass difference between the two neutral $B$-mesons. We use
$M_{12}^{B}=\frac{1}{2m_{B}}\langle B^{0}|{\cal L}_{\Delta
F=2}|\overline{B}^{0}\rangle$ and

$\langle
B^{0}|(\overline{d_{La}}\gamma^{\mu}b_{La})(\overline{d_{Lb}}\gamma_{\mu}b_{Lb})|\overline{B}^{0}\rangle=-\frac{1}{3}m_{B}^{2}f_{B}^{2}B_{B}.$
(46)

This leads to $\Delta
m_{B}/m_{B}=2|M_{12}^{B}|/m_{B}\sim(|z_{bd}|/3)(f_{B}/\Lambda_{\rm NP})^{2}$.
Analogous expressions hold for the other neutral mesons.

The experimental results for CP conserving and CP violating observables
related to neutral meson mixing (mass splittings and CP asymmetries in tree
level decays, respectively) are given in Table 1.

Table 1: Measurements related to neutral meson mixing Sector | CP-conserving | CP-violating
---|---|---
sd |  $\Delta m_{K}/m_{K}=7.0\times 10^{-15}$ |  $\epsilon_{K}=2.3\times 10^{-3}$
cu |  $\Delta m_{D}/m_{D}=8.7\times 10^{-15}$ |  $A_{\Gamma}/y_{\rm CP}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.2$
bd |  $\Delta m_{B}/m_{B}=6.3\times 10^{-14}$ |  $S_{\psi K}=+0.67\pm 0.02$
bs |  $\Delta m_{B_{s}}/m_{B_{s}}=2.1\times 10^{-12}$ |  $S_{\psi\phi}=-0.04\pm 0.09$

The measurements quoted in Table 1 lead, for a given value of $|z_{ij}|$ and
$z^{I}_{ij}\equiv{\cal I}m(z_{ij})$, to lower bounds on the scale
$\Lambda_{\rm NP}$. In Table 2 we give the bounds that correspond to
$|z_{ij}|=1$ and to $z^{I}_{ij}=1$. The bounds scale like $\sqrt{z_{ij}}$ and
$\sqrt{z^{I}_{ij}}$, respectively.

Table 2: Lower bounds on the scale of new physics $\Lambda_{\rm NP}$, in units of TeV. The bounds from CP conserving (violating) observables scale like $\sqrt{z_{ij}}$ ($\sqrt{z^{I}_{ij}}$). $ij$ | CP-conserving | CP-violating
---|---|---
sd |  $1\times 10^{3}$ |  $2\times 10^{4}$
cu |  $1\times 10^{3}$ |  $3\times 10^{3}$
bd |  $4\times 10^{2}$ |  $8\times 10^{2}$
bs |  $7\times 10^{1}$ |  $2\times 10^{2}$

We conclude that if the new physics has a generic flavor structure, that is
$z_{ij}={\cal O}(1)$, then its scale must be above $10^{3}-10^{4}$ TeV. If the
leading contributions involve electroweak loops, the lower bound is somewhat
lower, of order $10^{2}-10^{3}$ TeV. The bounds from the corresponding four-
fermi terms with LR structure, instead of the LL structure of Eq. (45), are
even stronger. If indeed $\Lambda_{\rm NP}\gg TeV$, it means that we have
misinterpreted the hints from the fine-tuning problem and the dark matter
puzzle.

There is, however, another way to look at these constraints:

$\displaystyle z_{sd}$ $\displaystyle\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$ $\displaystyle
8\times 10^{-7}\ (\Lambda_{\rm NP}/TeV)^{2},$ $\displaystyle z_{cu}$
$\displaystyle\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$ $\displaystyle 5\times 10^{-7}\
(\Lambda_{\rm NP}/TeV)^{2},$ $\displaystyle z_{bd}$
$\displaystyle\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$ $\displaystyle 5\times 10^{-6}\
(\Lambda_{\rm NP}/TeV)^{2},$ $\displaystyle z_{bs}$
$\displaystyle\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$ $\displaystyle 2\times 10^{-4}\
(\Lambda_{\rm NP}/TeV)^{2},$ (47) $\displaystyle z_{sd}^{I}$
$\displaystyle\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$ $\displaystyle 6\times 10^{-9}\
(\Lambda_{\rm NP}/TeV)^{2},$ $\displaystyle z_{cu}^{I}$
$\displaystyle\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$ $\displaystyle 1\times 10^{-7}\
(\Lambda_{\rm NP}/TeV)^{2},$ $\displaystyle z_{bd}^{I}$
$\displaystyle\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$ $\displaystyle 1\times 10^{-6}\
(\Lambda_{\rm NP}/TeV)^{2},$ $\displaystyle z_{bs}^{I}$
$\displaystyle\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$ $\displaystyle 2\times 10^{-5}\
(\Lambda_{\rm NP}/TeV)^{2}.$ (48)

It could be that the scale of new physics is of order TeV, but its flavor
structure is far from generic. Specifically, if new particles at the TeV scale
couple to the SM fermions, then there are two ways in which their
contributions to FCNC processes, such as neutral meson mixing, can be
suppressed: degeneracy and alignment. Either of these principles, or a
combination of both, signifies non-generic structure.

One can use the language of effective operators also for the SM, integrating
out all particles significantly heavier than the neutral mesons (that is, the
top, the Higgs and the weak gauge bosons). Thus, the scale is $\Lambda_{\rm
SM}\sim m_{W}$. Since the leading contributions to neutral meson mixings come
from box diagrams, the $z_{ij}$ coefficients are suppressed by
$\alpha_{2}^{2}$. To identify the relevant flavor suppression factor, one can
employ the spurion formalism. For example, the flavor transition that is
relevant to $B^{0}-\overline{B}{}^{0}$ mixing involves $\overline{d_{L}}b_{L}$
which transforms as $(8,1,1)_{SU(3)_{q}^{3}}$. The leading contribution must
then be proportional to $(Y^{u}Y^{u\dagger})_{13}\propto
y_{t}^{2}V_{tb}V_{td}^{*}$. Indeed, an explicit calculation, using VIA for the
matrix element and neglecting QCD corrections, gives (a detailed derivation
can be found in Appendix B of [17])

$\frac{2M_{12}^{B}}{m_{B}}\approx-\frac{\alpha_{2}^{2}}{12}\frac{f_{B}^{2}}{m_{W}^{2}}S_{0}(x_{t})(V_{tb}V_{td}^{*})^{2},$
(49)

where $x_{i}=m_{i}^{2}/m_{W}^{2}$ and

$S_{0}(x)=\frac{x}{(1-x)^{2}}\left[1-\frac{11x}{4}+\frac{x^{2}}{4}-\frac{3x^{2}\ln
x}{2(1-x)}\right].$ (50)

Similar spurion analyses, or explicit calculations, allow us to extract the
weak and flavor suppression factors that apply in the SM:

$\displaystyle{\cal I}m(z_{sd}^{\rm SM})$ $\displaystyle\sim$
$\displaystyle\alpha_{2}^{2}y_{t}^{2}|V_{td}V_{ts}|^{2}\sim 1\times 10^{-10},$
$\displaystyle z_{sd}^{\rm SM}$ $\displaystyle\sim$
$\displaystyle\alpha_{2}^{2}y_{c}^{2}|V_{cd}V_{cs}|^{2}\sim 5\times 10^{-9},$
$\displaystyle{\cal I}m(z_{cu}^{\rm SM})$ $\displaystyle\sim$
$\displaystyle\alpha_{2}^{2}y_{b}^{2}|V_{ub}V_{cb}|^{2}\sim 2\times 10^{-14},$
$\displaystyle z_{bd}^{\rm SM}$ $\displaystyle\sim$
$\displaystyle\alpha_{2}^{2}y_{t}^{2}|V_{td}V_{tb}|^{2}\sim 7\times 10^{-8},$
$\displaystyle z_{bs}^{\rm SM}$ $\displaystyle\sim$
$\displaystyle\alpha_{2}^{2}y_{t}^{2}|V_{ts}V_{tb}|^{2}\sim 2\times 10^{-6}.$
(51)

Note that we did not include $z_{cu}^{\rm SM}$ in the list. The reason is tha
it requires a more detailed consideration. The naively leading short distance
contribution is
$\propto\alpha_{2}^{2}(y_{s}^{4}/y_{c}^{2})|V_{cs}V_{us}|^{2}\sim 5\times
10^{-13}$. However, higher dimension terms can replace a $y_{s}^{2}$ factor
with $(\Lambda/m_{D})^{2}$ [18]. Moreover, long distance contributions are
expected to dominate. In particular, peculiar phase space effects [19, 20]
have been identified which are expected to enhance $\Delta m_{D}$ to within an
order of magnitude of the its measured value. The CP violating part, on the
other hand, is dominated by short distance physics.

It is clear then that contributions from new physics at $\Lambda_{\rm NP}\sim
1\ TeV$ should be suppressed by factors that are comparable or smaller than
the SM ones. Why does that happen? This is the new physics flavor puzzle.

The fact that the flavor structure of new physics at the TeV scale must be
non-generic means that flavor measurements are a good probe of the new
physics. Perhaps the best-studied example is that of supersymmetry. Here, the
spectrum of the superpartners and the structure of their couplings to the SM
fermions will allow us to probe the mechanism of dynamical supersymmetry
breaking.

###  0.5.2 The supersymmetric flavor puzzle

We consider, as an example, the contributions from the box diagrams involving
the squark doublets of the second and third generations, $\tilde{Q}_{L2,3}$,
to the $B_{s}-\overline{B_{s}}$ mixing amplitude. The contributions are
proportional to $K_{3i}^{d*}K^{d}_{2i}K_{3j}^{d*}K^{d}_{2j}$, where $K^{d}$ is
the mixing matrix of the gluino couplings to a left-handed down quark and
their supersymmetric squark partners ($\propto[(\delta^{d}_{LL})_{23}]^{2}$ in
the mass insertion approximation, described in Appendix .11.1). We work in the
mass basis for both quarks and squarks. A detailed derivation [21] is given in
Appendix .11.2. It gives:

$\displaystyle M_{12}^{s}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}^{2}m_{B_{s}}f_{B_{s}}^{2}B_{B_{s}}\eta_{\rm
QCD}}{108m_{\tilde{d}}^{2}}[11\tilde{f}_{6}(x)+4xf_{6}(x)]\frac{(\Delta\tilde{m}^{2}_{\tilde{d}})^{2}}{\tilde{m}_{d}^{4}}(K_{32}^{d*}K_{22}^{d})^{2}.$
(52)

Here $m_{\tilde{d}}$ is the average mass of the two squark generations,
$\Delta m^{2}_{\tilde{d}}$ is the mass-squared difference, and
$x=m_{\tilde{g}}^{2}/m_{\tilde{d}}^{2}$.

