11institutetext: J. Stefan Institute, Ljubljana, Slovenia
Department of Physics, University of Ljubljana, Ljubljana, Slovenia

# Flavour Physics and CP Violation

J. F. Kamenik

###### Abstract

These notes represent a summary of three lectures on flavour and CP violation,
given at the CERN’s European School of High Energy Physics in 2014. They cover
flavour physics within the standard model, phenomenology of CP violation in
meson mixing and decays, as well as constraints of flavour observables on
physics beyond the standard model. In preparing the lectures (and consequently
this summary) I drew heavily from several existing excellent and exhaustive
sets of lecture notes and reviews on flavour physics and CP violation [1]. The
reader is encouraged to consult those as well as the original literature for a
more detailed study.

##  1 What is flavour?

In the standard model (SM) the basic constituents of matter are excitations of
fermionic fields with spin $1/2$. In this context matter flavours refers to
several copies of the same gauge representation. Under the unbroken SM gauge
group $SU(3)_{c}\times U(1)_{\rm EM}$ these are

  * •

up-type quarks: $(3)_{2/3}$ : $u,~{}c,~{}t$,

  * •

down-type quarks: $(3)_{-1/3}$ : $d,~{}s,~{}b$,

  * •

chrged leptons: $(1)_{-1}$ : $e,~{}\mu,~{}\tau$,

  * •

neutrinos: $(1)_{0}$ : $\nu_{1},~{}\nu_{2},~{}\nu_{3}$,

where the colour representations are given in the brackets, while the electric
charges are written as subscripts. The different flavours of the same gauge
representation differ only in their masses.

Ordinary matter is essentially made up of the first generation: $u$ and $d$
quarks are bound within protons and neutrons, while the electrons form atoms;
finally “electron neutrinos", which are an admixture of $\nu_{1,2,3}$, are
produced in reactions inside stars. Second and third generation families are
produced only in high-energy particle collisions. They all decay via weak
interactions into first generation particles. One of the big open questions in
fundamental physics is why there are thee almost identical replicas of quarks
and leptons and which is the origin of their different masses?

Flavour physics refers to interactions that distinguish between flavours.
Within the SM these are weak and Yukawa (Higgs boson) interactions.

Flavour parameters are those that carry flavour indices. Within the SM these
are the nine masses of charged fermions and four mixing parameters (three
angles and one complex CP violating phase).111Adding Majorana mass terms for
neutrinos introduces three additional neutrino masses plus six mixing
parameters (three mixing angles and three phases).

Flavour universal interactions are those with couplings proportional to the
identity in flavour space. Within the SM these are strong and electromagnetic
interactions (and also weak interactions in the so-called interactions basis,
see below). Such interactions are sometimes also called flavour blind.

Flavour diagonal interactions are those whose couplings are diagonal (in the
matter mass basis), but not necessarily universal. Within the SM these are the
Yukawa interactions of the Higgs boson.

Flavour changing processes are those where the initial and final flavour-
numbers are different (a flavour number is the number of particles with a
certain flavour minus the number of anti-particles of the same flavour). We
can further specify flavour changing charged currents which involve both up-
and down-type quark flavours or both charged lepton and neutrino flavours.
Examples of such processes are the muon decay $\mu^{-}\to
e^{-}\nu_{i}\bar{\nu}_{j}$ or the muonic charged kaon decay
$K^{-}\to\mu^{-}\bar{\nu}_{i}$ (which corresponds to the quark-level
transition $s\bar{u}\to\mu^{-}\bar{\nu}_{i}$). Within the SM such processes
are mediated already by a single $W$ exchange at the tree level (the
amplitudes being proportional to the Fermi constant $G_{F}$). On the other
hand, flavour changing neutral currents (FCNCs) involve either up- or down-
type flavours but not both; and/or either charged lepton flavours or neutrino
flavours but not both. Examples of such processes are the radiative muon decay
$\mu^{-}\to e^{-}\gamma$ and the muonic decays of the neutral kaons,
$K_{L}\to\mu^{+}\mu^{-}$ ($s\bar{d}\to\mu^{+}\mu^{-}$ at the quark level).
Within the SM these processes occur at higher orders in the weak expansion
(i.e. via loops) and are often highly suppressed. In connection with flavour
changing interactions, one often speaks also of flavour violation.

###  1.1 Why is flavour interesting?

Flavour physics can discover new physics (NP) or probe it before it is
directly observed in high-energy experiments. Historical examples of this
include:

  * •

The smallness of the ratio
$\Gamma(K_{L}\to\mu^{+}\mu^{-})/\Gamma(K^{-}\to\mu^{-}\bar{\nu}_{i})$ lead to
the prediction of the charmed quark.

  * •

Furthermore, the measurement of the mass difference between the two neutral
kaons $\Delta m_{K}\equiv m_{K_{L}}-m_{K_{S}}$ lead to the prediction of the
charm quark mass.

  * •

Similarly, the mass difference between the two neutral $B$ mesons $\Delta
m_{B}\equiv m_{B^{0}_{H}}-m_{B^{0}_{L}}$ inferred a prediction of the top
quark mass almost two decades before top quarks (or more precisely, their
decay products) were directly observed in experiments.

  * •

Finally, the observation of the CP violating decay $K_{L}\to\pi^{+}\pi^{-}$
(i.e the measurement of $\epsilon_{K}$) lead to the prediction of the third
generation of matter.

CP violation: Within the SM there is a single CP violating parameter
determining the amount of CP violation in all flavour changing processes.
Successful baryogenesis would require new CP violating sources.

Solutions of the electroweak (EW) hierarchy problem (in the form of a
quadratic sensitivity of the EW scale to UV physics) require NP to appear at
or below the TeV scale. On the other hand, such NP with a generic flavour
structure would predict FCNCs orders of magnitude above the observed rates.
Conversely, flavour physics can probe NP scales up to
$\mathcal{O}(10^{5}~{}{\rm TeV})$. The resulting NP flavour puzzle refers to
the fact that NP at the TeV scale needs to exhibit approximate flavour
symmetries.

The SM flavour parameters are both hierarchical (i.e. $m_{u}\ll m_{c}\ll
m_{t}$) and mostly very small ($m_{f\neq t}\ll m_{W,Z,h}$) . The question
whether this points to some unknown underlying flavour dynamics is sometimes
called the SM flavour puzzle.

##  2 Flavour in the standard model

Any (local) quantum field theory model is specified by both (i) symmetries and
the pattern of their spontaneous breaking; as well as (ii) representations of
fermions and scalars. The SM Lagrangian ($\mathcal{L}_{\rm SM}$) is thus
completely determined by specifying the local (gauge) symmetry
$\mathcal{G}_{\rm local}^{\rm SM}=SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$
which is spontaneously broken to $\mathcal{G}_{\rm local}^{\rm SM}\to
SU(3)_{c}\times U(1)_{\rm EM}$ ; plus the relevant fermionic

$\displaystyle Q_{L}^{i}\sim(3,2)_{1/6}\,,$ $\displaystyle
U_{R}^{i}\sim(3,1)_{2/3}\,,$ $\displaystyle D_{R}^{i}\sim(3,1)_{-1/3}\,,$
$\displaystyle L_{L}^{i}\sim(1,2)_{-1/2}\,,$ (1)

(where $i=1,2,3$) and scalar

$\displaystyle\phi$ $\displaystyle\sim(1,2)_{1/2}\,,$
$\displaystyle\langle\phi^{0}\rangle$
$\displaystyle\equiv\frac{v}{\sqrt{2}}\simeq 174\rm GeV\,,$ (2)

representations. Above, the first (second) entries in the brackets denote the
$SU(3)_{c}$ ($SU(2)_{L}$) representations, while the $U(1)_{Y}$ charges are
given in the subscripts. Also, $\langle\ldots\rangle$ denotes a vacuum
condensate value. $\mathcal{L}_{\rm SM}$ can be conveniently split into three
parts

$\displaystyle\mathcal{L}_{\rm SM}=\mathcal{L}_{\rm kinetic}^{\rm
SM}+\mathcal{L}_{\rm EWSB}^{\rm SM}+\mathcal{L}_{\rm Yukawa}^{\rm SM}\,.$ (3)

The sum of the gauge-kinetic terms $\mathcal{L}_{\rm kinetic}^{\rm SM}$ is
simple and symmetric. It is completely specified by the SM local symmetry and
its matter representations. The three physical parameters associated with this
part of the theory are conventionally chosen to be the three gauge couplings
($g_{s}$, $g$ and $g^{\prime}$) . The EW symmetry breaking (EWSB) part
$\mathcal{L}_{\rm EWSB}^{\rm SM}$ contains two additional parameters. They can
be chosen to correspond to $v$ and the physical Higgs boson mass $m_{h}$.
Finally, all flavour dynamics is contained in $\mathcal{L}_{\rm Yukawa}^{\rm
SM}$ which also involves all the SM flavour parameters.

###  2.1 Interaction basis

It is convenient to start our discussion in a flavour basis where all the
gauge-kinetic terms are diagonal. This can always be achieved by applying
suitable unitary rotations on the matter fields. In this basis

$\displaystyle\mathcal{L}_{\rm kinetic}^{\rm SM}$
$\displaystyle=(D_{\mu}\phi)^{\dagger}(D^{\mu}\phi)+\sum_{i,j=1,2,3}\sum_{\psi=Q_{L},\ldots,E_{R}}\bar{\psi}^{i}i\not{D}\delta^{ij}\psi^{j}$
$\displaystyle-\frac{1}{4}\sum_{a=1,\ldots,8}G_{\mu\nu}^{a}G^{a,\mu\nu}-\frac{1}{4}\sum_{a=1,2,3}W_{\mu\nu}^{a}W^{a,\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}\,,$
(4)

where $G,~{}W,$ and $B$ denote the field strengths of the $SU(3)_{c}$,
$SU(2)_{L}$ and $U(1)_{Y}$ gauge interactions, respectively. The covariant
derivatives $D_{\mu}$ are defined as
$D_{\mu}=\partial_{\mu}+ig_{s}G_{\mu}^{a}L^{a}+igW_{\mu}^{b}T^{b}+ig^{\prime}B_{\mu}Y$
, where $L^{a}$, $T^{a}$ and $Y$ denote the $SU(3)_{c}$, $SU(2)_{L}$
generators and the $U(1)_{Y}$ charges, respectively. Note that in this basis,
$\mathcal{L}_{\rm kinetic}^{\rm SM}$ is manifestly flavour universal and CP
conserving. Similarly

$\displaystyle\mathcal{L}^{\rm SM}_{\rm
EWSB}=\mu^{2}\phi^{\dagger}\phi-\lambda(\phi^{\dagger}\phi)^{2}\,,$ (5)

is also CP and flavour conserving.222It is also symmetric under $SO(4)$
rotations of the four real scalar fields $\phi_{1,2,3,4}$ contained in
$\phi=(\phi_{1}+i\phi_{2},\phi_{3}+i\phi_{4})^{T}$. This approximate symmetry
of the SM is sometimes called the custodial symmetry. Thus both
$\mathcal{L}_{\rm kinetic}^{\rm SM}$ and trivially $\mathcal{L}_{\rm
EWSB}^{\rm SM}$ have a large flavour symmetry corresponding to the independent
unitary rotations in the flavour space of the five fermionic fields

$\displaystyle\mathcal{G}_{\rm flavour}^{\rm SM}$
$\displaystyle=U(3)^{5}=SU(3)_{q}^{3}\times SU(3)_{\ell}^{2}\times
U(1)^{5}\,,$ $\displaystyle SU(3)_{q}^{3}$ $\displaystyle=SU(3)_{Q}\times
SU(3)_{U}\times SU(3)_{D}\,,$ $\displaystyle SU(3)_{\ell}^{2}$
$\displaystyle=SU(3)_{L}\times SU(3)_{E}\,,$ $\displaystyle U(1)^{5}$
$\displaystyle=U(1)_{B}\times U(1)_{L}\times U(1)_{Y}\times U(1)_{\rm
PQ}\times U(1)_{E}\,.$ (6)

Among the $U(1)$ factors, $U(1)_{B,L}$ are the baryon and lepton number,
respectively. $U(1)_{Y}$ is gauged and broken spontanouesly by
$\langle\phi^{0}\rangle$ . On the other hand $U(1)_{\rm PQ}$ can be defined
such that only the Higgs and $D^{i}_{R},E^{i}_{R}$ are charged under it and
with opposite charges. It is thus broken only by the up-quark Yukawas. Finally
$U(1)_{E}$ refers to flavour universal phase rotations of $E^{i}_{R}$ alone
and is thus broken by the charged lepton Yukawas.