Eq. (52) can be translated into our generic language:

$\displaystyle\Lambda_{\rm NP}$ $\displaystyle=$ $\displaystyle
m_{\tilde{q}},$ (53) $\displaystyle z_{1}^{bs}$ $\displaystyle=$
$\displaystyle\frac{11\tilde{f}_{6}(x)+4xf_{6}(x)}{18}\alpha_{s}^{2}\left(\frac{\Delta\tilde{m}_{\tilde{d}}^{2}}{m_{\tilde{d}}^{2}}\right)^{2}(K_{32}^{d*}K_{22}^{d})^{2}\approx
10^{-4}(\delta^{LL}_{23})^{2},$

where, for the last approximation, we took the example of $x=1$ [and used,
correspondingly, $11\tilde{f}_{6}(1)+4f_{6}(1)=1/6$], and defined

$\delta^{LL}_{23}=\left(\frac{\Delta\tilde{m}_{\tilde{d}}^{2}}{m_{\tilde{d}}^{2}}\right)(K_{32}^{d*}K_{22}^{d}).$
(54)

Similar expressions can be derived for the dependence of
$K^{0}-\overline{K^{0}}$ on $(\delta^{d}_{MN})_{12}$, $B^{0}-\overline{B^{0}}$
on $(\delta^{d}_{MN})_{13}$, and $D^{0}-\overline{D^{0}}$ on
$(\delta^{u}_{MN})_{12}$. Then we can use the constraints of Eqs.
(0.5.1,0.5.1) to put upper bounds on $(\delta^{q}_{MN})_{ij}$. Some examples
are given in Table 3 (see Ref. [22] for details and list of references).

Table 3: The phenomenological upper bounds on $(\delta^{q}_{LL})_{ij}$ and $\langle\delta^{q}_{ij}\rangle=\sqrt{(\delta^{q}_{LL})_{ij}(\delta^{q}_{RR})_{ij}}$. Here $q=u,d$ and $M=L,R$. The constraints are given for $m_{\tilde{q}}=1$ TeV and $x=m_{\tilde{g}}^{2}/m_{\tilde{q}}^{2}=1$. We assume that the phases could suppress the imaginary part by a factor of $\sim 0.3$. Taken from Ref. [22]. $q$ | $ij$ | $(\delta^{q}_{LL})_{ij}$ | $\langle\delta^{q}_{ij}\rangle$
---|---|---|---
d | 12 | 0.03 | 0.002
d | 13 | 0.2 | 0.07
d | 23 | 0.2 | 0.07
u | 12 | 0.1 | 0.008

We learn that, in most cases, we need $\delta^{q}_{ij}/m_{\tilde{q}}\ll 1/{\rm
TeV}$. One can immediately identify three generic ways in which supersymmetric
contributions to neutral meson mixing can be suppressed:

  1. 1.

Heaviness: $m_{\tilde{q}}\gg 1\ TeV$;

  2. 2.

Degeneracy: $\Delta m^{2}_{\tilde{q}}\ll m_{\tilde{q}}^{2}$;

  3. 3.

Alignment: $K^{q}_{ij}\ll 1$.

When heaviness is the only suppression mechanism, as in split supersymmetry
[23], the squarks are very heavy and supersymmetry no longer solves the fine
tuning problem. (When the first two squark generations are mildly heavy and
the third generation is light, as in effective supersymmetry [24], the fine
tuning problem is still solved, but additional suppression mechanisms are
needed.) If we want to maintain supersymmetry as a solution to the fine tuning
problem, either degeneracy or alignment or a combination of the two is needed.
This means that the flavor structure of supersymmetry is not generic, as
argued in the previous section.

Take, for example, $(\delta^{d}_{LL})_{12}\leq 0.03$. Naively, one might
expect the alignment to be of order $(V_{cd}V_{cs}^{*})\sim 0.2$, which is far
from sufficient by itself. Barring a very precise alignment
($|K^{d}_{12}|\ll|V_{us}|$) [25, 26] and accidental cancelations, we are led
to conclude that the first two squark generations must be quasi-degenerate.
Actually, by combining the constraints from $K^{0}-\overline{K^{0}}$ mixing
and $D^{0}-\overline{D^{0}}$ mixing, one can show that this is the case
independently of assumptions about the alignment [27, 28, 29]. Analogous
conclusions can be drawn for many TeV-scale new physics scenarios: a strong
level of degeneracy is required (for definitions and detailed analysis, see
[30]).

Exercise 9: Does $K_{31}^{d}\sim|V_{ub}|$ suffice to satisfy the $\Delta
m_{B}$ constraint with neither degeneracy nor heaviness? (Use the two
generation approximation and ignore the second generation.)

Is there a natural way to make the squarks degenerate? Degeneracy requires
that the $3\times 3$ matrix of soft supersymmetry breaking mass-squared terms
$\tilde{m}^{2}_{Q_{L}}\simeq\tilde{m}^{2}_{\tilde{q}}{\bf 1}$. We have
mentioned already that flavor universality is a generic feature of gauge
interactions. Thus, the requirement of degeneracy is perhaps a hint that
supersymmetry breaking is gauge mediated to the MSSM fields.

###  0.5.3 Minimal flavor violation (MFV)

If supersymmetry breaking is gauge mediated, the squark mass matrices for
$SU(2)_{L}$\- doublet and $SU(2)_{L}$-singlet squarks have the following form
at the scale of mediation $m_{M}$:

$\displaystyle\tilde{M}^{2}_{U_{L}}(m_{M})$ $\displaystyle=$
$\displaystyle\left(m^{2}_{\tilde{Q}_{L}}+D_{U_{L}}\right){\bf
1}+M_{u}M_{u}^{\dagger},$ $\displaystyle\tilde{M}^{2}_{D_{L}}(m_{M})$
$\displaystyle=$
$\displaystyle\left(m^{2}_{\tilde{Q}_{L}}+D_{D_{L}}\right){\bf
1}+M_{d}M_{d}^{\dagger},$ $\displaystyle\tilde{M}^{2}_{U_{R}}(m_{M})$
$\displaystyle=$
$\displaystyle\left(m^{2}_{\tilde{U}_{R}}+D_{U_{R}}\right){\bf
1}+M_{u}^{\dagger}M_{u},$ $\displaystyle\tilde{M}^{2}_{D_{R}}(m_{M})$
$\displaystyle=$
$\displaystyle\left(m^{2}_{\tilde{D}_{R}}+D_{D_{R}}\right){\bf
1}+M_{d}^{\dagger}M_{d},$ (55)

where $D_{q_{A}}=(T_{3})_{q_{A}}-(Q_{\rm EM})_{q_{A}}s^{2}_{W}m_{Z}^{2}\cos
2\beta$ are the $D$-term contributions. Here, the only source of the
$SU(3)^{3}_{q}$ breaking are the SM Yukawa matrices.

This statement holds also when the renormalization group evolution is applied
to find the form of these matrices at the weak scale. Taking the scale of the
soft breaking terms $m_{\tilde{q}_{A}}$ to be somewhat higher than the
electroweak breaking scale $m_{Z}$ allows us to neglect the $D_{q_{A}}$ and
$M_{q}$ terms in (0.5.3). Then we obtain

$\displaystyle\tilde{M}^{2}_{Q_{L}}(m_{Z})$ $\displaystyle\sim$ $\displaystyle
m^{2}_{\tilde{Q}_{L}}\left(r_{3}{\bf
1}+c_{u}Y_{u}Y_{u}^{\dagger}+c_{d}Y_{d}Y_{d}^{\dagger}\right),$
$\displaystyle\tilde{M}^{2}_{U_{R}}(m_{Z})$ $\displaystyle\sim$ $\displaystyle
m^{2}_{\tilde{U}_{R}}\left(r_{3}{\bf 1}+c_{uR}Y_{u}^{\dagger}Y_{u}\right),$
$\displaystyle\tilde{M}^{2}_{D_{R}}(m_{Z})$ $\displaystyle\sim$ $\displaystyle
m^{2}_{\tilde{D}_{R}}\left(r_{3}{\bf 1}+c_{dR}Y_{d}^{\dagger}Y_{d}\right).$
(56)

Here $r_{3}$ represents the universal RGE contribution that is proportional to
the gluino mass ($r_{3}={\cal O}(6)\times(M_{3}(m_{M})/m_{\tilde{q}}(m_{M}))$)
and the $c$-coefficients depend logarithmically on $m_{M}/m_{Z}$ and can be of
${\cal O}(1)$ when $m_{M}$ is not far below the GUT scale.

Models of gauge mediated supersymmetry breaking (GMSB) provide a concrete
example of a large class of models that obey a simple principle called minimal
flavor violation (MFV) [31]. This principle guarantees that low energy flavor
changing processes deviate only very little from the SM predictions. The basic
idea can be described as follows. The gauge interactions of the SM are
universal in flavor space. The only breaking of this flavor universality comes
from the three Yukawa matrices, $Y^{u}$, $Y^{d}$ and $Y^{e}$. If this remains
true in the presence of the new physics, namely $Y^{u}$, $Y^{d}$ and $Y^{e}$
are the only flavor non-universal parameters, then the model belongs to the
MFV class.

Let us now formulate this principle in a more formal way, using the language
of spurions that we presented in section 0.3.2. The Standard Model with
vanishing Yukawa couplings has a large global symmetry (11,0.3.2). In this
section we concentrate only on the quarks. The non-Abelian part of the flavor
symmetry for the quarks is $SU(3)_{q}^{3}$ of Eq. (0.3.2) with the three
generations of quark fields transforming as follows:

$Q_{L}(3,1,1),\ \ U_{R}(1,3,1),\ \ D_{R}(1,1,3).$ (57)

The Yukawa interactions,

${\cal L}_{Y}=\overline{Q_{L}}Y^{d}D_{R}H+\overline{Q_{L}}Y^{u}U_{R}H_{c},$
(58)

($H_{c}=i\tau_{2}H^{*}$) break this symmetry. The Yukawa couplings can thus be
thought of as spurions with the following transformation properties under
$SU(3)_{q}^{3}$ [see Eq. (15)]:

$Y^{u}\sim(3,\bar{3},1),\qquad Y^{d}\sim(3,1,\bar{3}).$ (59)

When we say “spurions”, we mean that we pretend that the Yukawa matrices are
fields which transform under the flavor symmetry, and then require that all
the Lagrangian terms, constructed from the SM fields, $Y^{d}$ and $Y^{u}$,
must be (formally) invariant under the flavor group $SU(3)_{q}^{3}$. Of
course, in reality, ${\cal L}_{Y}$ breaks $SU(3)_{q}^{3}$ precisely because
$Y^{d,u}$ are not fields and do not transform under the symmetry.

The idea of minimal flavor violation is relevant to extensions of the SM, and
can be applied in two ways:

  1. 1.

If we consider the SM as a low energy effective theory, then all higher-
dimension operators, constructed from SM-fields and $Y$-spurions, are formally
invariant under $G_{\rm global}$.

  2. 2.

If we consider a full high-energy theory that extends the SM, then all
operators, constructed from SM and the new fields, and from $Y$-spurions, are
formally invariant under $G_{\rm global}$.

Exercise 10: Use the spurion formalism to argue that, in MFV models, the
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay amplitude is proportional to
$y_{t}^{2}V_{td}V_{ts}^{*}$.

Exercise 11: Find the flavor suppression factors in the $z_{i}^{bs}$
coefficients, if MFV is imposed, and compare to the bounds in Eq. (0.5.1).

Examples of MFV models include models of supersymmetry with gauge-mediation or
with anomaly-mediation of its breaking.

#### Testing MFV at the LHC

If the LHC discovers new particles that couple to the SM fermions, then it
will be able to test solutions to the new physics flavor puzzle such as MFV
[32]. Much of its power to test such frameworks is based on identifying top
and bottom quarks.

To understand this statement, we notice that the spurions $Y^{u}$ and $Y^{d}$
can always be written in terms of the two diagonal Yukawa matrices
$\lambda_{u}$ and $\lambda_{d}$ and the CKM matrix $V$, see Eqs. (17,18).
Thus, the only source of quark flavor changing transitions in MFV models is
the CKM matrix. Next, note that to an accuracy that is better than ${\cal
O}(0.05)$, we can write the CKM matrix as follows:

$V=\pmatrix{1}&0.230\\\ -0.2310\\\ 001\\\ .$ (60)

Exercise 12: The approximation (60) should be intuitively obvious to top-
physicists, but definitely counter-intuitive to bottom-physicists. (Some of
them have dedicated a large part of their careers to experimental or
theoretical efforts to determine $V_{cb}$ and $V_{ub}$.) What does the
approximation imply for the bottom quark? When we take into account that it is
only good to ${\cal O}(0.05)$, what would the implications be?