The Yukawa Lagrangian of the SM

$\displaystyle-\mathcal{L}_{\rm Yukawa}^{\rm SM}=Y_{d}^{ij}\bar{Q}_{L}^{i}\phi
D_{R}^{j}+Y_{u}^{ij}\bar{Q}_{L}^{i}\tilde{\phi}U_{R}^{j}+Y_{e}^{ij}\bar{L}^{i}\phi
E_{R}^{j}+\rm h.c.\,,$ (7)

where $\tilde{\phi}=i\sigma_{2}\phi$, is in general flavour dependent (if
$Y_{f}\not{\propto}\,\mathrm{I}$) and CP violating. The pattern of explicit
$\mathcal{G}_{\rm flavour}^{\rm SM}$ breaking by $Y_{f}\neq 0$ is as follows:

  * •

$U(1)_{E}$ is broken by $Y_{e}\neq 0$ ,

  * •

$U(1)_{\rm PQ}$ is broken by $Y_{u}\cdot Y_{d}\neq 0$ and $Y_{u}\cdot
Y_{e}\neq 0$ ,

  * •

$SU(3)_{Q}\times SU(3)_{U}\to U(1)_{u}\times U(1)_{c}\times U(1)_{t}$ is due
to $Y_{u}\not{\propto}\,\rm I$ ,

  * •

$SU(3)_{Q}\times SU(3)_{D}\to U(1)_{d}\times U(1)_{s}\times U(1)_{b}$ is due
to $Y_{d}\not{\propto}\,\rm I$ ,

  * •

the remaining $U(1)$ factors in the quark sector are broken by the fact that
$[Y_{u},Y_{d}]\neq 0$ down to $U(1)_{B}$ ,

  * •

finally, $SU(3)_{L}\times SU(3)_{E}\to U(1)_{e}\times U(1)_{\mu}\times
U(1)_{\tau}$ due to $Y_{e}\not{\propto}\,\rm I$ . The remaining factor group
also contains the global $U(1)_{L}$ .

Thus, the global symmetry of the SM in presence of the Yukawas is
$\mathcal{G}_{\rm global}^{\rm SM}(Y_{f}\neq 0)=U(1)_{B}\times U(1)_{e}\times
U(1)_{\mu}\times U(1)_{\tau}$ . In this language, flavour physics refers to
interactions which break the $SU(3)_{q}^{3}\times SU(3)_{\ell}^{2}$ and are
thus flavour violating.

Commonly, a spurion analysis is useful for parameter counting, identification
of suppression factors, and for the idea of minimal flavour violation (MFV)
[2]. In this approach we promote the SM Yukawas to non-dynamical fields with
well-defined transformation properties under $\mathcal{G}_{\rm flavour}^{\rm
SM}$

$\displaystyle Y_{u}$ $\displaystyle\sim(3,\bar{3},1)_{SU(3)_{q}^{3}}\,,$
$\displaystyle Y_{d}$ $\displaystyle\sim(3,1,\bar{3})_{SU(3)_{q}^{3}}\,,$
$\displaystyle Y_{e}$ $\displaystyle\sim(3,\bar{3})_{SU(3)_{\ell}^{2}}\,.$ (8)

In the following we will focus on the quark sector.

###  2.2 Counting the standard model quark flavour parameters

The flavour symmetry breaking pattern described above is useful in counting
the number of physical flavour parameters in the theory. In particular:

  1. 1.

Consider a theory with a global symmetry group $\mathcal{G}_{f}$ with $N_{\rm
total}$ generators.

  2. 2.

Add interactions with $N_{\rm general}$ parameters, breaking
$\mathcal{G}_{f}\to\mathcal{H}_{f}$ with $N_{\rm total}-N_{\rm broken}$
generators.

  3. 3.

Then the $N_{\rm broken}$ generators can be used to rotate away $N_{\rm
broken}$ number of symmetry breaking parameters.

  4. 4.

The number of remaining physical parameters is thus $N_{\rm physical}=N_{\rm
general}-N_{\rm broken}$ .

We can apply this recipe to the SM breaking of $U(3)_{Q}\times U(3)_{U}\times
U(3)_{D}\to U(1)_{B}$. In this case the three $U(3)$ group rotations are
described by unitary $3\times 3$ matrices containing three real angles and six
phases each. Thus schematically $N_{\rm total}=3\times(3+6i)$ . Consequently
$N_{\rm broken}=N_{\rm total}-1i=9+17i$ . The two quark Yukawas are general
$3\times 3$ matrices containing nine complex parameters ($N_{\rm
general}=2\times(9+9i)$). Finally, the number of physical parameters is
$N_{\rm physical}=N_{\rm general}-N_{\rm broken}=9+1i$, representing six quark
masses, three mixing angles and a single CP violating phase.

###  2.3 Discrete symmetries of the standard model

Any local Lorentz invariant quantum field theory conserves CPT [3]. It follows
that in these theories (including the SM) T violation equals CP violation.
There is no reason, a priori, for C, P and CP to be related to flavour
physics. However, in the SM (and apparently in Nature) this is so. In the SM C
and P are violated maximally: left-handed and right-handed fermion fields
furnish different gauge representations, while C and P both change the
chirality of fermion fields. This maximal C and P violation within the SM is
also independent of the values of the SM parameters. On the other hand, the CP
violation within the SM does depend on the (Yukawa) parameters. The
hermiticity of the Lagrangian namely implies

$\displaystyle
Y_{ij}\bar{\psi}^{i}_{L}\phi\psi_{R}^{j}+Y_{ij}^{*}\bar{\psi}_{R}^{j}\phi^{\dagger}\psi_{L}^{i}\overset{\rm
CP}{\to}Y_{ij}\bar{\psi}_{R}^{j}\phi^{\dagger}\psi_{L}^{i}+Y_{ij}^{*}\bar{\psi}_{L}^{i}\phi\psi_{R}^{j}\,.$
(9)

Thus, the Yukawa Lagrangian will be CP symmetric if $Y_{ij}=Y_{ij}^{*}$. More
precisely, the requirement for CP conservation can be written in terms of the
Jarlskog invariant ($J$) [4] as

$\displaystyle J\equiv{\rm Im}[{\rm
det}(Y_{d}Y_{d}^{\dagger},Y_{u}Y_{u}^{\dagger})]=0\,.$ (10)

###  2.4 Mass basis

Upon replacing ${\rm Re}(\phi^{0})\to(v+h)/\sqrt{2}$, Yukawa interactions give
rise to fermion mass matrices

$\displaystyle M_{q}=\frac{v}{\sqrt{2}}Y_{q}\,.$ (11)

The mass bassis corresponds, by definition, to diagonal mass matrices. The
unitary transformations between any two bases which leave the gauge-kinetic
terms invariant are

$\displaystyle Q_{L}$ $\displaystyle\to V_{Q}Q_{L}\,,$ $\displaystyle U_{R}$
$\displaystyle\to V_{U}U_{R}\,,$ $\displaystyle D_{R}$ $\displaystyle\to
V_{D}D_{R}\,.$ (12)

The Yukawa matrices on the other hand transform as

$\displaystyle Y_{u}$ $\displaystyle\to V_{Q}Y_{u}V_{U}^{\dagger}\,,$
$\displaystyle Y_{d}$ $\displaystyle\to V_{Q}Y_{d}V_{D}^{\dagger}\,.$ (13)

The diagonalization of $M_{Q}$ requires bi-unitary transformations

$\displaystyle V_{Q}^{u}M_{u}V_{U}^{\dagger}$ $\displaystyle=M_{u}^{\rm
diag}=\frac{v}{\sqrt{2}}\lambda_{u}\,;$ $\displaystyle\lambda_{u}$
$\displaystyle={\rm diag}(y_{u},y_{c},y_{t})\,,$ $\displaystyle
V_{Q}^{d}M_{d}V_{D}^{\dagger}$ $\displaystyle=M_{d}^{\rm
diag}=\frac{v}{\sqrt{2}}\lambda_{d}\,;$ $\displaystyle\lambda_{d}$
$\displaystyle={\rm diag}(y_{d},y_{s},y_{b})\,.$ (14)

While $V_{U,D}$ are unphysical (they leave the gauge-kinetic terms invariant),
$V_{Q}^{u,d}$ produce a physical effect. In particular, since
$[M_{u},M_{d}]\neq 0$, a nontrivial mixing matrix
$V_{Q}^{u}V_{Q}^{d\dagger}\equiv V_{\rm CKM}\neq\rm 1$ (due to Cabibbo,
Kobayashi and Maskawa [5]) modifies the charged weak gauge interactions. The
resulting SM flavour Lagrangian in the mass basis is thus

$\displaystyle\mathcal{L}_{m}^{F}$
$\displaystyle=\left(\bar{q}_{i}\not{D}q^{j}\delta_{ij}\right)_{\rm
NC}+\frac{g}{\sqrt{2}}\bar{u}_{L}^{i}\not{W}^{+}V^{ij}_{\rm
CKM}d_{L}^{j}+\bar{u}_{L}^{i}\lambda^{ij}_{u}u_{R}^{j}\left(\frac{v+h}{\sqrt{2}}\right)+\bar{d}_{L}^{i}\lambda^{ij}_{d}d_{R}^{j}\left(\frac{v+h}{\sqrt{2}}\right)+\rm
h.c.\,,$ (15)

where $(u^{i}_{L},d^{i}_{L})\equiv Q_{L}^{T}$ and NC refers to neutral
currents (interactions with gluons, the photon and the Z boson).