We learn that the third generation of quarks is decoupled, to a good
approximation, from the first two. This, in turn, means that any new particle
that couples to an odd number of the SM quarks (think, for example, of heavy
quarks in vector-like representations of $G_{\rm SM}$), decay into either
third generation quark, or to non-third generation quark, but not to both. For
example, in Ref. [32], MFV models with additional charge $-1/3$, $SU(2)_{\rm
L}$-singlet quarks – $B^{\prime}$ – were considered. A concrete test of MFV
was proposed, based on the fact that the largest mixing effect involving the
third generation is of order $|V_{cb}|^{2}\sim 0.002$: Is the following
prediction, concerning events of $B^{\prime}$ pair production, fulfilled:

$\frac{\Gamma(B^{\prime}\overline{B^{\prime}}\to
Xq_{1,2}q_{3})}{\Gamma(B^{\prime}\overline{B^{\prime}}\to
Xq_{1,2}q_{1,2})+\Gamma(B^{\prime}\overline{B^{\prime}}\to
Xq_{3}q_{3})}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}10^{-3}.$ (61)

If not, then MFV is excluded. One could similarly test various versions of
minimal lepton flavor violation (MLFV) [33, 34, 35, 36, 37, 38].

Analogous tests can be carried out in the supersymmetric framework [39, 43,
44, 40, 41, 42, 45]. Here, there is also a generic prediction that, in each of
the three sectors ($Q_{L},U_{R},D_{R}$), squarks of the first two generations
are quasi-degenerate, and do not decay into third generation quarks. Squarks
of the third generation can be separated in mass (though, for small
$\tan\beta$, the degeneracy in the $\tilde{D}_{R}$ sector is threefold), and
decay only to third generation quarks.

We conclude that measurements at the LHC related to new particles that couple
to the SM fermions are likely to teach us much more about flavor physics.

##  0.6 The Standard Model flavor puzzle

The SM has thirteen flavor parameters: six quark Yukawa couplings, four CKM
parameters (three angles and a phase), and three charged lepton Yukawa
couplings. (One can use fermions masses instead of the fermion Yukawa
couplings, $Y_{f}=\sqrt{2}m_{f}/v$.) The orders of magnitudes of these
thirteen dimensionless parameters are as follows:

$\displaystyle Y_{t}$ $\displaystyle\sim$ $\displaystyle 1,\ \ \ Y_{c}\sim
10^{-2},\ \ \ Y_{u}\sim 10^{-5},$ $\displaystyle Y_{b}$ $\displaystyle\sim$
$\displaystyle 10^{-2},\ \ \ Y_{s}\sim 10^{-3},\ \ \ Y_{d}\sim 10^{-4},$
$\displaystyle Y_{\tau}$ $\displaystyle\sim$ $\displaystyle 10^{-2},\ \ \
Y_{\mu}\sim 10^{-3},\ \ \ Y_{e}\sim 10^{-6},$ $\displaystyle|V_{us}|$
$\displaystyle\sim$ $\displaystyle 0.2,\ \ |V_{cb}|\sim 0.04,\ \ |V_{ub}|\sim
0.004,\ \ \ \delta_{\rm KM}\sim 1.$ (62)

Only two of these parameters are clearly of ${\cal O}(1)$, the top-Yukawa and
the KM phase. The other flavor parameters exhibit smallness and hierarchy.
Their values span six orders of magnitude. It may be that this set of
numerical values are just accidental. More likely, the smallness and the
hierarchy have a reason. The question of why there is smallness and hierarchy
in the SM flavor parameters constitutes “The Standard Model flavor puzzle."

The motivation to think that there is indeed a structure in the flavor
parameters is strengthened by considering the values of the four SM parameters
that are not flavor parameters, namely the three gauge couplings and the Higgs
self-coupling:

$g_{s}\sim 1,\ \ g\sim 0.6,\ \ e\sim 0.3,\ \ \lambda\sim 0.2.$ (63)

This set of values does seem to be a random distribution of order-one numbers,
as one would naively expect.

A few examples of mechanisms that were proposed to explain the observed
structure of the flavor parameters are the following:

  * •

An approximate Abelian symmetry (“The Froggatt-Nielsen mechanism" [46]);

  * •

An approximate non-Abelian symmetry (see e.g. [47]);

  * •

Conformal dynamics (“The Nelson-Strassler mechanism" [48]);

  * •

Location in an extra dimension [49].

We will take as an example the Froggatt-Nielsen mechanism.

###  0.6.1 The Froggatt-Nielsen mechanism

Small numbers and hierarchies are often explained by approximate symmetries.
For example, the small mass splitting between the charged and neural pions
finds an explanation in the approximate isospin (global $SU(2)$) symmetry of
the strong interactions.

Approximate symmetries lead to selection rules which account for the size of
deviations from the symmetry limit. Spurion analysis is particularly
convenient to derive such selection rules. The Froggatt-Nielsen mechanism
postulates a $U(1)_{H}$ symmetry, that is broken by a small spurion
$\epsilon_{H}$. Without loss of generality, we assign $\epsilon_{H}$ a
$U(1)_{H}$ charge of $H(\epsilon_{H})=-1$. Each SM field is assigned a
$U(1)_{H}$ charge. In general, different fermion generations are assigned
different charges, hence the term ‘horizontal symmetry.’ The rule is that each
term in the Lagrangian, made of SM fields and the spurion should be formally
invariant under $U(1)_{H}$.

The approximate $U(1)_{H}$ symmetry thus leads to the following selection
rules:

$\displaystyle Y^{u}_{ij}$ $\displaystyle=$
$\displaystyle\epsilon_{H}^{|H(\bar{Q}_{i})+H(U_{j})+H(\phi_{u})|},$
$\displaystyle Y^{d}_{ij}$ $\displaystyle=$
$\displaystyle\epsilon_{H}^{|H(\bar{Q}_{i})+H(D_{j})+H(\phi_{d})|},$
$\displaystyle Y^{e}_{ij}$ $\displaystyle=$
$\displaystyle\epsilon_{H}^{|H(\bar{L}_{i})+H(E_{j})-H(\phi_{d})|}.$ (64)

As a concrete example, we take the following set of charges:

$\displaystyle H(\bar{Q}_{i})$ $\displaystyle=$ $\displaystyle
H(U_{i})=H(E_{i})=(2,1,0),$ $\displaystyle H(\bar{L}_{i})$ $\displaystyle=$
$\displaystyle H(D_{i})=(0,0,0),$ $\displaystyle H(\phi_{u})$ $\displaystyle=$
$\displaystyle H(\phi_{d})=0.$ (65)

It leads to the following parametric suppressions of the Yukawa couplings:

$Y^{u}\sim\pmatrix{\epsilon}^{4}&\epsilon^{3}\epsilon^{2}\\\
\epsilon^{3}\epsilon^{2}\epsilon\\\ \epsilon^{2}\epsilon 1\\\ ,\ \
Y^{d}\sim(Y^{e})^{T}\sim\pmatrix{\epsilon}^{2}&\epsilon^{2}\epsilon^{2}\\\
\epsilon\epsilon\epsilon\\\ 111\\\ .$ (66)

We emphasize that for each entry we give the parametric suppression (that is
the power of $\epsilon$), but each entry has an unknown (complex) coefficient
of order one, and there are no relations between the order one coefficients of
different entries.

The structure of the Yukawa matrices dictates the parametric suppression of
the physical observables:

$\displaystyle Y_{t}$ $\displaystyle\sim$ $\displaystyle 1,\ \ \
Y_{c}\sim\epsilon^{2},\ \ \ Y_{u}\sim\epsilon^{4},$ $\displaystyle Y_{b}$
$\displaystyle\sim$ $\displaystyle 1,\ \ \ Y_{s}\sim\epsilon,\ \ \
Y_{d}\sim\epsilon^{2},$ $\displaystyle Y_{\tau}$ $\displaystyle\sim$
$\displaystyle 1,\ \ \ Y_{\mu}\sim\epsilon,\ \ \ Y_{e}\sim\epsilon^{2},$
$\displaystyle|V_{us}|$ $\displaystyle\sim$ $\displaystyle\epsilon,\ \
|V_{cb}|\sim\epsilon,\ \ |V_{ub}|\sim\epsilon^{2},\ \ \ \delta_{\rm KM}\sim
1.$ (67)

For $\epsilon\sim 0.05$, the parametric suppressions are roughly consistent
with the observed hierarchy. In particular, this set of charges predicts that
the down and charged lepton mass hierarchies are similar, while the up
hierarchy is the square of the down hierarchy. These features are roughly
realized in Nature.

Exercise 13: Derive the parametric suppression and approximate numerical
values of $Y^{u}$, its eigenvalues, and the three angles of $V_{L}^{u}$, for
$H(Q_{i})=4,2,0$, $H(U_{i})=3,2,0$ and $\epsilon_{H}=0.2$

Could we explain any set of observed values with such an approximate symmetry?
If we could, then the FN mechanism cannot be really tested. The answer however
is negative. Consider, for example, the quark sector. Naively, we have 11
$U(1)_{H}$ charges that we are free to choose. However, the $U(1)_{Y}\times
U(1)_{B}\times U(1)_{\rm PQ}$ symmetry implies that there are only 8
independent choices that affect the structure of the Yukawa couplings. On the
other hand, there are 9 physical parameters. Thus, there should be a single
relation between the physical parameters that is independent of the choice of
charges. Assuming that the sum of charges in the exponents of Eq. (0.6.1) is
of the same sign for all 18 combinations, the relation is

$|V_{ub}|\sim|V_{us}V_{cb}|,$ (68)

which is fulfilled to within a factor of 2. There are also interesting
inequalities (here $i<j$):

$|V_{ij}|\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}m(U_{i})/m(U_{j}),\
m(D_{i})/m(D_{j}).$ (69)

All six inequalities are fulfilled. Finally, if we order the up and the down
masses from light to heavy, then the CKM matrix is predicted to be $\sim{\bf
1}$, namely the diagonal entries are not parametrically suppressed. This
structure is also consistent with the observed CKM structure.

###  0.6.2 The flavor of neutrinos

Five neutrino flavor parameters have been measured in recent years (see e.g.
[50]): two mass-squared differences,

$\Delta m^{2}_{21}=(7.5\pm 0.2)\times 10^{-5}\ {\rm eV}^{2},\ \ \ |\Delta
m^{2}_{32}|=(2.5\pm 0.1)\times 10^{-3}\ {\rm eV}^{2},$ (70)

and the three mixing angles,

$|U_{e2}|=0.55\pm 0.01,\ \ |U_{\mu 3}|=0.64\pm 0.02,\ \ |U_{e3}|=0.15\pm
0.01.$ (71)

These parameters constitute a significant addition to the thirteen SM flavor
parameters and provide, in principle, tests of various ideas to explain the SM
flavor puzzle.

The numerical values of the parameters show various surprising features:

  * •

$|U_{\mu 3}|>{\rm any}\ |V_{ij}|$;

  * •

$|U_{e2}|>{\rm any}\ |V_{ij}|$;

  * •

$|U_{e3}|$ is not particularly small ($|U_{e3}|\not\ll|U_{e2}U_{\mu 3}|$);

  * •

$m_{2}/m_{3}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}1/6>{\rm any}\ m_{i}/m_{j}$ for
charged fermions.

These features can be summarized by the statement that, in contrast to the
charged fermions, neither smallness nor hierarchy have been observed so far in
the neutrino related parameters.

One way of interpretation of the neutrino data comes under the name of
neutrino mass anarchy [51, 52, 53]. It postulates that the neutrino mass
matrix has no structure, namely all entries are of the same order of
magnitude. Normalized to an effective neutrino mass scale, $v^{2}/\Lambda_{\rm
seesaw}$, the various entries are random numbers of order one. Note that
anarchy means neither hierarchy nor degeneracy.

If true, the contrast between neutrino mass anarchy and quark and charged
lepton mass hierarchy may be a deep hint for a difference between the flavor
physics of Majorana and Dirac fermions. The source of both anarchy and
hierarchy might, however, be explained by a much more mundane mechanism. In
particular, neutrino mass anarchy could be a result of a FN mechanism, where
the three left-handed lepton doublets carry the same FN charge. In that case,
the FN mechanism predict parametric suppression of neither neutrino mass
ratios nor leptonic mixing angles, which is quite consistent with (70) and
(71). Indeed, the viable FN model presented in Section 0.6.1 belongs to this
class.

Another possible interpretation of the neutrino data is to take
$m_{2}/m_{3}\sim|U_{e3}|\sim 0.15$ to be small, and require that they are
parametrically suppressed (while the other two mixing angles are order one).
Such a situation is impossible to accommodate in a large class of FN models
[54].

The same data, and in particular the proximity of $|U_{e2}|$ to
$1/\sqrt{3}\simeq 0.58$ and the proximity of $|U_{\mu 3}|$ to
$1/\sqrt{2}\simeq 0.71$ led to a very different interpretation. This
interpretation, termed ‘tribimaximal mixing’ (TBM), postulates that the
leptonic mixing matrix is parametrically close to the following special form
[55]:

$|U|_{\rm TBM}=\pmatrix{}{2}{\sqrt{6}}&\frac{1}{\sqrt{3}}0\\\
\frac{1}{\sqrt{6}}\frac{1}{\sqrt{3}}\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{6}}\frac{1}{\sqrt{3}}\frac{1}{\sqrt{2}}\\\ .$ (72)

Such a form is suggestive of discrete non-Abelian symmetries, and indeed
numerous models based on an $A_{4}$ symmetry have been proposed [56, 57]. A
significant feature of of TBM is that the third mixing angle should be close
to $|U_{e3}|=0$. Until recently, there have been only upper bounds on
$|U_{e3}|$, consistent with the models in the literature. In the last year,
however, a value of $|U_{e3}|$ close to the previous upper bound has been
established [58], see Eq. (71). Such a large value (and the consequent
significant deviation of $|U_{\mu 3}|$ from maximal bimixing) puts in serious
doubt the TBM idea. Indeed, it is difficult in this framework, if not
impossible, to account for $\Delta m^{2}_{12}/\Delta
m^{2}_{23}\sim|U_{e3}|^{2}$ without fine-tuning [59].

##  0.7 Higgs physics: the new flavor arena

A Higgs-like boson $h$ has been discovered by the ATLAS and CMS experiments at
the LHC [60, 61]. The fact that for the $f=\gamma\gamma$ and $f=ZZ^{*}$ final
states, the experiments measure

$R_{f}\equiv\frac{\sigma(pp\to h){\rm BR}(h\to f)}{[\sigma(pp\to h){\rm
BR}(h\to f)]^{\rm SM}},$ (73)

of order one (see e.g. [62]),

$\displaystyle R_{ZZ^{*}}$ $\displaystyle=$ $\displaystyle 1.1\pm 0.2,$ (74)
$\displaystyle R_{\gamma\gamma}$ $\displaystyle=$ $\displaystyle 1.1\pm 0.2,$
(75)

is suggestive that the $h$-production via gluon-gluon fusion proceeds at a
rate similar to the Standard Model (SM) prediction, giving a strong indication
that $Y_{t}$, the $ht\bar{t}$ Yukawa coupling, is of order one. This first
determination of $Y_{t}$ signifies a new arena for the exploration of flavor
physics.

In the future, measurements of $R_{b\bar{b}}$ and $R_{\tau^{+}\tau^{-}}$ will
allow us to extract additional flavor parameters: $Y_{b}$, the $hb\bar{b}$
Yukawa coupling, and $Y_{\tau}$, the $h\tau^{+}\tau^{-}$ Yukawa coupling. For
the latter, the current allowed range is already quite restrictive:

$R_{\tau^{+}\tau^{-}}=1.0\pm 0.4.$ (76)

It may well be that the values of $Y_{b}$ and/or $Y_{\tau}$ will deviate from
their SM values. The most likely explanation of such deviations will be that
there are more than one Higgs doublets, and that the doublet(s) that couple to
the down and charged lepton sectors are not the same as the one that couples
to the up sector.

A more significant test of our understanding of flavor physics, which might
provide a window into new flavor physics, will come further in the future,
when $R_{\mu^{+}\mu^{-}}$ is measured. (At present, there is an upper bound,
$R_{\mu^{+}\mu^{-}}<9.8$.) The ratio

$X_{\mu^{+}\mu^{-}}\equiv\frac{{\rm BR}(h\to\mu^{+}\mu^{-})}{{\rm
BR}(h\to\tau^{+}\tau^{-})},$ (77)

is predicted within the SM with impressive theoretical cleanliness. To leading
order, it is given by $X_{\mu^{+}\mu^{-}}=m_{\mu}^{2}/m_{\tau}^{2}$, and the
corrections of order $\alpha_{W}$ and of order $m_{\mu}^{2}/m_{\tau}^{2}$ to
this leading result are known. It is an interesting question to understand
what can be learned from a test of this relation [63, 64].

It is also possible to search for the SM-forbidden decay modes,
$h\to\mu^{\pm}\tau^{\mp}$ [65, 66, 67, 68]. A measurement of, or an upper
bound on

$X_{\mu\tau}\equiv\frac{{\rm BR}(h\to\mu^{+}\tau^{-})+{\rm
BR}(h\to\mu^{-}\tau^{+})}{{\rm BR}(h\to\tau^{+}\tau^{-})},$ (78)

would provide additional information relevant to flavor physics. Thus, a
broader question is to understand the implications for flavor physics of
measurements of $R_{\tau^{+}\tau^{-}}$, $X_{\mu^{+}\mu^{-}}$ and $X_{\mu\tau}$
[63].

Let us take as an example how we can use the set of these three measurements
if there is a single light Higgs boson. A violation of the SM relation
$Y_{ij}^{\rm SM}=\frac{\sqrt{2}m_{i}}{v}\delta_{ij}$, is a consequence of
nonrenormalizable terms. The leading ones are the $d=6$ terms. In the
interaction basis, we have

$\displaystyle{\cal L}_{Y}^{d=4}$ $\displaystyle=$
$\displaystyle-\lambda_{ij}\bar{f}_{L}^{i}f_{R}^{j}\phi+{\rm h.c.},$ (79)
$\displaystyle{\cal L}_{Y}^{d=6}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{\prime}_{ij}}{\Lambda^{2}}\bar{f}_{L}^{i}f_{R}^{j}\phi(\phi^{\dagger}\phi)+{\rm
h.c.}\,,$

where expanding around the vacuum we have $\phi=(v+h)/\sqrt{2}$. Defining
$V_{L,R}$ via

$\sqrt{2}m=V_{L}\left(\lambda+\frac{v^{2}}{2\Lambda^{2}}\lambda^{\prime}\right)V_{R}^{\dagger}v,$
(80)

where $m={\rm diag}(m_{e},m_{\mu},m_{\tau})$, and defining $\hat{\lambda}$ via

$\hat{\lambda}=V_{L}\lambda^{\prime}V_{R}^{\dagger},$ (81)

we obtain

$Y_{ij}=\frac{\sqrt{2}m_{i}}{v}\delta_{ij}+\frac{v^{2}}{\Lambda^{2}}\hat{\lambda}_{ij}.$
(82)

To proceed, one has to make assumptions about the structure of
$\hat{\lambda}$. In what follows, we consider first the assumption of minimal
flavor violation (MFV) and then a Froggatt-Nielsen (FN) symmetry.

###  0.7.1 MFV

MFV requires that the leptonic part of the Lagrangian is invariant under an
$SU(3)_{L}\times SU(3)_{E}$ global symmetry, with the left-handed lepton
doublets transforming as $(3,1)$, the right-handed charged lepton singlets
transforming as $(1,3)$ and the charged lepton Yukawa matrix $Y$ is a spurion
transforming as $(3,\bar{3})$.

Specifically, MFV means that, in Eq. (79),

$\lambda^{\prime}=a\lambda+b\lambda\lambda^{\dagger}\lambda+{\cal
O}(\lambda^{5}),$ (83)

where $a$ and $b$ are numbers. Note that, if $V_{L}$ and $V_{R}$ are the
diagonalizing matrices for $\lambda$, $V_{L}\lambda
V_{R}^{\dagger}=\lambda^{\rm diag}$, then they are also the diagonalizing
matrices for $\lambda\lambda^{\dagger}\lambda$,
$V_{L}\lambda\lambda^{\dagger}\lambda V_{R}^{\dagger}=(\lambda^{\rm
diag})^{3}$. Then, Eqs. (80), (81) and (82) become

$\displaystyle\frac{\sqrt{2}m}{v}$ $\displaystyle=$
$\displaystyle\left(1+\frac{av^{2}}{2\Lambda^{2}}\right)\lambda^{\rm
diag}+\frac{bv^{2}}{2\Lambda^{2}}(\lambda^{\rm diag})^{3},$
$\displaystyle\hat{\lambda}$ $\displaystyle=$ $\displaystyle a\lambda^{\rm
diag}+b(\lambda^{\rm
diag})^{3}=a\frac{\sqrt{2}m}{v}+\frac{2\sqrt{2}bm^{3}}{v^{3}},$ $\displaystyle
Y_{ij}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}m_{i}}{v}\delta_{ij}\left[1+\frac{av^{2}}{\Lambda^{2}}+\frac{2bm_{i}^{2}}{\Lambda^{2}}\right],$
(84)

where, in the expressions for $\hat{\lambda}$ and $Y$, we included only the
leading universal and leading non-universal corrections to the SM relations.

We learn the following points about the Higgs-related lepton flavor parameters
in this class of models:

  1. 1.

$h$ has no flavor off-diagonal couplings:

$Y_{\mu\tau},Y_{\tau\mu}=0.$ (85)

  2. 2.

The values of the diagonal couplings deviate from their SM values. The
deviation is small, of order $v^{2}/\Lambda^{2}$:

$Y_{\tau}\approx\left(1+\frac{av^{2}}{\Lambda^{2}}\right)\
\frac{\sqrt{2}m_{\tau}}{v}.$ (86)

  3. 3.