##  3 Testing the CKM description of flavour

Let us recap the main features of quark flavour conversion in the SM: (i) it
only proceeds via the three CKM angles; (ii) is mediated by charged current
electroweak interactions; and (iii) these charged current interactions involve
exclusively left-handed fermion fields.

###  3.1 Parametrisation of the CKM matrix

We start by fixing the permutation of quark generations via mass ordering. The
resulting CKM matrix has the form

$V_{\rm CKM}=\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\\ V_{cd}&V_{cs}&V_{cb}\\\
V_{td}&V_{ts}&V_{tb}\end{pmatrix}\,.$ (16)

Experimentally, $V_{\rm CKM}$ exhbits a strong hierarchical pattern in off-
diagonal elements [6]

$\displaystyle|V_{ud}|$ $\displaystyle\simeq|V_{cs}|\simeq|V_{tb}|\simeq 1\,,$
$\displaystyle|V_{us}|$ $\displaystyle\simeq|V_{cd}|\simeq 0.22\,,$
$\displaystyle|V_{cb}|$ $\displaystyle\simeq|V_{ts}|\simeq 4\times 10^{-2}\,,$
$\displaystyle|V_{ub}|$ $\displaystyle\simeq|V_{td}|\simeq 5\times 10^{-3}\,.$
(17)

Such structure can be made explicit in the Wolfenstein expansion [7] in
$\lambda\equiv|V_{us}|\simeq 0.22$

$V_{\rm
CKM}=\begin{pmatrix}1-\frac{\lambda^{2}}{2}&\lambda&A\lambda^{3}(\rho-i\eta)\\\
-\lambda&1-\frac{\lambda^{2}}{2}&A\lambda^{2}\\\
A\lambda^{3}(1-\rho-i\eta)&-A\lambda^{2}&1\end{pmatrix}+\mathcal{O}(\lambda^{4})\,.$
(18)

The four parameters in this parametrisation $\lambda$, $A$, $\rho$ and $\eta$
can be mapped exactly to the four physical CKM parameters at any order in the
$\lambda$ expansion. All are of the order $\mathcal{O}(0.1-1)$ and the CP
violating phase is encoded in the imaginary contribution proportional to
$\eta$. Current experimental precision already requires that in
phenomenological applications, expansion at least to order
$\mathcal{O}(\lambda^{4})$ should be taken into account.

###  3.2 Unitarity of the CKM

Being a unitary matrix, one can derive unitarity conditions on the rows and
columns of the CKM matrix, in particular

$\displaystyle\sum_{k}V_{ik}^{*}V_{jk}$ $\displaystyle=\delta_{ij}\,,$
$\displaystyle\sum_{k}V_{ki}^{*}V_{kj}$ $\displaystyle=\delta_{ij}\,.$ (19)

Phenomenologically, the most interesting condition applies for $i=1$ and $j=3$

$V_{ud}V_{ub}^{*}+V_{cd}V_{cb}^{*}+V_{td}V_{tb}^{*}=0\,,$ (20)

simply because all the three terms on the left hand side are of the same order
in $\lambda$. The equation defines a triangle in the complex plane.
Normalizing one of the sides to unity

$\frac{V_{ud}V_{ub}^{*}}{V_{cd}V_{cb}^{*}}+\frac{V_{td}V_{tb}^{*}}{V_{cd}V_{cb}^{*}}+1=0\,,$
(21)

one can re-express it in terms of the Wolfenstein parameters (up to
$\mathcal{O}(\lambda^{5})$)

$[\bar{\rho}+i\bar{\eta}]+[(1-\bar{\rho})-i\bar{\eta}]+1=0\,,$ (22)

where $\bar{\rho}=\rho(1-\lambda^{2}/2)+\mathcal{O}(\lambda^{4})$ and
$\bar{\eta}=\eta(1-\lambda^{2}/2)+\mathcal{O}(\lambda^{4})$ . The angles
(denoted by $\alpha,~{}\beta$ and $\gamma$ in Fig. 1) and sides of this
triangle are invariant under phase transformations of quark fields and are
observable quantities.

###  3.3 Self consistency of the CKM assumption

The CKM description of quark flavour conversion has been tested experimentally
to great precision. In particular

  * •

$|V_{us}|~{}(\lambda)$ can be extracted from the semileptonic kaon decay
$K\to\pi\ell\nu$ with a precision of three per-mille: $\lambda=0.2253(9)$ [6]
.

  * •

$|V_{cb}|~{}(A)$ can be determined from semileptonic $B$ meson decay width
measurements $B\to X_{c}\ell\nu$ to a precision of two percent: $A=0.822(12)$
[6, 8] .

  * •

Then, $|V_{ub}|\propto\sqrt{\bar{\rho}^{2}+\bar{\eta}^{2}}$ can be extracted
using charmless semileptonic decays of $B$ mesons $B\to X_{u}\ell\nu$ .

  * •

The time-dependent CP asymmetry in the decay $B\to\psi K_{S}$ ($S_{\psi
K_{S}}\simeq\sin
2\beta=2\bar{\eta}(1-\bar{\rho})/[(1-\bar{\rho})^{2}+\bar{\eta}^{2}]$) has
been measured to great precision at the $B$ factory experiments Belle and
BaBar.

  * •

The rates $B\to DK$ decays depend on the phase
$\exp(i\gamma)=(\rho+i\eta)/(\rho^{2}+\eta^{2})$ .

  * •

Similarly, the rates of $B\to\pi\pi,\rho\pi,\rho\rho$ depend on the angle
$\alpha=\pi-\beta-\gamma$ .

  * •

The ratio of neutral $B$ and $B_{s}$ meson mass diferences $\Delta
m_{d}/\Delta
m_{s}\propto|V_{td}/V_{ts}|^{2}=\lambda^{2}\left[(1-\bar{\rho})^{2}+\bar{\eta}^{2}\right]$
exhibits another non-trivial constraint in the $(\bar{\rho},\bar{\eta})$
plane.

  * •

Finally, CP violation in $K\to\pi\pi$ decays ($\epsilon_{K}$) depends in a
complicate way on $(\bar{\rho},\bar{\eta})$.

Combined, these measurements lead to an impressive agreement with the best fit
ranges for $\rho$ and $\eta$ (see also Fig. 1 and Ref. [9]) [8]

$\displaystyle\rho$ $\displaystyle=0.130\pm 0.024\,,$
$\displaystyle\eta=0.362\pm 0.014\,.$ (23)

Figure 1: Result of the SM CKM fit projected onto the $\bar{\rho}-\bar{\eta}$
plane, as obtained by the CKMfitter [8] collaboration. Shown shaded are the
95% C.L. regions selected by the given observables.

Note that $|\eta|\gtrsim|\rho|$ implies that the CKM phase defined in this way
is $\mathcal{O}(1)$ . We can also conclude that, very likely, CP violation in
flavour changing processes is dominated by the CKM phase and that the
Kobayashi-Maskawa mechanism of CP violation is at work. Again one can define a
reparametrisation invariant measure of CP violation

${\rm
Im}[V_{ij}V_{kj}^{*}V_{kl}V_{il}^{*}]=J_{KM}\sum\epsilon_{ikm}\epsilon_{jln}\,,$
(24)

where $J_{KM}=\lambda^{6}A^{2}\eta=\mathcal{O}(10^{-5})$ . Written in this
form it is clear the CP violation in the SM is suppressed by small mixing
among the quark generations. The Jarlskog determinant in the SM can then be
written compactly as

$J=J_{KM}\prod_{i>j}\frac{m_{i}^{2}-m_{j}^{2}}{v^{2}}=\mathcal{O}(10^{-22})\,.$
(25)

We see that compared to $J_{KM}$, $J$ is further suppressed by the large quark
mass hierarchies.

##  4 Closer look at CP violation in neutral meson mixing and decays

For simplicity, we will focus on the neutral $B$ meson sistem with the flavour
eigenstates $B^{0}\sim\bar{b}d$ and $\bar{B}^{0}\sim b\bar{d}$. Since in
general, these are not CP eigenstates, we have

$\displaystyle CP|B^{0}\rangle$
$\displaystyle=e^{i\xi_{B}}|\bar{B}^{0}\rangle\,,$ $\displaystyle
CP|\bar{B}^{0}\rangle$ $\displaystyle=e^{-i\xi_{B}}|B^{0}\rangle\,.$ (26)

Stating from an initial superposition state at $t=0$
$|\psi(0)\rangle=a(0)|B^{0}\rangle+b(0)|\bar{B}^{0}\rangle$, the time
evolution of such a system can in general be described as

$\displaystyle|\psi(t)\rangle=a(t)|B^{0}\rangle+b(t)|\bar{B}^{0}\rangle+c_{1}(t)|f_{1}\rangle+c_{2}(t)|f_{2}\rangle+\ldots\,,$
(27)

where $f_{1,2,\ldots}$ denote the $B^{0}$ and $\bar{B}^{0}$ decay products. If
we are only interested in $a(t)$ and $b(t)$, we can construct an effective
description of the time evolution in terms of a non-hermition Hamiltonian

$\mathcal{H}=M+i\frac{\Gamma}{2}\,,$ (28)

where $M$ and $\Gamma$ are time-independent, Hermitian $2\times 2$ matrices,
describing possible oscillations and decays, respectively. The dispersive part
$M$ recieves contributions from off-shell intermediate states, while $\Gamma$
is the absorptive part and given by a sum over possible on-shell intermediate
states. The time-evolution is then described by

$i\frac{d}{dt}\begin{pmatrix}a(t)\\\ b(t)\end{pmatrix}=H\begin{pmatrix}a(t)\\\
b(t)\end{pmatrix}\,,$ (29)

with the eigenvectors $|B_{L,H}\rangle=p_{L,H}|B^{0}\rangle\pm
q_{L,H}|\bar{B}^{0}\rangle$ , and where $|p_{L,H}|^{2}+|q_{L,H}|^{2}=1$ .
Imposing CPT, one obtains $M_{11}=M_{22}$, $\Gamma_{11}=\Gamma_{22}$, and
consequently $p_{L}=p_{H}\equiv p$ and $q_{L}=q_{H}\equiv q$ . If CP is
conserved one furthermore obtains that ${\rm Arg}(M_{12})={\rm
Arg}(\Gamma_{12})$ and thus $|q/p|=1$ .

Conventionally, on defines the following CP conserving oscillation parameters

$\displaystyle m$ $\displaystyle\equiv\frac{M_{L}+M_{H}}{2}\,,$
$\displaystyle\Gamma$ $\displaystyle\equiv\frac{\Gamma_{L}+\Gamma_{H}}{2}\,,$
$\displaystyle\Delta m$ $\displaystyle\equiv M_{H}-M_{L}\,,$
$\displaystyle\Delta\Gamma$ $\displaystyle\equiv\Gamma_{H}-\Gamma_{L}\,,$ (30)

or equivalently $x\equiv\Delta m/\Gamma$ and $y\equiv\Delta\Gamma/2\Gamma$ .

The time evolution of the neutral meson system can finally be parametrized in
terms of states $|B^{0}(t)\rangle$ corresponding to $|B^{0}\rangle$ at initial
time $t=0$, and $|\bar{B}^{0}(t)\rangle$ corresponding to
$|\bar{B}^{0}\rangle$ at $t=0$

$\displaystyle|B^{0}(t)\rangle$
$\displaystyle=g_{+}(t)|B^{0}\rangle-\frac{q}{p}g_{-}(t)|\bar{B}^{0}\rangle\,,$
$\displaystyle|\bar{B}^{0}(t)\rangle$
$\displaystyle=g_{+}(t)|\bar{B}^{0}\rangle-\frac{q}{p}g_{-}(t)|B^{0}\rangle\,,$
(31)

where

$g_{\pm}\equiv\frac{1}{2}\left(e^{-m_{H}t-\Gamma_{H}t/2}\pm
e^{-m_{L}t-\Gamma_{L}t/2}\right)\,.$ (32)

The decay of the two mass eigenstates to some some final state $f$ after time
t is described by the decay amplitudes

$\displaystyle\langle f|\mathcal{H}|B^{0}\rangle\equiv A_{f}\,,$
$\displaystyle\langle\bar{f}|\mathcal{H}|B^{0}\rangle\equiv A_{\bar{f}}\,.$
(33)

The time-dependent decay rates are then given by

$\displaystyle\frac{d\Gamma(|B^{0}(0)\rangle)\to|f(t)\rangle}{dt}=\mathcal{N}_{0}e^{-\Gamma\,t}|A_{f}|^{2}\times$
$\displaystyle\left\\{\frac{1+|\lambda_{f}|^{2}}{2}\cosh\frac{\Delta\Gamma\,t}{2}+\frac{1-|\lambda_{f}|^{2}}{2}\cos\Delta
m\,t\right.$ $\displaystyle\left.+{\rm
Re}\lambda_{f}\sinh\frac{\Delta\Gamma\,t}{2}-{\rm Im}\lambda_{f}\sin\Delta
m\,t\right\\}\,,$
$\displaystyle\frac{d\Gamma(|\bar{B}^{0}(0)\rangle)\to|f(t)\rangle}{dt}=\mathcal{N}_{0}e^{-\Gamma\,t}|\bar{A}_{f}|^{2}\times$
$\displaystyle\left\\{\frac{1+|\bar{\lambda}_{f}|^{2}}{2}\cosh\frac{\Delta\Gamma\,t}{2}+\frac{1-|\bar{\lambda}_{f}|^{2}}{2}\cos\Delta
m\,t\right.$ $\displaystyle\left.+{\rm
Re}\bar{\lambda}_{f}\sinh\frac{\Delta\Gamma\,t}{2}-{\rm
Im}\bar{\lambda}_{f}\sin\Delta m\,t\right\\}\,,$ (34)

where $\mathcal{N}_{0}$ is the overall flux normalization,

$\displaystyle\lambda_{f}$
$\displaystyle\equiv\frac{q}{p}\frac{\bar{A}_{f}}{A_{f}}\,,$
$\displaystyle\bar{\lambda}_{f}$
$\displaystyle\equiv\frac{p}{q}\frac{A_{f}}{\bar{A}_{f}}=\frac{1}{\lambda_{f}}\,,$
(35)

and analogously for decays to $\bar{f}$ . The various terms in the above tiem-
evolution can be understood as follwing

  * •

Terms proportional to $|A_{f}|^{2}$, $|\bar{A}_{f}|^{2}$ describe a decay
without net oscillation.

  * •

Terms proportional to $|\lambda_{f}|^{2}$, $|\bar{\lambda}_{f}|^{2}$ describe
a decays following net oscillations.

  * •

Terms proportional to $\sin\Delta mt$, $\sinh\Delta\Gamma t/2$ describe
interference between the above two cases.

  * •

CP violation in interference is possible only if ${\rm Im}(\lambda_{f})\neq 0$
.

Such effects can be observed in neutral $B$ meson decays to CP eigenstates via
a time-dependent CP asymmetry

$A_{f_{CP}}(t)\equiv\frac{\frac{d\Gamma}{dt}\left[\bar{B}^{0}(0)\to
f_{CP}(t)\right]-\frac{d\Gamma}{dt}\left[B^{0}(0)\to
f_{CP}(t)\right]}{\frac{d\Gamma}{dt}\left[\bar{B}^{0}(0)\to
f_{CP}(t)\right]+\frac{d\Gamma}{dt}\left[B^{0}(0)\to f_{CP}(t)\right]}\,.$
(36)

In the $B$ (and also $B_{s}$) system experimentally $\Delta\Gamma\ll\Delta m$
and so $|q/p|\simeq 1$ . In this limit, the full expression for $A_{f}$
greatly simplifies and can be written as

$A_{f}(t)=S_{f}\sin(\Delta m\,t)-C_{f}\cos(\Delta m\,t)\,,$ (37)

where

$\displaystyle S_{f}$ $\displaystyle\equiv\frac{2\,{\rm
Im}(\lambda_{f})}{1+|\lambda_{f}|^{2}}\,,$ $\displaystyle C_{f}$
$\displaystyle\equiv\frac{1-|\lambda_{f}|^{2}}{1+|\lambda_{f}|^{2}}\,.$ (38)

###  4.1 Phases in decay amplitudes

Consider the decay $B\to f$ described by the amplitude $A_{f}$ and its CP
conjugate process $\bar{B}\to\bar{f}$ associated with the amplitude
$\bar{A}_{\bar{f}}$. Any complex parameter in the theory Lagrangian entering
the two amplitudes will appear complex conjugated after CP and will thus
appear with opposite signs in $A_{f}$ and $\bar{A}_{\bar{f}}$ . The associated
CP odd phases are conventionally called weak phases. In the SM they are
induced via $W$ exchanges. Note that single amplitude phases are convention
dependent and thus not physical. Only differences between phases of different
amplitudes are physical.

On the other hand, on-shell intermediate states in scattering or decay
ampitudes can produce phase changes even if the relevant Lagrangian is real.
These are thus independent of CP. They will appear with same signs in both
$A_{f}$ and $\bar{A}_{\bar{f}}$. These CP even phases are often reffered to as
strong phases. In the SM they are due to strong interaction induced re-
scattering. Again, only relative phases between amplitudes are physical.

In general, one can thus write both decay amplitudes as

$\displaystyle A_{f}$
$\displaystyle=|a_{1}|e^{i(\delta_{1}+\phi_{1})}+|a_{2}|e^{i(\delta_{2}+\phi_{2})}+\ldots\,,$
$\displaystyle\bar{A}_{\bar{f}}$
$\displaystyle=|a_{1}|e^{i(\delta_{1}-\phi_{1})}+|a_{2}|e^{i(\delta_{2}-\phi_{2})}+\ldots\,,$
(39)

where $a_{1,2,\ldots}$ are contributions to the amplitude with different
phases, $\delta_{1,2\ldots}$ are the strong phases and $\phi_{1,2\ldots}$ are
the weak phases.

###  4.2 CP violation in $B\to\psi K_{S}$

To a good approximation, the $B\to\psi K_{S}$ decays are described by a just
single weak decay amplitude to a CP eigenstate (with CP eigenvalue $\eta_{f}$)

$\displaystyle A_{f}$ $\displaystyle=|a_{f}|e^{i(\delta_{f}+\phi_{f})}\,,$
$\displaystyle\bar{A}_{f}$
$\displaystyle=|a_{f}|e^{i(\delta_{f}-\phi_{f})}\eta_{f}\,,$ (40)

so that $\lambda_{f}=\eta_{f}(q/p)\exp(-2i\phi_{f})$ . In the neutral $B$
system $|\Gamma_{12}|\ll|M_{12}|$, since it is due to $\mathcal{O}(G_{F}^{2})$
long distance effects, which are suppressed by small CKM elements (a fact also
verified experimentally since $\Delta\Gamma\ll\Delta m$). Then one can write

$\left(\frac{q}{p}\right)^{2}=\frac{M_{12}^{*}-\frac{i}{2}\Gamma_{12}^{*}}{M_{12}-\frac{i}{2}\Gamma_{12}}\simeq
e^{2i\xi_{B}}\,,$ (41)

and thus $\lambda_{f}\simeq\eta_{f}\exp[i(\xi_{B}-2\phi_{f})]$, leading to a
simple expression for the time-dependent CP asymmetry $A_{f}(t)$, in
particular

$S_{f_{CP}}\simeq\eta_{f}\sin(\xi_{B}-2\phi_{f})\,.$ (42)

In the SM, $\xi_{B}$ and $\phi_{f}$ are exactly computable in terms of the CKM
elements. In particular

$\xi_{B}=-{\rm Arg}(M_{12})\simeq-{\rm Arg}[(V_{tb}^{*}V_{td})^{2}]=-{\rm
Arg}\left(\frac{V_{tb}^{*}V_{td}}{V_{tb}V_{td}^{*}}\right)\,,$ (43)

while

$-e^{-2i\phi_{f}}=\frac{\bar{A}^{(B)}_{\psi K_{S}}}{A^{(B)}_{\psi
K_{S}}}=-\frac{V_{cb}V_{cs}^{*}\,a_{T}+\ldots}{V_{cb}^{*}V_{cs}\,a_{T}+\ldots}e^{i\xi_{K}}\simeq-\frac{V_{cb}V_{cs}^{*}}{V_{cb}^{*}V_{cs}}\frac{V^{*}_{cd}V_{cs}}{V_{cd}V^{*}_{cs}}\,.$
(44)

In the above equation, the dots denote additional amplitudes suppressed by
small coefficients and CKM elements. Also, in the second step we have taken
into account the phase projection of the neutral kaon flavour eigenstates onto
the mass eigenstate $K_{S}$ due to $K-\bar{K}$ oscillations (analogous to Eq.
(43)). Combining Eqs. (43) and (44) we thus obtain

$\lambda_{\psi
K_{S}}^{(B)}\simeq\frac{V_{tb}^{*}V_{td}}{V_{tb}V_{td}^{*}}\frac{V_{cb}V_{cd}^{*}}{V_{cb}^{*}V_{cd}}=-e^{-2i\beta}\,.$
(45)

The observable $S_{\psi K_{S}}^{(B)}\simeq\sin 2\beta$ (note that
$C^{(B)}_{\psi K_{S}}\simeq 0$) demonstrates CP violation in interference
between the mixing and decay amplitudes. Experimentally, it has been measured
to an accuracy of $\sim 1\%$ at the $B$ factories [6].

###  4.3 CP violation in $B_{s}$ mixing

Establishing CP violation in the $B_{s}$ system is considerably more
challenging. The golden channel is the decay $B_{s}\to\psi\phi$ . Since it is
an admixture of different CP eigenestates (represented by the different
polarizations of the two vector mesons in the final state), an angular
analysis is required for the extraction of the CP violating phase. In
addition, $B_{s}$ oscillations are much faster than those of $B_{d}$, namely

$\frac{\Delta m_{s}}{\Delta
m_{d}}\sim\frac{|M_{12}^{s}}{|M_{12}^{d}|}\propto\left|\frac{V_{ts}}{V_{td}}\right|^{2}\sim
30\,.$ (46)

Finally, $\Delta\Gamma_{s}$ effects in the time evolution cannot be neglected
compared to $\Delta m_{s}$. In the SM
$\lambda_{\psi\phi}^{(B_{s})}=-\exp[{i(\xi_{B_{s}}-2\phi_{\psi\phi})}]$
evaluates to [10]

$\left[S_{\psi\phi}^{(B_{s})}\right]_{\rm SM}=2{\rm
Arg}\frac{V_{tb}^{*}V_{ts}}{V_{cb}^{*}V_{cs}}=0.036(1)\,,$ (47)

which is still small compared to the currently attainable experimental
precision of $S_{\psi\phi}^{(B_{s})}=-0.02(4)$ [11].