The ratio between the Yukawa couplings to different charged lepton flavors
deviates from its SM value. The deviation is, however, very small, of order
$m_{\ell}^{2}/\Lambda^{2}$:

$\frac{Y_{\mu}}{Y_{\tau}}=\frac{m_{\mu}}{m_{\tau}}\left(1-\frac{2b(m_{\tau}^{2}-m_{\mu}^{2})}{\Lambda^{2}}\right).$
(87)

The predictions of the SM with MFV non-renormalizable terms are then the
following:

$\displaystyle\left(\frac{\sigma(pp\to h)^{\rm SM}}{\sigma(pp\to
h)}\frac{\Gamma_{\rm tot}}{\Gamma_{\rm tot}^{\rm
SM}}\right)R_{\tau^{+}\tau^{-}}$ $\displaystyle=$ $\displaystyle
1+2av^{2}/\Lambda^{2},$ $\displaystyle X_{\mu^{+}\mu^{-}}$ $\displaystyle=$
$\displaystyle(m_{\mu}/m_{\tau})^{2}(1-4bm_{\tau}^{2}/\Lambda^{2}),$
$\displaystyle X_{\tau\mu}$ $\displaystyle=$ $\displaystyle 0.$ (88)

Thus, MFV will be excluded if experiments observe the $h\to\mu\tau$ decay. On
the other hand, MFV allows for a universal deviation of ${\cal
O}(v^{2}/\Lambda^{2})$ of the flavor-diagonal dilepton rates, and a smaller
non-universal deviation of ${\cal O}(m_{\tau}^{2}/\Lambda^{2})$.

###  0.7.2 FN

An attractive explanation of the smallness and hierarchy in the Yukawa
couplings is provided by the Froggatt-Nielsen (FN) mechanism [46]. In this
framework, a $U(1)_{H}$ symmetry, under which different generations carry
different charges, is broken by a small parameter $\epsilon_{H}$. Without loss
of generality, $\epsilon_{H}$ is taken to be a spurion of charge $-1$. Then,
various entries in the Yukawa mass matrices are suppressed by different powers
of $\epsilon_{H}$, leading to smallness and hierarchy.

Specifically for the leptonic Yukawa matrix, taking $h$ to be neutral under
$U(1)_{H}$, $H(h)=0$, we have

$\lambda_{ij}\propto\epsilon_{H}^{H(E_{j})-H(L_{i})}\,.$ (89)

We emphasize that the FN mechanism dictates only the parametric suppression.
Each entry has an arbitrary order one coefficient. The resulting parametric
suppression of the masses and leptonic mixing angles is given by [69]

$m_{\ell_{i}}/v\sim\epsilon_{H}^{H(E_{i})-H(L_{i})}\,,\ \ \
|U_{ij}|\sim\epsilon_{H}^{H(L_{j})-H(L_{i})}\,.$ (90)

Since $H(\phi^{\dagger}\phi)=0$, the entries of the matrix $\lambda^{\prime}$
have the same parametric suppression as the corresponding entries in $\lambda$
[26], though the order one coefficients are different:

$\lambda^{\prime}_{ij}={\cal O}(1)\times\lambda_{ij}.$ (91)

This structure allows us to estimate the entries of $\hat{\lambda}_{ij}$ in
terms of physical observables:

$\displaystyle\hat{\lambda}_{33}$ $\displaystyle\sim$ $\displaystyle
m_{\tau}/v,$ $\displaystyle\hat{\lambda}_{22}$ $\displaystyle\sim$
$\displaystyle m_{\mu}/v,$ $\displaystyle\hat{\lambda}_{23}$
$\displaystyle\sim$ $\displaystyle|U_{23}|(m_{\tau}/v),$
$\displaystyle\hat{\lambda}_{32}$ $\displaystyle\sim$
$\displaystyle(m_{\mu}/v)/|U_{23}|.$ (92)

We learn the following points about the Higgs-related lepton flavor parameters
in this class of models:

  1. 1.

$h$ has flavor off-diagonal couplings:

$\displaystyle Y_{\mu\tau}$ $\displaystyle=$ $\displaystyle{\cal
O}\left(\frac{|U_{23}|vm_{\tau}}{\Lambda^{2}}\right),$ $\displaystyle
Y_{\tau\mu}$ $\displaystyle=$ $\displaystyle{\cal
O}\left(\frac{vm_{\mu}}{|U_{23}|\Lambda^{2}}\right).$ (93)

  2. 2.

The values of the diagonal couplings deviate from their SM values:

$Y_{\tau}\approx\frac{\sqrt{2}m_{\tau}}{v}\ \left[1+{\cal
O}\left(\frac{v^{2}}{\Lambda^{2}}\right)\right].$ (94)

  3. 3.

The ratio between the Yukawa couplings to different charged lepton flavors
deviates from its SM value:

$\frac{Y_{\mu}}{Y_{\tau}}=\frac{m_{\mu}}{m_{\tau}}\left[1+{\cal
O}\left(\frac{v^{2}}{\Lambda^{2}}\right)\right].$ (95)

The predictions of the SM with FN-suppressed non-renormalizable terms are then
the following:

$\displaystyle\left(\frac{\sigma(pp\to h)^{\rm SM}}{\sigma(pp\to
h)}\frac{\Gamma_{\rm tot}}{\Gamma_{\rm tot}^{\rm
SM}}\right)R_{\tau^{+}\tau^{-}}$ $\displaystyle=$ $\displaystyle 1+{\cal
O}(v^{2}/\Lambda^{2}),$ $\displaystyle X_{\mu^{+}\mu^{-}}$ $\displaystyle=$
$\displaystyle(m_{\mu}/m_{\tau})^{2}(1+{\cal O}(v^{2}/\Lambda^{2})),$
$\displaystyle X_{\tau\mu}$ $\displaystyle=$ $\displaystyle{\cal
O}(v^{4}/\Lambda^{4}).$ (96)

Thus, FN will be excluded if experiments observe deviations from the SM of the
same size in both flavor-diagonal and flavor-changing $h$ decays. On the other
hand, FN allows non-universal deviations of ${\cal O}(v^{2}/\Lambda^{2})$ in
the flavor-diagonal dilepton rates, and a smaller deviation of ${\cal
O}(v^{4}/\Lambda^{4})$ in the off-diagonal rate.

##  0.8 Conclusions

(i) Measurements of CP violating $B$-meson decays have established that the
Kobayashi-Maskawa mechanism is the dominant source of the observed CP
violation.

(ii) Measurements of flavor changing $B$-meson decays have established the the
Cabibbo-Kobayashi-Maskawa mechanism is a major player in flavor violation.

(iii) The consistency of all these measurements with the CKM predictions
sharpens the new physics flavor puzzle: If there is new physics at, or below,
the TeV scale, then its flavor structure must be highly non-generic.

(iv) Measurements of neutrino flavor parameters have not only not clarified
the standard model flavor puzzle, but actually deepened it. Whether they imply
an anarchical structure, or a tribimaximal mixing, it seems that the neutrino
flavor structure is very different from that of quarks.

(v) If the LHC experiments, ATLAS and CMS, discover new particles that couple
to the Standard Model fermions, then, in principle, they will be able to
measure new flavor parameters. Consequently, the new physics flavor puzzle is
likely to be understood.

(vi) If the flavor structure of such new particles is affected by the same
physics that sets the flavor structure of the Yukawa couplings, then the LHC
experiments (and future flavor factories) may be able to shed light also on
the standard model flavor puzzle.

(vii) The recently discovered Higgs-like boson provides an opportunity to make
progress in our understanding of the flavor puzzle(s).

The huge progress in flavor physics in recent years has provided answers to
many questions. At the same time, new questions arise. The LHC era is likely
to provide more answers and more questions.

##  .9 The CKM matrix

The CKM matrix $V$ is a $3\times 3$ unitary matrix. Its form, however, is not
unique:

$(i)$ There is freedom in defining $V$ in that we can permute between the
various generations. This freedom is fixed by ordering the up quarks and the
down quarks by their masses, i.e. $(u_{1},u_{2},u_{3})\to(u,c,t)$ and
$(d_{1},d_{2},d_{3})\to(d,s,b)$. The elements of $V$ are written as follows:

$V=\pmatrix{V}_{ud}&V_{us}V_{ub}\\\ V_{cd}V_{cs}V_{cb}\\\
V_{td}V_{ts}V_{tb}\\\ .$ (97)

$(ii)$ There is further freedom in the phase structure of $V$. This means that
the number of physical parameters in $V$ is smaller than the number of
parameters in a general unitary $3\times 3$ matrix which is nine (three real
angles and six phases). Let us define $P_{q}$ ($q=u,d$) to be diagonal unitary
(phase) matrices. Then, if instead of using $V_{qL}$ and $V_{qR}$ for the
rotation (21) to the mass basis we use $\tilde{V}_{qL}$ and $\tilde{V}_{qR}$,
defined by $\tilde{V}_{qL}=P_{q}V_{qL}$ and $\tilde{V}_{qR}=P_{q}V_{qR}$, we
still maintain a legitimate mass basis since $M_{q}^{\rm diag}$ remains
unchanged by such transformations. However, $V$ does change:

$V\to P_{u}VP_{d}^{*}.$ (98)

This freedom is fixed by demanding that $V$ has the minimal number of phases.
In the three generation case $V$ has a single phase. (There are five phase
differences between the elements of $P_{u}$ and $P_{d}$ and, therefore, five
of the six phases in the CKM matrix can be removed.) This is the Kobayashi-
Maskawa phase $\delta_{\rm KM}$ which is the single source of CP violation in
the quark sector of the Standard Model [1].

The fact that $V$ is unitary and depends on only four independent physical
parameters can be made manifest by choosing a specific parametrization. The
standard choice is [70]

$V=\pmatrix{c}_{12}c_{13}&s_{12}c_{13}s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}c_{23}c_{13}\\\
,$ (99)

where $c_{ij}\equiv\cos\theta_{ij}$ and $s_{ij}\equiv\sin\theta_{ij}$. The
$\theta_{ij}$’s are the three real mixing parameters while $\delta$ is the
Kobayashi-Maskawa phase. It is known experimentally that $s_{13}\ll s_{23}\ll
s_{12}\ll 1$. It is convenient to choose an approximate expression where this
hierarchy is manifest. This is the Wolfenstein parametrization, where the four
mixing parameters are $(\lambda,A,\rho,\eta)$ with $\lambda=|V_{us}|=0.23$
playing the role of an expansion parameter and $\eta$ representing the CP
violating phase [71, 72]:

$V=\pmatrix{1}-\frac{1}{2}\lambda^{2}-\frac{1}{8}\lambda^{4}&\lambda
A\lambda^{3}(\rho-i\eta)\\\
-\lambda+\frac{1}{2}A^{2}\lambda^{5}[1-2(\rho+i\eta)]1-\frac{1}{2}\lambda^{2}-\frac{1}{8}\lambda^{4}(1+4A^{2})A\lambda^{2}\\\
A\lambda^{3}[1-(1-\frac{1}{2}\lambda^{2})(\rho+i\eta)]-A\lambda^{2}+\frac{1}{2}A\lambda^{4}[1-2(\rho+i\eta)]1-\frac{1}{2}A^{2}\lambda^{4}\\\
\;.$ (100)

A very useful concept is that of the unitarity triangles. The unitarity of the
CKM matrix leads to various relations among the matrix elements, e.g.

$\displaystyle V_{ud}V_{us}^{*}+V_{cd}V_{cs}^{*}+V_{td}V_{ts}^{*}=0,$ (101)
$\displaystyle V_{us}V_{ub}^{*}+V_{cs}V_{cb}^{*}+V_{ts}V_{tb}^{*}=0,$ (102)
$\displaystyle V_{ud}V_{ub}^{*}+V_{cd}V_{cb}^{*}+V_{td}V_{tb}^{*}=0.$ (103)

Each of these three relations requires the sum of three complex quantities to
vanish and so can be geometrically represented in the complex plane as a
triangle. These are “the unitarity triangles", though the term “unitarity
triangle" is usually reserved for the relation (103) only. The unitarity
triangle related to Eq. (103) is depicted in Fig. 5.