###  4.4 CP violation in $B$ decays to CP conjugate states

This form of measurements is interesting if $B^{0}\to\bar{f}$ and
$\bar{B}^{0}\to f$ transitions are forbidden. In this case
$|A_{f}|=|\bar{A}_{\bar{f}}|$ and $|A_{\bar{f}}|=|\bar{A}_{f}|=0$ and one can
define a CP asymmetry

$\frac{\frac{d\Gamma}{dt}\left[\bar{B}^{0}(0)\to
f(t)\right]-\frac{d\Gamma}{dt}\left[B^{0}(0)\to\bar{f}(t)\right]}{\frac{d\Gamma}{dt}\left[\bar{B}^{0}(0)\to
f(t)\right]+\frac{d\Gamma}{dt}\left[B^{0}(0)\to\bar{f}(t)\right]}=\frac{\left|\frac{p}{q}\right|^{2}-\left|\frac{q}{p}\right|^{2}}{\left|\frac{p}{q}\right|^{2}+\left|\frac{q}{p}\right|^{2}}\simeq{\rm
Im}\left(\frac{\Gamma_{12}}{M_{12}}\right)+\mathcal{O}\left(\left|\frac{\Gamma_{12}}{M_{12}}\right|^{2}\right)\,,$
(48)

where in the last step we have again used the fact the
$|\Gamma_{12}|\ll|M_{12}|$ in the $B$ system. Note that the above asymmetry is
accessible with a time-independent measurement. It also represents CP
violation in mixing, simetimes termed indirect CP violation . An illustrative
example is the wrong sign semileptonic decay asymmetry

$a_{SL}^{(d)}=\frac{\Gamma(\bar{B}^{0}\to X\ell^{+}\nu)-\Gamma(B^{0}\to
X\ell^{-}\bar{\nu})}{\Gamma(\bar{B}^{0}\to X\ell^{+}\nu)+\Gamma(B^{0}\to
X\ell^{-}\bar{\nu})}\,,$ (49)

with the SM expectation of $a_{SL}^{(d)}=-8(2)\times 10^{-4}$ [10] .

###  4.5 CP violation in charged $B$ decays

The possibility of CP violation in charged $B$ decays is of special interest
in the case of $B^{\pm}\to DK^{\pm}$, since $D-\bar{D}$ oscillations allow for
interference of two tree-level dominated decay amplitudes, in particular

$\displaystyle B^{-}$ $\displaystyle\to D^{0}K^{-}:~{}b\to c\bar{u}s\,,$
$\displaystyle B^{-}$ $\displaystyle\to\bar{D}^{0}K^{-}:~{}b\to\bar{c}us\,.$
(50)

The resulting phenomenology is particularly transparent by focusing on
subsequent $D$ decays to CP eigenstates [12]

$\displaystyle D^{0}$ $\displaystyle\to f_{CP}:~{}c\to
d\bar{d}u\,,s\bar{s}u\,,$ $\displaystyle\bar{D}^{0}$ $\displaystyle\to
f_{CP}:~{}\bar{c}\to d\bar{d}\bar{u}\,,s\bar{s}\bar{u}\,.$ (51)

In the SM the ratio of the two decay amplitudes is then

$\frac{A^{B}_{(D\to f)K}}{A^{B}_{(\bar{D}\to
f)K}}=\frac{V_{cb}^{*}V_{us}a^{B}_{DK}}{V_{ub}^{*}V_{cs}a^{B}_{\bar{D}K}}e^{i(\delta^{B}_{DK}-\delta^{B}_{\bar{D}K})}\eta_{f}\frac{V_{cd}V_{ud}^{*}}{V_{cd}^{*}V_{ud}}\frac{a^{D}_{f}}{a^{\bar{D}}_{f}}e^{i(\delta^{D}_{f}-\delta^{\bar{D}}_{f})}\simeq\eta_{f}r_{B}e^{i(\delta_{B}-\gamma)}\,,$
(52)

where we have used the definition of the angle $\gamma\equiv{\rm
Arg}(-V_{ud}V_{ub}^{*}/V_{cd}V_{cb}^{*})\simeq 70^{\circ}$ [6] and have
collected the hadronic amplitude ratios into $r_{B}$ and the associated strong
phases in $\delta_{B}$ .

The virtue of these modes is that in principle all unknown parameters can be
determined by measuring several available decay rates only, which are CP even
quantities. In particular

$\displaystyle A(B^{-}\to f_{+}K^{-})$
$\displaystyle=A_{0}\left[1+r_{B}e^{i(\delta_{B}-\gamma)}\right]\,,$
$\displaystyle A(B^{-}\to f_{-}K^{-})$
$\displaystyle=A_{0}\left[1-r_{B}e^{i(\delta_{B}-\gamma)}\right]\,,$
$\displaystyle A(B^{+}\to f_{+}K^{-})$
$\displaystyle=A_{0}\left[1+r_{B}e^{i(\delta_{B}+\gamma)}\right]\,,$
$\displaystyle A(B^{+}\to f_{-}K^{-})$
$\displaystyle=A_{0}\left[1-r_{B}e^{i(\delta_{B}+\gamma)}\right]\,.$ (53)

can be used to extract the three hadronic parameters ($A_{0}$, $r_{B}$ and
$\delta_{B}$) as well as $\gamma$ . Since no $B$ mixing is involved, these
measurements are sensitive to CP violation in decay also termed direct CP
violation. The determination of $\gamma$ in this way is theoretically
extremely clean, in particular, since CP violation in $D-\bar{D}$ mixing is
negligible. Experimentally, it is advantageous to have both a large $r_{B}$
and large $\delta_{B}$. Therefore, it is welcome that such approach can be
adapted also for D-decay products, which are non CP eigenstates [13].

##  5 Flavour and New Physics

Let us first consider how much NP can still contribute to flavour observables,
given the current experimental and theoretical precision. For example, given
the good agreement of SM tree-level mediated processes with experiment, one
can perform basic tests of CKM unitarity. Taking only the moduli of the first
row CKM elements:

  * •

$|V_{ud}|$ which can be extracted from $0^{+}\to 0^{+}e\nu$ super-allowed
nuclear $\beta$ decays, yielding $|V_{ud}|=0.97425(22)$ [6] ;

  * •

$|V_{us}|$ which is determined from the semileptonic kaon decays
$K^{+}\to\pi^{+}\ell\nu$, yielding $|V_{us}|=0.2237(13)$ [6] ;

  * •

finally, $|V_{ub}|$ which is measured using charmless semileptonic $B$ decays
$B\to X_{u}\ell\nu$, yelding $|V_{ub}|=4.2(5)\times 10^{-3}$ [6] ;

one can form the following CKM unitarity constraint [14]

$|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}-1=-0.0008(7)\,.$ (54)

Using the measurements of the Fermi constant from the muon life-time, one can
further reinterpret these constraints as tests of the charged current
universality between leptonic and semileptonic weak processes at the per-mille
level. In light of this, it is reasonable to consider NP contributions to
observables which are (loop, CKM) suppressed in the SM. Then one can use the
CKM determination from tree-level observables, in particular $|V_{ud}|$,
$|V_{us}|$, $|V_{cb}|$ and $|V_{ub}|$ as well as $\gamma$ from $B\to DK$
decays (and/or $\alpha$ from tree-level dominated $B\to\pi\pi$ decays) . This
finally allows to predict SM contributions also to loop suppressed
observables, greatly enhancing their sensitivity to NP.

###  5.1 New physics in $B-\bar{B}$ mixing

In the following we will assume a presence of heavy NP – such that it would
only contribute to dispersive $B-\bar{B}$ amplitudes. In that case, the most
general modification of $M_{12}$ can be parametrised as

$M_{12}=M_{12}^{\rm SM}r_{d}^{2}e^{2i\theta_{d}}$ (55)

where the NP parameters $r_{d}$ and $\theta_{d}$ signify a change of the
magnitude and phase with respect to the SM prediction, respectively. Such
effects of NP can then be easily translated to all relevant $B$ mixing
observables as

$\displaystyle\Delta m_{B}$ $\displaystyle=r_{d}^{2}\left(\Delta
m_{B}\right)^{\rm SM}\,,$ $\displaystyle S_{\psi K_{S}}^{(B)}$
$\displaystyle\simeq\sin(2\beta+2\theta_{d})\,,$ $\displaystyle a_{SL}^{(d)}$
$\displaystyle=-{\rm Re}\left(\frac{\Gamma_{12}}{M_{12}}\right)^{\rm
SM}\frac{\sin 2\theta_{d}}{r_{d}^{2}}+{\rm
Im}\left(\frac{\Gamma_{12}}{M_{12}}\right)^{\rm SM}\frac{\cos
2\theta_{d}}{r_{d}^{2}}\,.$ (56)

One can compare these expectations to the current experimental measurements of
[6]

$\displaystyle\Delta m_{B}$ $\displaystyle=51.0(4)\times 10^{10}/s\,,$
$\displaystyle S_{\psi K_{S}}^{(B)}$ $\displaystyle=0.671(24)\,,$
$\displaystyle a_{SL}^{(d)}$ $\displaystyle=-0.2(7)\times 10^{-3}\,,$ (57)

where the SM expectation with tree-level CKM inputs for $[S_{\Psi
K_{S}}^{(B)}]^{\rm SM}_{\rm tree}=0.76(4)$ [8] . We can immediately draw the
following conclusions

  * •

NP in $M_{12}$ with a large phase relative to $\beta$ is constrained to
$20\%-30\%$ of the SM contribution. Thus, CKM clearly dominates CP violation
in $B-\bar{B}$ mixing. (A similar conclusion can be made for the case of
$K^{0}-\bar{K}^{0}$ system: the measured value of
$\epsilon_{K}=1.596(13)\times 10^{-3}$ constrains CP violating NP in kaon
mixing amplitudes to be subdominant. The power of this constraint is however
presently limited by theory uncertainties.)

  * •

NP in $M_{12}$ with a phase that is aligned to $\beta$ is constrained to be at
most comparable to the SM contribution. (Again a similar conclusion can be
made for the case of $K^{0}-\bar{K}^{0}$ system: the measured value of $\Delta
m_{K}=52.93(9)\times 10^{8}/s$ constrains CP conserving NP in kaon mixing
amplitudes to be comparable to SM estimates, which however contrary to the $B$
case carry sizable theory uncertainties.)

As a comparison, in the case of $B_{s}$ mixing, NP can be at most comparable
to the SM contribution regardless of the phase since $S_{\psi\phi}^{\rm
SM}\lesssim\delta S_{\psi\phi}^{\rm exp}$ .

###  5.2 The new physics flavour puzzle

The SM is not a complete theory of Nature.

  1. 1.

It does not include a (quantum) description of gravity. Thus its validity is
limited below the Planck scale $m_{\rm Planck}\simeq 10^{19}$ GeV.

  2. 2.

It does not include neutrino masses. This further limits its validity down to
below the maximal scale at which new degrees of freedom can accommodate at
least two massive neutrinos $m_{\rm see-saw}\lesssim 10^{15}$ GeV.

  3. 3.

The fine-tuning of the EW symmetry breaking scale compared to the large scales
in the above points 1. and 2. suggests NP already at scales of the order $4\pi
v\sim 1$ TeV .333Incidentally, the TeV mass scale can also be associated with
the explanation of the cosmological dark matter, if the later is in the form
of a thermal particle relic.

Given the SM is merely an effective field theory valid below a cut-off energy
scale $\Lambda$, one needs to consider additional terms in the theory
Lagrangian consisting of SM field operators with canonical dimensions $d>4$:

$\mathcal{L}=\mathcal{L}_{\rm
SM}+\sum_{d>4}\sum_{n}\frac{c_{n}^{(d)}}{\Lambda^{d-4}}\mathcal{O}_{n}^{(d)}\,.$
(58)

In a natural theory one expects $c_{n}^{(d)}\sim\mathcal{O}(1)$ unless the
relevant operators are forbidden or suppressed by symmetries. For
$\Lambda\sim$ TeV and without imposing additional symmetries beyond the gauged
SM ones, the above condition is severely violated for several
$\mathcal{O}_{n}^{(6)}$ , which contribute to flavour changing processes. This
constitutes the so-called NP flavour puzzle , which can be articulated through
the following question: If there is NP at the TeV scale, why haven’t we seen
its effects in flavour observables? Naively, one could argue, that the same it
true for NP violating baryon and lepton numbers. However, $B$ and $L$ are
(classically) exact accidental symmetries of the SM, while in the SM the
flavour symmetry is already broken explicitly.

###  5.3 Bounds on new physics from $\Delta F=2$ processes

The NP flavour puzzle can be demonstrated perhaps most dramatically in the
case $\Delta F=2$ FCNCs. In the SM the dispersive contributions to $\Delta
F=2$ processes of down-quarks are typically dominated by box diagrams with the
top quarks appearing in the loop. These contributions can be schematically
written as

$M_{12}^{\rm
SM}=\frac{G_{F}^{2}m_{t}^{2}}{16\pi^{2}}\left(V_{ti}^{*}V_{tj}\right)^{2}\langle\bar{M}|(\bar{d}_{L}^{i}\gamma_{\mu}d_{L}^{j})^{2}|M\rangle
F\left(\frac{m_{t}^{2}}{m_{W}^{2}}\right)+\ldots\,,$ (59)

where $M=K^{0},~{}B^{0},~{}B_{s}$, $d^{i,j}$ denote meson valence quarks,
$F(x)\sim\mathcal{O}(1)$ is the relevant loop function normalized to
$F(\infty)=1$ , while the dots denote corrections due to charm quark
contributions, which are numerically relevant only in the case of $K-\bar{K}$
mixing. Note that the prefactor can be rewritten completely in terms of the
fundamental flavour parameters (Yukawas) in the unbroken theory

$\frac{G_{F}^{2}m_{t}^{2}}{16\pi^{2}}\left(V_{ti}^{*}V_{tj}\right)^{2}=\frac{(Y_{u}Y_{u}^{*})_{ij}}{128\pi^{2}m_{t}^{2}}\,,$
(60)

which can be interpreted as due to Goldstone Higgs exchanges in the gaugeless
$(g\to 0)$ limit of the SM.

The relevant hadronic matrix elements between the external $M$ and $\bar{M}$
mesons can be written as

$\langle\bar{M}|(\bar{d}^{i}_{L}\gamma_{\mu}d_{L}^{j})(\bar{d}^{i}_{L}\gamma^{\mu}d_{L}^{j})|M\rangle=\frac{2}{3}f_{M}^{2}m_{M}^{2}\hat{B}_{M}\,,$
(61)

where the relevant meson decay constant $f_{M}$ is defined via $\langle
0|d^{i}\gamma_{\mu}\gamma_{5}d^{j}|M(p)\rangle\equiv ip_{\mu}f_{M}$ , while
$\hat{B}_{M}\sim\mathcal{O}(1)$ is called the bag parameter. These two
hadronic quantities can be computed numerically using lattice QCD methods.The
tremendous progress in these calculations over the past 30 years is reflected
in the precise values of [15]

$\displaystyle f_{B}$ $\displaystyle=0.186(4)~{}{\rm GeV}\,,$
$\displaystyle\hat{B}_{B}$ $\displaystyle=1.27(10)\,,$ $\displaystyle
f_{B_{s}}$ $\displaystyle=0.224(5)~{}{\rm GeV}\,,$
$\displaystyle\hat{B}_{B_{s}}$ $\displaystyle=1.33(6)\,,$ $\displaystyle
f_{K}$ $\displaystyle=0.1563(9)~{}{\rm GeV}\,,$ $\displaystyle\hat{B}_{K}$
$\displaystyle=0.7661(99)\,.$ (62)

With these inputs we can use the experimental measurements of neutral meson
mixing observables to constrain possible NP contributions of the form

$\displaystyle\mathcal{L}_{\rm NP}^{\Delta F=2}$
$\displaystyle=\frac{c_{sd}}{\Lambda^{2}}\left(\bar{d}_{L}\gamma^{\mu}s_{L}\right)^{2}+\frac{c_{bd}}{\Lambda^{2}}\left(\bar{b}_{L}\gamma^{\mu}s_{L}\right)^{2}+\frac{c_{bs}}{\Lambda^{2}}\left(\bar{s}_{L}\gamma^{\mu}b_{L}\right)^{2}$
$\displaystyle+\frac{c_{cu}}{\Lambda^{2}}\left(\bar{u}_{L}\gamma^{\mu}c_{L}\right)^{2}+\frac{c_{tu}}{\Lambda^{2}}\left(\bar{u}_{L}\gamma^{\mu}t_{L}\right)^{2}+\frac{c_{tc}}{\Lambda^{2}}\left(\bar{c}_{L}\gamma^{\mu}t_{L}\right)^{2}\,.$
(63)

The effects of such NP on neutral meson oscillations can namely be completely
encoded into

$\frac{M_{12}^{M}}{m_{M}}\sim c_{ij}\left(\frac{f_{M}}{\Lambda}\right)^{2}\,,$
(64)

which leads to the following set of current experimental constraints [9, 16]

$\displaystyle\frac{\Delta m_{K}}{m_{K}}$ $\displaystyle\sim 7\times 10^{-15}$
$\displaystyle\Rightarrow$ $\displaystyle\frac{\Lambda}{\sqrt{|c_{sd}|}}$
$\displaystyle\gtrsim 10^{3}~{}{\rm TeV}$ $\displaystyle{\rm or}$
$\displaystyle|c_{sd}|$ $\displaystyle\lesssim 10^{-6}\left(\frac{\Lambda}{\rm
TeV}\right)^{2}\,,$ $\displaystyle\frac{\Delta m_{D}}{m_{D}}$
$\displaystyle\sim 9\times 10^{-15}$ $\displaystyle\Rightarrow$
$\displaystyle\frac{\Lambda}{\sqrt{|c_{cu}|}}$ $\displaystyle\gtrsim
10^{3}~{}{\rm TeV}$ $\displaystyle{\rm or}$ $\displaystyle|c_{cu}|$
$\displaystyle\lesssim 10^{-6}\left(\frac{\Lambda}{\rm TeV}\right)^{2}\,,$
$\displaystyle\frac{\Delta m_{B}}{m_{B}}$ $\displaystyle\sim 6\times 10^{-14}$
$\displaystyle\Rightarrow$ $\displaystyle\frac{\Lambda}{\sqrt{|c_{bd}|}}$
$\displaystyle\gtrsim 4\times 10^{2}~{}{\rm TeV}$ $\displaystyle{\rm or}$
$\displaystyle|c_{bd}|$ $\displaystyle\lesssim 5\times
10^{-6}\left(\frac{\Lambda}{\rm TeV}\right)^{2}\,,$ $\displaystyle\frac{\Delta
m_{B_{s}}}{m_{B_{s}}}$ $\displaystyle\sim 2\times 10^{-12}$
$\displaystyle\Rightarrow$ $\displaystyle\frac{\Lambda}{\sqrt{|c_{bs}|}}$
$\displaystyle\gtrsim 70~{}{\rm TeV}$ $\displaystyle{\rm or}$
$\displaystyle|c_{bs}|$ $\displaystyle\lesssim 2\times
10^{-4}\left(\frac{\Lambda}{\rm TeV}\right)^{2}\,.$ (65)

Furthermore, in case of maximal CP violating phases in $c_{ij}$, one obtains
even stronger constraints

$\displaystyle\epsilon_{K}$ $\displaystyle\sim 0.0023$
$\displaystyle\Rightarrow$ $\displaystyle\frac{\Lambda}{\sqrt{|{\rm
Im}(c_{sd})|}}$ $\displaystyle\gtrsim 2\times 10^{4}~{}{\rm TeV}$
$\displaystyle{\rm or}$ $\displaystyle|{\rm Im}(c_{sd})|$
$\displaystyle\lesssim 6\times 10^{-10}\left(\frac{\Lambda}{\rm
TeV}\right)^{2}\,,$ $\displaystyle\frac{A_{\Gamma}}{y_{CP}}$
$\displaystyle\lesssim 0.2$ $\displaystyle\Rightarrow$
$\displaystyle\frac{\Lambda}{\sqrt{|{\rm Im}(c_{cu})|}}$ $\displaystyle\gtrsim
3\times 10^{3}~{}{\rm TeV}$ $\displaystyle{\rm or}$ $\displaystyle|{\rm
Im}(c_{cu})|$ $\displaystyle\lesssim 10^{-7}\left(\frac{\Lambda}{\rm
TeV}\right)^{2}\,,$ $\displaystyle S_{\psi K_{S}}$ $\displaystyle\sim 0.67$
$\displaystyle\Rightarrow$ $\displaystyle\frac{\Lambda}{\sqrt{|{\rm
Im}(c_{bd})|}}$ $\displaystyle\gtrsim 8\times 10^{2}~{}{\rm TeV}$
$\displaystyle{\rm or}$ $\displaystyle|{\rm Im}(c_{bd})|$
$\displaystyle\lesssim 10^{-6}\left(\frac{\Lambda}{\rm TeV}\right)^{2}\,,$
$\displaystyle S_{\psi\phi}$ $\displaystyle\sim 0.1$
$\displaystyle\Rightarrow$ $\displaystyle\frac{\Lambda}{\sqrt{|{\rm
Im}(c_{bs})|}}$ $\displaystyle\gtrsim 70~{}{\rm TeV}$ $\displaystyle{\rm or}$
$\displaystyle|{\rm Im}(c_{bs})|$ $\displaystyle\lesssim 2\times
10^{-4}\left(\frac{\Lambda}{\rm TeV}\right)^{2}\,.$ (66)

The two main messages one can draw from such an analysis are that (1) NP with
a generic flavour structure is irrelevant for EW hierarchy, since flavour
measurements in this case require $\Lambda\gg$ TeV; and (2) in case of TeV NP,
its flavour structure needs to be far from generic.