Figure 5: Graphical representation of the unitarity constraint
$V_{ud}V_{ub}^{*}+V_{cd}V_{cb}^{*}+V_{td}V_{tb}^{*}=0$ as a triangle in the
complex plane.

The rescaled unitarity triangle is derived from (103) by (a) choosing a phase
convention such that $(V_{cd}V_{cb}^{*})$ is real, and (b) dividing the
lengths of all sides by $|V_{cd}V_{cb}^{*}|$. Step (a) aligns one side of the
triangle with the real axis, and step (b) makes the length of this side 1. The
form of the triangle is unchanged. Two vertices of the rescaled unitarity
triangle are thus fixed at (0,0) and (1,0). The coordinates of the remaining
vertex correspond to the Wolfenstein parameters $(\rho,\eta)$. The area of the
rescaled unitarity triangle is $|\eta|/2$.

Depicting the rescaled unitarity triangle in the $(\rho,\eta)$ plane, the
lengths of the two complex sides are

$R_{u}\equiv\left|{V_{ud}V_{ub}\over
V_{cd}V_{cb}}\right|=\sqrt{\rho^{2}+\eta^{2}},\ \ \
R_{t}\equiv\left|{V_{td}V_{tb}\over
V_{cd}V_{cb}}\right|=\sqrt{(1-\rho)^{2}+\eta^{2}}.$ (104)

The three angles of the unitarity triangle are defined as follows [73, 74]:

$\alpha\equiv\arg\left[-{V_{td}V_{tb}^{*}\over V_{ud}V_{ub}^{*}}\right],\ \ \
\beta\equiv\arg\left[-{V_{cd}V_{cb}^{*}\over V_{td}V_{tb}^{*}}\right],\ \ \
\gamma\equiv\arg\left[-{V_{ud}V_{ub}^{*}\over V_{cd}V_{cb}^{*}}\right].$ (105)

They are physical quantities and can be independently measured by CP
asymmetries in $B$ decays. It is also useful to define the two small angles of
the unitarity triangles (102,101):

$\beta_{s}\equiv\arg\left[-{V_{ts}V_{tb}^{*}\over V_{cs}V_{cb}^{*}}\right],\ \
\ \beta_{K}\equiv\arg\left[-{V_{cs}V_{cd}^{*}\over V_{us}V_{ud}^{*}}\right].$
(106)

##  .10 CPV in $B$ decays to final CP eigenstates

We define decay amplitudes of $B$ (which could be charged or neutral) and its
CP conjugate $\overline{B}$ to a multi-particle final state $f$ and its CP
conjugate $\overline{f}$ as

$A_{f}=\langle f|{\cal H}|B\rangle\quad,\quad\overline{A}_{f}=\langle f|{\cal
H}|\overline{B}\rangle\quad,\quad A_{\overline{f}}=\langle\overline{f}|{\cal
H}|B\rangle\quad,\quad\overline{A}_{\overline{f}}=\langle\overline{f}|{\cal
H}|\overline{B}\rangle\;,$ (107)

where ${\cal H}$ is the Hamiltonian governing weak interactions. The action of
CP on these states introduces phases $\xi_{B}$ and $\xi_{f}$ according to

$\displaystyle CP\ |B\rangle$ $\displaystyle=$ $\displaystyle
e^{+i\xi_{B}}\,|\overline{B}\rangle\quad,\quad CP\
|f\rangle=e^{+i\xi_{f}}\,|\overline{f}\rangle\;,$ $\displaystyle CP\
|\overline{B}\rangle$ $\displaystyle=$ $\displaystyle
e^{-i\xi_{B}}\,|B\rangle\quad,\quad CP\
|\overline{f}\rangle=e^{-i\xi_{f}}\,|f\rangle\ ,$ (108)

so that $(CP\ )^{2}=1$. The phases $\xi_{B}$ and $\xi_{f}$ are arbitrary and
unphysical because of the flavor symmetry of the strong interaction. If CP is
conserved by the dynamics, $[CP\ ,{\cal H}]=0$, then $A_{f}$ and
$\overline{A}_{\overline{f}}$ have the same magnitude and an arbitrary
unphysical relative phase

$\overline{A}_{\overline{f}}=e^{i(\xi_{f}-\xi_{B})}\,A_{f}\;.$ (109)

A state that is initially a superposition of ${B^{0}}$ and
$\overline{B}{}^{0}$, say

$|\psi(0)\rangle=a(0)|{B^{0}}\rangle+b(0)|\overline{B}{}^{0}\rangle\;,$ (110)

will evolve in time acquiring components that describe all possible decay
final states $\\{f_{1},f_{2},\ldots\\}$, that is,

$|\psi(t)\rangle=a(t)|{B^{0}}\rangle+b(t)|\overline{B}{}^{0}\rangle+c_{1}(t)|f_{1}\rangle+c_{2}(t)|f_{2}\rangle+\cdots\;.$
(111)

If we are interested in computing only the values of $a(t)$ and $b(t)$ (and
not the values of all $c_{i}(t)$), and if the times $t$ in which we are
interested are much larger than the typical strong interaction scale, then we
can use a much simplified formalism [75]. The simplified time evolution is
determined by a $2\times 2$ effective Hamiltonian ${\cal H}$ that is not
Hermitian, since otherwise the mesons would only oscillate and not decay. Any
complex matrix, such as ${\cal H}$, can be written in terms of Hermitian
matrices $M$ and $\Gamma$ as

${\cal H}=M-\frac{i}{2}\,\Gamma\;.$ (112)

$M$ and $\Gamma$ are associated with
$({B^{0}},\overline{B}{}^{0})\leftrightarrow({B^{0}},\overline{B}{}^{0})$
transitions via off-shell (dispersive) and on-shell (absorptive) intermediate
states, respectively. Diagonal elements of $M$ and $\Gamma$ are associated
with the flavor-conserving transitions ${B^{0}}\to{B^{0}}$ and
$\overline{B}{}^{0}\to\overline{B}{}^{0}$ while off-diagonal elements are
associated with flavor-changing transitions
${B^{0}}\leftrightarrow\overline{B}{}^{0}$.

The eigenvectors of ${\cal H}$ have well defined masses and decay widths. We
introduce complex parameters $p$ and $q$ to specify the components of the
strong interaction eigenstates, ${B^{0}}$ and $\overline{B}{}^{0}$, in the
light ($B_{L}$) and heavy ($B_{H}$) mass eigenstates:

$|B_{L,H}\rangle=p|{B^{0}}\rangle\pm q|\overline{B}{}^{0}\rangle$ (113)

with the normalization $|p|^{2}+|q|^{2}=1$. The special form of Eq. (113) is
related to the fact that CPT imposes $M_{11}=M_{22}$ and
$\Gamma_{11}=\Gamma_{22}$. Solving the eigenvalue problem gives

$\left(\frac{q}{p}\right)^{2}=\frac{M_{12}^{\ast}-(i/2)\Gamma_{12}^{\ast}}{M_{12}-(i/2)\Gamma_{12}}\;.$
(114)

If either CP or T is a symmetry of ${\cal H}$, then $M_{12}$ and $\Gamma_{12}$
are relatively real, leading to

$\left(\frac{q}{p}\right)^{2}=e^{2i\xi_{B}}\quad\Rightarrow\quad\left|\frac{q}{p}\right|=1\;,$
(115)

where $\xi_{B}$ is the arbitrary unphysical phase introduced in Eq. (.10).

The real and imaginary parts of the eigenvalues of ${\cal H}$ corresponding to
$|B_{L,H}\rangle$ represent their masses and decay-widths, respectively. The
mass difference $\Delta m_{B}$ and the width difference $\Delta\Gamma_{B}$ are
defined as follows:

$\Delta m_{B}\equiv M_{H}-M_{L},\ \ \
\Delta\Gamma_{B}\equiv\Gamma_{H}-\Gamma_{L}.$ (116)

Note that here $\Delta m_{B}$ is positive by definition, while the sign of
$\Delta\Gamma_{B}$ is to be experimentally determined. The average mass and
width are given by

$m_{B}\equiv{M_{H}+M_{L}\over 2},\ \ \
\Gamma_{B}\equiv{\Gamma_{H}+\Gamma_{L}\over 2}.$ (117)

It is useful to define dimensionless ratios $x$ and $y$:

$x\equiv{\Delta m_{B}\over\Gamma_{B}},\ \ \ y\equiv{\Delta\Gamma_{B}\over
2\Gamma_{B}}.$ (118)

Solving the eigenvalue equation gives

$(\Delta m_{B})^{2}-{1\over
4}(\Delta\Gamma_{B})^{2}=(4|M_{12}|^{2}-|\Gamma_{12}|^{2}),\ \ \ \ \Delta
m_{B}\Delta\Gamma_{B}=4{\cal R}e(M_{12}\Gamma_{12}^{*}).$ (119)

All CP-violating observables in $B$ and $\overline{B}$ decays to final states
$f$ and $\overline{f}$ can be expressed in terms of phase-convention-
independent combinations of $A_{f}$, $\overline{A}_{f}$, $A_{\overline{f}}$
and $\overline{A}_{\overline{f}}$, together with, for neutral-meson decays
only, $q/p$. CP violation in charged-meson decays depends only on the
combination $|\overline{A}_{\overline{f}}/A_{f}|$, while CP violation in
neutral-meson decays is complicated by
${B^{0}}\leftrightarrow\overline{B}{}^{0}$ oscillations and depends,
additionally, on $|q/p|$ and on
$\lambda_{f}\equiv(q/p)(\overline{A}_{f}/A_{f})$.

For neutral $D$, $B$, and $B_{s}$ mesons, $\Delta\Gamma/\Gamma\ll 1$ and so
both mass eigenstates must be considered in their evolution. We denote the
state of an initially pure $|{B^{0}}\rangle$ or $|\overline{B}{}^{0}\rangle$
after an elapsed proper time $t$ as $|{B^{0}}_{\mathrm{phys}}(t)\rangle$ or
$|\overline{B}{}^{0}_{\mathrm{phys}}(t)\rangle$, respectively. Using the
effective Hamiltonian approximation, we obtain

$\displaystyle|{B^{0}}_{\rm phys}(t)\rangle$ $\displaystyle=$ $\displaystyle
g_{+}(t)\,|{B^{0}}\rangle-\frac{q}{p}\ g_{-}(t)|\overline{B}{}^{0}\rangle,$
$\displaystyle|\overline{B}{}^{0}_{\rm phys}(t)\rangle$ $\displaystyle=$
$\displaystyle g_{+}(t)\,|\overline{B}{}^{0}\rangle-\frac{p}{q}\
g_{-}(t)|{B^{0}}\rangle\;,$ (120)

where

$g_{\pm}(t)\equiv\frac{1}{2}\left(e^{-im_{H}t-\frac{1}{2}\Gamma_{H}t}\pm
e^{-im_{L}t-\frac{1}{2}\Gamma_{L}t}\right).$ (121)

One obtains the following time-dependent decay rates:

$\displaystyle\frac{d\Gamma[{B^{0}}_{\rm phys}(t)\to f]/dt}{e^{-\Gamma t}{\cal
N}_{f}}$ $\displaystyle=$
$\displaystyle\left(|A_{f}|^{2}+|(q/p)\overline{A}_{f}|^{2}\right)\cosh(y\Gamma
t)+\left(|A_{f}|^{2}-|(q/p)\overline{A}_{f}|^{2}\right)\cos(x\Gamma t)$ (122)
$\displaystyle+$ $\displaystyle 2\,{\cal
R}e((q/p)A_{f}^{\ast}\overline{A}_{f})\sinh(y\Gamma t)-2\,{\cal
I}m((q/p)A_{f}^{\ast}\overline{A}_{f})\sin(x\Gamma t)\;,$
$\displaystyle\frac{d\Gamma[\overline{B}{}^{0}_{\rm phys}(t)\to
f]/dt}{e^{-\Gamma t}{\cal N}_{f}}$ $\displaystyle=$
$\displaystyle\left(|(p/q)A_{f}|^{2}+|\overline{A}_{f}|^{2}\right)\cosh(y\Gamma
t)-\left(|(p/q)A_{f}|^{2}-|\overline{A}_{f}|^{2}\right)\cos(x\Gamma t)$ (123)
$\displaystyle+$ $\displaystyle 2\,{\cal
R}e((p/q)A_{f}\overline{A}^{\ast}_{f})\sinh(y\Gamma t)-2\,{\cal
I}m((p/q)A_{f}\overline{A}^{\ast}_{f})\sin(x\Gamma t)\;,$

where ${\cal N}_{f}$ is a common normalization factor. Decay rates to the CP-
conjugate final state $\overline{f}$ are obtained analogously, with ${\cal
N}_{f}={\cal N}_{\overline{f}}$ and the substitutions $A_{f}\to
A_{\overline{f}}$ and $\overline{A}_{f}\to\overline{A}_{\overline{f}}$ in Eqs.
(122,123). Terms proportional to $|A_{f}|^{2}$ or $|\overline{A}_{f}|^{2}$ are
associated with decays that occur without any net
$B\leftrightarrow\overline{B}$ oscillation, while terms proportional to
$|(q/p)\overline{A}_{f}|^{2}$ or $|(p/q)A_{f}|^{2}$ are associated with decays
following a net oscillation. The $\sinh(y\Gamma t)$ and $\sin(x\Gamma t)$
terms of Eqs. (122,123) are associated with the interference between these two
cases. Note that, in multi-body decays, amplitudes are functions of phase-
space variables. Interference may be present in some regions but not others,
and is strongly influenced by resonant substructure.

One possible manifestation of CP-violating effects in meson decays [76] is in
the interference between a decay without mixing, ${B^{0}}\to f$, and a decay
with mixing, ${B^{0}}\to\overline{B}{}^{0}\to f$ (such an effect occurs only
in decays to final states that are common to ${B^{0}}$ and
$\overline{B}{}^{0}$, including all CP eigenstates). It is defined by

${\cal I}m(\lambda_{f})\neq 0\;,$ (124)

with

$\lambda_{f}\equiv\frac{q}{p}\frac{\overline{A}_{f}}{A_{f}}\;.$ (125)

This form of CP violation can be observed, for example, using the asymmetry of
neutral meson decays into final CP eigenstates $f_{CP\ }$

${\cal A}_{f_{CP\ }}(t)\equiv\frac{d\Gamma/dt[\overline{B}{}^{0}_{\rm
phys}(t)\to f_{CP\ }]-d\Gamma/dt[{B^{0}}_{\rm phys}(t)\to f_{CP\
}]}{d\Gamma/dt[\overline{B}{}^{0}_{\rm phys}(t)\to f_{CP\
}]+d\Gamma/dt[{B^{0}}_{\rm phys}(t)\to f_{CP\ }]}\;.$ (126)

For $\Delta\Gamma=0$ and $|q/p|=1$ (which is a good approximation for $B$
mesons), ${\cal A}_{f_{CP\ }}$ has a particularly simple form [77, 78, 79]:

$\displaystyle{\cal A}_{f}(t)$ $\displaystyle=$ $\displaystyle
S_{f}\sin(\Delta mt)-C_{f}\cos(\Delta mt),$ $\displaystyle S_{f}$
$\displaystyle\equiv$ $\displaystyle\frac{2\,{\cal
I}m(\lambda_{f})}{1+|\lambda_{f}|^{2}},\ \ \
C_{f}\equiv\frac{1-|\lambda_{f}|^{2}}{1+|\lambda_{f}|^{2}}\;,$ (127)

Consider the $B\to f$ decay amplitude $A_{f}$, and the CP conjugate process,
$\overline{B}\to\overline{f}$, with decay amplitude
$\overline{A}_{\overline{f}}$. There are two types of phases that may appear
in these decay amplitudes. Complex parameters in any Lagrangian term that
contributes to the amplitude will appear in complex conjugate form in the CP-
conjugate amplitude. Thus their phases appear in $A_{f}$ and
$\overline{A}_{\overline{f}}$ with opposite signs. In the Standard Model,
these phases occur only in the couplings of the $W^{\pm}$ bosons and hence are
often called “weak phases”. The weak phase of any single term is convention
dependent. However, the difference between the weak phases in two different
terms in $A_{f}$ is convention independent. A second type of phase can appear
in scattering or decay amplitudes even when the Lagrangian is real. Their
origin is the possible contribution from intermediate on-shell states in the
decay process. Since these phases are generated by CP-invariant interactions,
they are the same in $A_{f}$ and $\overline{A}_{\overline{f}}$. Usually the
dominant rescattering is due to strong interactions and hence the designation
“strong phases” for the phase shifts so induced. Again, only the relative
strong phases between different terms in the amplitude are physically
meaningful.

The ‘weak’ and ‘strong’ phases discussed here appear in addition to the
‘spurious’ CP-transformation phases of Eq. (109). Those spurious phases are
due to an arbitrary choice of phase convention, and do not originate from any
dynamics or induce any CP violation. For simplicity, we set them to zero from
here on.

It is useful to write each contribution $a_{i}$ to $A_{f}$ in three parts: its
magnitude $|a_{i}|$, its weak phase $\phi_{i}$, and its strong phase
$\delta_{i}$. If, for example, there are two such contributions,
$A_{f}=a_{1}+a_{2}$, we have

$\displaystyle A_{f}$ $\displaystyle=$
$\displaystyle|a_{1}|e^{i(\delta_{1}+\phi_{1})}+|a_{2}|e^{i(\delta_{2}+\phi_{2})},$
$\displaystyle\overline{A}_{\overline{f}}$ $\displaystyle=$
$\displaystyle|a_{1}|e^{i(\delta_{1}-\phi_{1})}+|a_{2}|e^{i(\delta_{2}-\phi_{2})}.$
(128)

Similarly, for neutral meson decays, it is useful to write

$M_{12}=|M_{12}|e^{i\phi_{M}}\quad,\quad\Gamma_{12}=|\Gamma_{12}|e^{i\phi_{\Gamma}}\;.$
(129)

Each of the phases appearing in Eqs. (.10,129) is convention dependent, but
combinations such as $\delta_{1}-\delta_{2}$, $\phi_{1}-\phi_{2}$,
$\phi_{M}-\phi_{\Gamma}$ and $\phi_{M}+\phi_{1}-\overline{\phi}_{1}$ (where
$\overline{\phi}_{1}$ is a weak phase contributing to $\overline{A}_{f}$) are
physical.

In the approximations that only a single weak phase contributes to decay,
$A_{f}=|a_{f}|e^{i(\delta_{f}+\phi_{f})}$, and that $|\Gamma_{12}/M_{12}|=0$,
we obtain $|\lambda_{f}|=1$ and the CP asymmetries in decays to a final CP
eigenstate $f$ [Eq. (126)] with eigenvalue $\eta_{f}=\pm 1$ are given by

${\cal A}_{f_{CP\ }}(t)={\cal I}m(\lambda_{f})\;\sin(\Delta mt)\;\
\mathrm{with}\ \ {\cal I}m(\lambda_{f})=\eta_{f}\sin(\phi_{M}+2\phi_{f}).$
(130)

Note that the phase so measured is purely a weak phase, and no hadronic
parameters are involved in the extraction of its value from ${\cal
I}m(\lambda_{f})$.

##  .11 Supersymmetric flavor violation

###  .11.1 Mass insertions

Supersymmetric models provide, in general, new sources of flavor violation. We
here present the formalism of mass insertions. We do that for the charged
sleptons, but the formalism is straightforwardly adapted for squarks.

The supersymmetric lepton flavor violation is most commonly analyzed in the
basis in which the charged lepton mass matrix and the gaugino vertices are
diagonal. In this basis, the slepton masses are not necessarily flavor-
diagonal, and have the form

$\widetilde{\ell}_{Mi}^{*}(M^{2}_{\widetilde{\ell}})^{MN}_{ij}\widetilde{\ell}_{Nj}=(\widetilde{\ell}_{Li}^{*}\
\widetilde{\ell}_{Rk}^{*})\left(\begin{array}[]{cc}M^{2}_{Lij}&A_{il}v_{d}\\\
A_{jk}v_{d}&M^{2}_{Rkl}\end{array}\right)\left(\begin{array}[]{c}\widetilde{\ell}_{Lj}\\\
\widetilde{\ell}_{Rl}\end{array}\right),$ (131)

where $M,N=L,R$ label chirality, and $i,j,k,l=1,2,3$ are generational indices.
$M^{2}_{L}$ and $M^{2}_{R}$ are the supersymmetry breaking slepton masses-
squared. The $A$ parameters enter in the trilinear scalar couplings
$A_{ij}\phi_{d}\widetilde{\ell}_{Li}\widetilde{\ell}_{Rj}^{*}$, where
$\phi_{d}$ is the down-type Higgs boson, and $v_{d}=\langle\phi_{d}\rangle$.
We neglect small flavor-conserving terms involving $\tan\beta=v_{u}/v_{d}$.

In this basis, charged LFV takes place through one or more slepton mass
insertion. Each mass insertion brings with it a factor of

$\delta^{MN}_{ij}\equiv(M^{2}_{\widetilde{\ell}})^{MN}_{ij}/\tilde{m}^{2},$
(132)

where $\tilde{m}^{2}$ is the representative slepton mass scale. Physical
processes therefore constrain

$(\delta^{MN}_{ij})_{\rm eff}\sim{\rm
max}\left[\delta^{MN}_{ij},\delta^{MP}_{ik}\delta^{PN}_{kj},\ldots,(i\leftrightarrow
j)\right].$ (133)

For example,

$(\delta^{LR}_{12})_{\rm eff}\sim{\rm
max}\left[A_{12}v_{d}/\tilde{m}^{2},M^{2}_{L1k}A_{k2}v_{d}/\tilde{m}^{4},A_{1k}v_{d}M^{2}_{Rk2}/\tilde{m}^{4},\ldots,(1\leftrightarrow
2)\right].$ (134)

Note that contributions with two or more insertions may be less suppressed
than those with only one.

It is useful to express the $\delta^{MN}_{ij}$ mass insertions in terms of
parameters in the mass basis. We can write, for example,

$\delta^{LL}_{ij}=\frac{1}{\tilde{m}^{2}}\sum_{\alpha}K^{L}_{i\alpha}K^{L*}_{j\alpha}\Delta\tilde{m}^{2}_{L\alpha}.$
(135)

Here, we ignore $L-R$ mixing, so that $K^{L}_{i\alpha}$ is the mixing angle in
the coupling of a neutralino to $\ell_{Li}-\widetilde{\ell}_{L\alpha}$ (with
$\ell_{i}=e,\mu,\tau$ denoting charged lepton mass eigenstates and
$\widetilde{\ell}_{\alpha}=\widetilde{\ell}_{1},\widetilde{\ell}_{2},\widetilde{\ell}_{3}$
denoting charged slepton mass eigenstates), and
$\Delta\tilde{m}^{2}_{L\alpha}=m^{2}_{\widetilde{\ell}_{L\alpha}}-\tilde{m}^{2}$.
Using the unitarity of the mixing matrix $K^{L}$, we can write

$\tilde{m}^{2}\delta^{LL}_{ij}=\sum_{\alpha}K^{L}_{i\alpha}K^{L*}_{j\alpha}(\Delta\tilde{m}^{2}_{L\alpha}+\tilde{m}^{2})=(M^{2}_{\widetilde{\ell}})^{LL}_{ij},$
(136)

thus reproducing the definition (132).