##  6 Conclusions

The absence of significant deviations from the SM in quark flavour physics is
a key constraint on any extension of the SM. At the same time there are still
various open questions regarding the flavour structure of the standard model
itself that can be possibly addressed only at low energies, using flavour
phyiscs measurements. The set of flavour observables to be measured with
higher precision in the search for indirect hints of NP is limited, but not
necessarily small. For example, we still have only limited knowledge about CP
violation in the $B_{s}$ and $D$ systems. In addition, despite significant
recent progress, new-physics effects could still be hidden in certain rare
kaon, $D$ and $B$ decays [17]. The experimental progress on these, as expected
from the LHCb [18] in LHC run II, Belle II [19] and other upcoming flavour
experiments will thus be invaluable.

### Acknowledgements

I wish to thank the organizers of the CERN’s 2014 European School of High-
Energy Physics for the invitation to lecture in this remarkable setting. I am
also grateful to the students and discussion leaders for the stimulating
questions and discussions.

## References

  * [1] A. J. Buras, hep-ph/9806471; G. C. Branco, L. Lavoura and J. P. Silva, Int. Ser. Monogr. Phys. 103 (1999) 1; U. Nierste, arXiv:0904.1869 [hep-ph]; O. Gedalia and G. Perez, arXiv:1005.3106 [hep-ph]; Y. Grossman, CERN Yellow Report CERN-2010-002, 111-144 [arXiv:1006.3534 [hep-ph]]; Y. Nir, CERN Yellow Report CERN-2010-001, 279-314 [arXiv:1010.2666 [hep-ph]]; G. Isidori, arXiv:1302.0661 [hep-ph].
  * [2] R. S. Chivukula and H. Georgi, Phys. Lett. B 188, 99 (1987); A. J. Buras, Acta Phys. Polon. B 34, 5615 (2003); G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B 645, 155 (2002).
  * [3] R. F. Streater and A. S. Wightman, Princeton, USA: Princeton Univ. Pr. (2000) 207 p.
  * [4] C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039.
  * [5] N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531; M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
  * [6] J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D 86 (2012) 010001.
  * [7] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945.
  * [8] J. Charles et al. [CKMfitter Group], Eur. Phys. J. C 41, 1-131 (2005), updated results and plots available at: http://ckmfitter.in2p3.fr
  * [9] M. Bona et al. [UTfit Collaboration], JHEP 0803 (2008) 049 [arXiv:0707.0636 [hep-ph]].
  * [10] A. Lenz, U. Nierste, J. Charles, S. Descotes-Genon, A. Jantsch, C. Kaufhold, H. Lacker and S. Monteil et al., Phys. Rev. D 83 (2011) 036004 [arXiv:1008.1593 [hep-ph]].
  * [11] D. Martinez Santos [LHCb Collaboration], LHCb-TALK-2014-337, Cern, Switzerland, 15 - 17 Oct 2014.
  * [12] M. Gronau and D. London, Phys. Lett. B 253 (1991) 483; M. Gronau and D. Wyler, Phys. Lett. B 265 (1991) 172.
  * [13] D. Atwood, I. Dunietz and A. Soni, Phys. Rev. Lett. 78 (1997) 3257 [hep-ph/9612433]; A. Giri, Y. Grossman, A. Soffer and J. Zupan, Phys. Rev. D 68 (2003) 054018 [hep-ph/0303187].
  * [14] P. A. Boyle, J. M. Flynn, N. Garron, A. J ttner, C. T. Sachrajda, K. Sivalingam and J. M. Zanotti, JHEP 1308 (2013) 132 [arXiv:1305.7217, arXiv:1305.7217 [hep-lat]].
  * [15] S. Aoki, Y. Aoki, C. Bernard, T. Blum, G. Colangelo, M. Della Morte, S. D rr and A. X. El Khadra et al., Eur. Phys. J. C 74 (2014) 9, 2890 [arXiv:1310.8555 [hep-lat]].
  * [16] G. Isidori, Y. Nir and G. Perez, Ann. Rev. Nucl. Part. Sci. 60 (2010) 355 [arXiv:1002.0900 [hep-ph]]; A. Lenz, U. Nierste, J. Charles, S. Descotes-Genon, H. Lacker, S. Monteil, V. Niess and S. T’Jampens, Phys. Rev. D 86 (2012) 033008 [arXiv:1203.0238 [hep-ph]]; V. Bertone et al. [ETM Collaboration], JHEP 1303 (2013) 089 [Erratum-ibid. 1307 (2013) 143] [arXiv:1207.1287 [hep-lat]]; N. Carrasco et al. [ETM Collaboration], JHEP 1403 (2014) 016 [arXiv:1308.1851 [hep-lat]]; N. Carrasco, M. Ciuchini, P. Dimopoulos, R. Frezzotti, V. Gimenez, V. Lubicz, G. C. Rossi and F. Sanfilippo et al., Phys. Rev. D 90 (2014) 014502 [arXiv:1403.7302 [hep-lat]].
  * [17] M. Artuso, D. M. Asner, P. Ball, E. Baracchini, G. Bell, M. Beneke, J. Berryhill and A. Bevan et al., Eur. Phys. J. C 57 (2008) 309 [arXiv:0801.1833 [hep-ph]]; J. F. Kamenik, Mod. Phys. Lett. A 29 (2014) 22, 1430021.
  * [18] R. Aaij et al. [LHCb Collaboration], Eur. Phys. J. C 73 (2013) 2373 [arXiv:1208.3355 [hep-ex]].
  * [19] T. Aushev, W. Bartel, A. Bondar, J. Brodzicka, T. E. Browder, P. Chang, Y. Chao and K. F. Chen et al., arXiv:1002.5012 [hep-ex].


# Sentiment-based Engagement Strategies for intuitive Human-Robot Interaction

Thorsten Hempel  , Laslo Dinges  and Ayoub Al-Hamadi
Neuro-Information Technology, Faculty of Electrical Engineering and
Information Technology,
Otto von Guericke University, Magdeburg, Germany
{thorsten.hempel, laslo.dinges<EMAIL_ADDRESS>
https://orcid.org/0000-0002-3621-7194 https://orcid.org/0000-0003-0364-8067
https://orcid.org/0000-0002-3632-2402

###### Abstract

Emotion expressions serve as important communicative signals and are crucial
cues in intuitive interactions between humans. Hence, it is essential to
include these fundamentals in robotic behavior strategies when interacting
with humans to promote mutual understanding and to reduce misjudgements. We
tackle this challenge by detecting and using the emotional state and attention
for a sentiment analysis of potential human interaction partners to select
well-adjusted engagement strategies. This way, we pave the way for more
intuitive human-robot interactions, as the robot’s action conforms to the
person’s mood and expectation. We propose four different engagement strategies
with implicit and explicit communication techniques that we implement on a
mobile robot platform for initial experiments.

##  1 Introduction

Once utopian, robots are increasingly moving from industrial and laboratory
settings to the real-world in order to assist humans in everyday tasks. In
this process, human-robot interaction became a central point of interest in
robotic research that investigates the manifold challenges in performing
interactive and collaborative tasks. As a general principle, the robot needs
to be acquired with the necessary skills to enable intuitive interactions with
arbitrary human interaction partners, regardless of the corresponding task,
the human’s intention and communication behavior. To this end, a fundamental
objective is the anticipation of appropriate strategies to proactively
approach or evade people based on the situation-specific context, such as the
person’s mood, attitude [Müller-Abdelrazeq et al., 2019, Elprama et al., 2016]
and ressentiments towards robots [Naneva et al., 2020, Miller et al., 2021].
First, it requires the determination of people’s interest in interacting at
all [Satake et al., 2009, Finke et al., 2005]. Thereupon, the robot has to
find an appropriate strategy for either engaging people to establish an
interaction or to avoid a confrontation if desired. In order to reach these
underlying decisions, the robot has to cope with a row of sub-tasks:
estimating people’s focus of attention [Dini et al., 2017], predicting their
mental state, and carrying out a (dis-)engagement strategy [Avelino et al.,
2021].

Especially, the estimation of the interaction willingness is a very
challenging task, as this mental state tends to provide only vague social
signals. As an alternative approach, many methods [Oertel et al., 2020] go
after the detection of engagement in order to determine if a person already
approached the robot and waits for the robot’s reaction. This state expresses
itself more clearly, e.g., in voice commands [Foster et al., 2017], gestures
and proxemics, but it undermines the proactive approach and neglects
situations where the human partner is in need of help, but too uncertain or
not aware of the robot.

In this work, we close the gap between the lack of proactive behavior and
advanced mental state predictions by introducing a sentiment-based engagement
strategy.

By combining the focus of attention and emotions, we derive a sentiment
analysis that allows us to select fine-grained behavior patterns that exceed
current binary engagement and disengagement strategies. We implement our
approach on a mobile robot platform and execute our strategies using explicit
and implicit embodied communication. Our models are trained on minimum sized
networks in order to perform our method on mobile systems with limited
computational resources.

##  2 Related Work

In recent years, proxemic rules have been a popular tool to draw conclusions
about the different states of human-robot interactions [Mumm and Mutlu, 2011]
and how spatial zones can be leverages to improve them [Syrdal et al., 2007].
[Repiso et al., 2018] predict appropriate encounter points to achieve a
natural engagement with a group of people. Similarly, [Satake et al., 2009]
proactively approach detected people to offer help. [Fischer et al., 2021]
verbally greet people and dynamically adapt the voice volume based on the
distance to the target person. Following [Kendon, 1990], [Carvalho et al.,
2021] apply the Kendon’s greeting model to approach people while tracking the
human mental states of the interaction using multiple features, such as gaze,
facial expressions, and gestures. Yet, all of these methods neglect if the
human counterpart is actually interested and ready to initiate an interaction.
This is problematic, as in the case of negative attitude, the robot’s
interaction efforts can be perceived as rude and annoying [Bröhl et al.,
2019]. An initial attempt to address this was presented by [Kato et al.,
2015], who only approach and offer help if they detect signs of attention
towards the robot. But this still doesn’t incorporate the actual mental
attitude of the human counterpart towards the interaction itself.

The recent advances in the area of deep-learning opened up new possibilities
for the visual analysis of humans, starting from general detection tasks to
the estimation of facial micro expressions. This allows the aggregation of
additional relevant information, such as emotions [Chuah and Yu, 2021,
Spezialetti et al., 2020], to better understand the behavior and intentions of
the human interaction partner. To the best of our knowledge, we are the first
to leverage this advancement by predicting sentiment states based on visual
emotion estimation to improve interaction strategy for robots. We summarize
our main contributions as follows:

  * •

We propose an image-based sentiment analysis to gauge the current interaction
preference of potential human interaction partners.