In many cases, a two generation effective framework is useful. To understand
that, consider a case where (no summation over $i,j,k$)

$\displaystyle|K^{L}_{ik}K^{L*}_{jk}|$ $\displaystyle\ll$
$\displaystyle|K^{L}_{ij}K^{L*}_{j}|,$
$\displaystyle|K^{L}_{ik}K^{L*}_{jk}\Delta
m^{2}_{\widetilde{\ell}_{Lk}\widetilde{\ell}_{Li}}|$ $\displaystyle\ll$
$\displaystyle|K^{L}_{ij}K^{L*}_{j}\Delta
m^{2}_{\widetilde{\ell}_{Lj}\widetilde{\ell}_{Li}}|,$ (137)

where $\Delta
m^{2}_{\widetilde{\ell}_{j}\widetilde{\ell}_{i}}=m^{2}_{\widetilde{\ell}_{Lj}}-m^{2}_{\widetilde{\ell}_{Li}}$.
Then, the contribution of the intermediate $\widetilde{\ell}_{k}$ can be
neglected and, furthermore, to a good approximation
$K^{L}_{ii}K^{L*}_{ji}+K^{L}_{ij}K^{L*}_{jj}=0$. For these cases, we obtain

$\delta^{LL}_{ij}=\frac{\Delta
m^{2}_{\widetilde{\ell}_{Lj}\widetilde{\ell}_{Li}}}{\tilde{m}^{2}}K^{L}_{ij}K^{L*}_{jj}.$
(138)

###  .11.2 Neutral meson mixing

We consider the squark-gluino box diagram contribution to
$D^{0}-\overline{D}^{0}$ mixing amplitude that is proportional to
$K_{2i}^{u}K^{u*}_{1i}K_{2j}^{u}K^{u*}_{1j}$, where $K^{u}$ is the mixing
matrix of the gluino couplings to left-handed up quarks and their up squark
partners. (In the language of the mass insertion approximation, we calculate
here the contribution that is $\propto[(\delta^{u}_{LL})_{12}]^{2}$.) We work
in the mass basis for both quarks and squarks.

The contribution is given by

$M_{12}^{D}=-i\frac{4\pi^{2}}{27}\alpha_{s}^{2}m_{D}f_{D}^{2}B_{D}\eta_{\rm
QCD}\sum_{i,j}(K_{2i}^{u}K_{1i}^{u*}K_{2j}^{u}K_{1j}^{u*})(11\tilde{I}_{4ij}+4\tilde{m}_{g}^{2}I_{4ij}).$
(139)

where

$\displaystyle\tilde{I}_{4ij}$ $\displaystyle\equiv$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}\frac{p^{2}}{(p^{2}-\tilde{m}_{g}^{2})^{2}(p^{2}-\tilde{m}_{i}^{2})(p^{2}-\tilde{m}_{j}^{2})}$
(140) $\displaystyle=$
$\displaystyle\frac{i}{(4\pi)^{2}}\left[\frac{\tilde{m}_{g}^{2}}{(\tilde{m}_{i}^{2}-\tilde{m}_{g}^{2})(\tilde{m}_{j}^{2}-\tilde{m}_{g}^{2})}\right.$
$\displaystyle+\left.\frac{\tilde{m}_{i}^{4}}{(\tilde{m}_{i}^{2}-\tilde{m}_{j}^{2})(\tilde{m}_{i}^{2}-\tilde{m}_{g}^{2})^{2}}\ln\frac{\tilde{m}_{i}^{2}}{\tilde{m}_{g}^{2}}+\frac{\tilde{m}_{j}^{4}}{(\tilde{m}_{j}^{2}-\tilde{m}_{i}^{2})(\tilde{m}_{j}^{2}-\tilde{m}_{g}^{2})^{2}}\ln\frac{\tilde{m}_{j}^{2}}{\tilde{m}_{g}^{2}}\right],$
$\displaystyle I_{4ij}$ $\displaystyle\equiv$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{(p^{2}-\tilde{m}_{g}^{2})^{2}(p^{2}-\tilde{m}_{i}^{2})(p^{2}-\tilde{m}_{j}^{2})}$
(141) $\displaystyle=$
$\displaystyle\frac{i}{(4\pi)^{2}}\left[\frac{1}{(\tilde{m}_{i}^{2}-\tilde{m}_{g}^{2})(\tilde{m}_{j}^{2}-\tilde{m}_{g}^{2})}\right.$
$\displaystyle+\left.\frac{\tilde{m}_{i}^{2}}{(\tilde{m}_{i}^{2}-\tilde{m}_{j}^{2})(\tilde{m}_{i}^{2}-\tilde{m}_{g}^{2})^{2}}\ln\frac{\tilde{m}_{i}^{2}}{\tilde{m}_{g}^{2}}+\frac{\tilde{m}_{j}^{2}}{(\tilde{m}_{j}^{2}-\tilde{m}_{i}^{2})(\tilde{m}_{j}^{2}-\tilde{m}_{g}^{2})^{2}}\ln\frac{\tilde{m}_{j}^{2}}{\tilde{m}_{g}^{2}}\right].$

We now follow the discussion in refs. [21, 80]. To see the consequences of the
super-GIM mechanism, let us expand the expression for the box integral around
some value $\tilde{m}^{2}_{q}$ for the squark masses-squared:

$\displaystyle I_{4}(\tilde{m}_{g}^{2},\tilde{m}_{i}^{2},\tilde{m}_{j}^{2})$
$\displaystyle=$ $\displaystyle
I_{4}(\tilde{m}_{g}^{2},\tilde{m}_{q}^{2}+\delta\tilde{m}_{i}^{2},\tilde{m}_{q}^{2}+\delta\tilde{m}_{j}^{2})$
(142) $\displaystyle=$ $\displaystyle
I_{4}(\tilde{m}_{g}^{2},\tilde{m}_{q}^{2},\tilde{m}_{q}^{2})+(\delta\tilde{m}_{i}^{2}+\delta\tilde{m}_{j}^{2})I_{5}(\tilde{m}_{g}^{2},\tilde{m}_{q}^{2},\tilde{m}_{q}^{2},\tilde{m}_{q}^{2})$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left[(\delta\tilde{m}_{i}^{2})^{2}+(\delta\tilde{m}_{j}^{2})^{2}+2(\delta\tilde{m}_{i}^{2})(\delta\tilde{m}_{j}^{2})\right]I_{6}(\tilde{m}_{g}^{2},\tilde{m}_{q}^{2},\tilde{m}_{q}^{2},\tilde{m}_{q}^{2},\tilde{m}_{q}^{2})+\cdots,$

where

$I_{n}(\tilde{m}_{g}^{2},\tilde{m}_{q}^{2},\ldots,\tilde{m}_{q}^{2})\equiv\int\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{(p^{2}-\tilde{m}_{g}^{2})^{2}(p^{2}-\tilde{m}_{q}^{2})^{n-2}},$
(143)

and similarly for $\tilde{I}_{4ij}$. Note that
$I_{n}\propto(\tilde{m}_{q}^{2})^{n-2}$ and
$\tilde{I}_{n}\propto(\tilde{m}_{q}^{2})^{n-3}$. Thus, using
$x\equiv\tilde{m}_{g}^{2}/\tilde{m}_{q}^{2}$, it is customary to define

$I_{n}\equiv\frac{i}{(4\pi)^{2}(\tilde{m}_{q}^{2})^{n-2}}f_{n}(x),\ \ \ \
\tilde{I}_{n}\equiv\frac{i}{(4\pi)^{2}(\tilde{m}_{q}^{2})^{n-3}}\tilde{f}_{n}(x).$
(144)

The unitarity of the mixing matrix implies that

$\sum_{i}(K_{2i}^{u}K_{1i}^{u*}K_{2j}^{u}K_{1j}^{u*})=\sum_{j}(K_{2i}^{u}K_{1i}^{u*}K_{2j}^{u}K_{1j}^{u*})=0.$
(145)

Consequently, the terms that are proportional $f_{4},\tilde{f}_{4},f_{5}$ and
$\tilde{f}_{5}$ vanish in their contribution to $M_{12}$. When
$\delta\tilde{m}_{i}^{2}\ll\tilde{m}_{q}^{2}$ for all $i$, the leading
contributions to $M_{12}$ come from $f_{6}$ and $\tilde{f}_{6}$. We learn that
for quasi-degenerate squarks, the leading contribution is quadratic in the
small mass-squared difference. The functions $f_{6}(x)$ and $\tilde{f}_{6}(x)$
are given by

$\displaystyle f_{6}(x)$ $\displaystyle=$ $\displaystyle\frac{6(1+3x)\ln
x+x^{3}-9x^{2}-9x+17}{6(1-x)^{5}},$ $\displaystyle\tilde{f}_{6}(x)$
$\displaystyle=$ $\displaystyle\frac{6x(1+x)\ln
x-x^{3}-9x^{2}+9x+1}{3(1-x)^{5}}.$ (146)

For example, with $x=1$, $f_{6}(1)=-1/20$ and $\tilde{f}_{6}=+1/30$; with
$x=2.33$, $f_{6}(2.33)=-0.015$ and $\tilde{f}_{6}=+0.013$.

To further simplify things, let us consider a two generation case. Then

$\displaystyle M_{12}^{D}$ $\displaystyle\propto$ $\displaystyle
2(K_{21}^{u}K_{11}^{u*})^{2}(\delta\tilde{m}_{1}^{2})^{2}+2(K_{22}^{u}K_{12}^{u*})^{2}(\delta\tilde{m}_{2}^{2})^{2}+(K_{21}^{u}K_{11}^{u*}K_{22}^{u}K_{12}^{u*})(\delta\tilde{m}_{1}^{2}+\delta\tilde{m}_{2}^{2})^{2}$
(147) $\displaystyle=$
$\displaystyle(K^{u}_{21}K_{11}^{u*})^{2}(\tilde{m}_{2}^{2}-\tilde{m}_{1}^{2})^{2}.$

We thus rewrite Eq. (139) for the case of quasi-degenerate squarks:

$M_{12}^{D}=\frac{\alpha_{s}^{2}m_{D}f_{D}^{2}B_{D}\eta_{\rm
QCD}}{108\tilde{m}_{q}^{2}}[11\tilde{f}_{6}(x)+4xf_{6}(x)]\frac{(\Delta\tilde{m}^{2}_{21})^{2}}{\tilde{m}_{q}^{4}}(K_{21}^{u}K_{11}^{u*})^{2}.$
(148)

For example, for $x=1$, $11\tilde{f}_{6}(x)+4xf_{6}(x)=+0.17$. For $x=2.33$,
$11\tilde{f}_{6}(x)+4xf_{6}(x)=+0.003$.

## Acknowledgements

I thank my students – Yonit Hochberg, Daniel Grossman, Aielet Efrati and
Avital Dery – for many useful discussions. The research of Y.N. is supported
by the I-CORE Program of the Planning and Budgeting Committee and the Israel
Science Foundation (grant No 1937/12), by the Israel Science Foundation (grant
No 579/11), and by the German–Israeli Foundation (GIF) (Grant No
G-1047-92.7/2009).

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