  * •

We propose four different behavior strategies (Engage, Attract, Ignore, Avoid)
that comprises a number of different explicit and implicit communication
modalities.

  * •

We design ultra lightweight models to implement our method and integrate it on
a mobile robot platform.

  * •

We conduct initial experiments in laboratory settings for an early evaluation.

##  3 Method

This section explains each step of our proposed method for sentiment-based
interaction behavior.

The sentiment analysis is based on two main features: head pose and emotion
estimation. The head pose estimation is a leading indicator to gauge the
current visual focus of attention of the person. The emotion estimation
predicts the current emotional state of the person. Together, it allows the
assumption about a person that a) is seeking for interaction, b) is indecisive
about it or c) doesn’t want to engage with the robot at all. Determining each
sentiment state enables to perform a dedicated robot behavior that corresponds
with the expected reaction from the person and, thus, improves an intuitive
interaction. Figure 2 illustrates an overview of our proposed system. In the
following subsection, we will give details about each component of the system
and its interplay.

###  3.1 Robot platform

Figure 1: The TIAGo robot platform with the key operations used in our method.

We use the TIAGo robot from PAL Robotics, a mobile service robot for indoor
environments. It is equipped with an RGB-D camera mounted inside its head,
that can be yawed and pitched to dynamically perceive the environment. For
grasping and moving objects, the robot has a manipulator arm with a 5 finger
gripper. The torso can be lifted to adjust the robot’s height, and integrated
speakers allow the output of voice commands. These abilities allow the
combination of explicit and implicit communication modalities that we use in
our method.

Figure 2: Method overview of our proposed system. Detected faces are used to
estimate the head pose and the corresponding emotions based on facial
expressions. Both information is used to determine the emotional state with
the corresponding visual focus of the person’s attention that indicates the
sentiment and allows a fine-tuned behavior strategy.

  * •

Head Following: The robot’s head aims at and follows a moving person as an
implicit signal of attentiveness without actively approaching.

  * •

Body Following: If a person exceeds the range of the robot’s moving head, the
entire robot base rotates in order to keep track of the person. This behavior
communicates an attention commitment and is therefore an even stronger signal
as the head following.

  * •

Torso Lifting: Lifting the torso is a rather subtle communication cue, as it
is perceived similarly to the social convention of standing up while greet
someone.

  * •

Speech: Approaching people with speech represents an explicit communication
strategy, that stronger commits them to a reaction. Hence, speech should be
only used if the approached person is highly expected to welcome an
interaction.

###  3.2 Visual Attention

Typical indicators for gauging the human visual attention are the gaze and,
more coarsely, the head pose. As the gaze depends on the eyes, which takes
only a small part in face images, it is prone to errors. Therefore, we
estimate the visual attention of the surrounding persons based on a head pose
prediction algorithm. At first, we locate faces in the image stream using an
ultra light SSD face detector. Then, the face crops are further processed by
the 6DRepNet [Hempel et al., 2022], that uses a rotation representation to
directly regress yaw, pitch, and roll angle of the faces. To assure real-time
processing capabilities, we replace the original 6DRepNet backbone with the
efficient MobileNetv3-Small [Howard et al., 2019]. The new head pose
prediction model is able to maintain 90% of its accuracy compared to the
original model, but is downsized to only one tenth of its parameters count.

###  3.3 Emotion Estimation

In order to assess the attitude towards robots as well as differentiated
subjective states such as dislike or willingness to cooperate, basic emotions
can be used as indicating features. Naturally, these are frequently
communicated by facial expressions. Hence, automatic facial expression
recognition is an essential method to generate features for Human-Robot-
Collaboration and attention prediction.

Previous facial expression recognition methods adhere to the established
pipeline of face detection, landmark extraction, and action unit (AU). Then,
emotions are classified using these AUs as feature vectors [Wegrzyn et al.,
2017, Vinkemeier et al., 2018, Werner et al., 2017]. However, end-to-end
learning on a single but more comprehensive, and better generalizing database,
often outperforms such traditional approaches. The AffectNet database
[Mollahosseini et al., 2017] for example contains about 500k in-the-wild
samples for neutral and the basic emotions happy, disgust, fear, surprise,
anger, and sadness. Furthermore, it is the only database, that also includes
ground truth of valence and arousal (VA), which are, in contrast to the basic
emotion classes, continuous labels $\in\left[-1,1\right]$ which can be used
for regression tasks. VA is less intuitively interpretable by humans, but it
allows differentiation between facial expressions of varying degrees. This is
particularly useful for capturing modest but long-term changes in expressed
emotion. In addition, a multitask network (simultaneous training of
classification and regression) also improved the accuracy of the
classification. Furthermore, Valence summarizes essential information about
several emotions in one parameter (negative, neutral, positive), which may
facilitate the development of heuristic behavior rules for some scenarios.

In our former work, we proposed a multitask network based on YOLOv3 [Dinges et
al., 2021]. For the current paper, we compared EfficientNetV2-S, ResNet18, and
MobileNetV3-Large. Despite the poorer converging training loss, MobileNet,
which is optimized for limited hardware, achieved almost the same accuracy on
the test set as EfficientNet (deviation $<1\%$). In addition, we used a
weighted random sampler to handle the imbalanced dataset and re-train the
network on the training and test set to generate an application model with
increased robustness to unseen data.

Figure 3: Behavior strategies based on the emotional and attentive state.

###  3.4 Sentiment Analysis and Approaching Strategy

Detecting emotional expressions combined with the visual focus of attention
allows perceiving important communicative signals. As in human-human
interaction, you may be more likely to greet a stranger who smiles at you but
avoid eye contact if that stranger is scowling. We transfer these principles
to determine states of sentiment based on the combination of the different
emotions and visual attention. Is a person showing visual signs of fear or
anger, while he is attentive towards the robot, it indicates a negative
attitude towards the robot. In this case, a passive robot behavior without
hectic movements can help to defuse the situation and to calm the person. If a
person is feared, but not attentive towards the robot, the reason for the fear
might not be the robot. Executing a more active robot behavior to show
awareness could be therefore a more suitable strategy. Hence, our sentiment
analysis allows us to compartmentalize the common hard binary engagement
strategy into soft behavior. Instead of choosing between engaging and not
engaging, we choose between four different behaviors: Engage, Attract, Avoid,
Ignore.

  * •

Engage: Engaging will extend the torso of the robot to signal the awareness of
the person’s presence. Additionally, voice commands will greet the people to
actively initiate an interaction. This strategy is called, when the attention
is directed at the robot with clear positive emotional attitude.

  * •

Attract: Positive emotional attitude without attention towards the robot leads
to more passive behavior. The goal is to signal the offer for interaction and
help without actively committing to an interaction. This is done by extending
the torso to show awareness, facing the person, but without verbally greeting.
This way, a non-verbal offer for support (especially in case of uncertainty
and sadness) is subtly given, that can be easily accepted or ignored.

  * •

Avoid: The avoiding strategy is executed when strong negative emotions (Fear,
Disgust, Anger) are facing the robot. In this case, the robot avoids eye
contact (keeps the face outside the camera center) and takes a passive part.
Especially in case of fear, the goal is to increase the feeling of safeness by
avoiding unexpected motions.

  * •

Ignore: If the negative emotions are not directed at the robot, the robot
mostly ignores the person, only following it with its head to capture changes
in emotional or attention states.

Figure 3 shows all combinations of emotional and attention states with
corresponding behavior strategy.

##  4 Implementation

We used the ROS111https://www.ros.org/ middleware to implement our proposed
solution on our robot platform, as this also offers a simplified integration
into other ROS-based robot platforms. Face detection, head pose and emotion
estimation are executed in separate nodes, that forward the necessary
information to perform the sentiment analysis and the action controls. While
the latter is performed on the robot itself, the inference of the neural
networks is outsourced to a separate notebook that is mounted on the shoulder
of the robot. This ensures the complete mobility of the robot, while
exploiting additional computational resources. The notebook has a Quadro RTX
4000 processing all models in parallel with the following inference times:

  * •

Face Detection: 6.7 ms

  * •

Head Pose Estimation: 1.4 ms

  * •

Emotion Estimation: 6.3 ms

The visual perception can be therefore performed in real time and is able to
continuously track people in the surroundings.

Figure 4: Sentiment analysis in a laboratory setting. The person is tracked
using a face detector, followed by a head pose and emotion estimation. As the
person shows attentiveness and happy, a positive sentiment towards the robot
is expected and a proactive initiation strategy selected.

##  5 Experiment

We conducted an initial experiment in a laboratory setting to test our
implemented method on the TIAGo robot platform. The objective of the
experiment was to gain experience about the model’s performances in real
application scenarios. Figure 4 shows a snapshot from one of our tests. The
robot platform detects a face in the environment and predicts head pose and
emotional state. The focus of attention is on the robot (indicated by the
green arrow) in conjunction with expressing happiness, so the robot initiates
an interaction by following the engage strategy.

While the models provide very robust predictions, the implementation is
currently entirely based on single shot detections. Yet, over the duration of
face tracking, facial expressions change intermediately and short distractions
make the head pose occasionally oscillating. This leads to abrupt changes of
states within short time periods. Hence, to improve the robustness of the
sentiment analysis, conclusions about the current state should incorporate the
predictions from multiple time frames.

##  6 Conclusion

In this paper, we present a sentiment-based methodology to improve the
intuitiveness and reasoning in human-robot interactions. Our method focuses on
the visual perception of mental states and the focus of attention to derive a
sentiment analysis for suitable interaction initiation strategies. We take
also into account the case where people are not interested in an engagement at
all, which has been mainly neglected in the recent literature. To implement
our method, we trained lightweight perception models that we integrate on a
mobile robot platform. Finally, we conducted initial experiments using a
mobile robot platform to analyze the performance of our models and the overall
system.

In the future, we will extend our work by embedding our sentiment analysis in
a temporal-probabilistic framework to improve the robustness in real-world
scenarios. Here, we will also incorporate the valence and arousal predictions
to evaluate the emotion’s intensity and transitions between emotional states.
Further enhancements offer the inclusion of the robot’s mobility to actively
engage and disengage people or to add additional interaction modalities, such
as following people by command, getting out of the way of passing people, or
approaching a place that offers a better overview of the surroundings.

Finally, we will conduct a study with test subjects to evaluate our approach
in terms of intuitiveness and sympathy.

## ACKNOWLEDGEMENTS

This work is funded and supported by the Federal Ministry of Education and
Research of Germany (BMBF) (AutoKoWAT-3DMAt under grant Nr. 13N16336) and
German Research Foundation (DFG) under grants Al 638/13-1, Al 638/14-1 and Al
638/15-1.

## REFERENCES

  * Avelino et al., 2021 Avelino, J., Garcia-Marques, L., Ventura, R., and Bernardino, A. (2021).  Break the ice: a survey on socially aware engagement for human–robot first encounters.  International Journal of Social Robotics, 13:1851 – 1877.
  * Bröhl et al., 2019 Bröhl, C., Nelles, J., Brandl, C., Mertens, A., and Nitsch, V. (2019).  Human-robot collaboration acceptance model: Development and comparison for germany, japan, china and the usa.  I. J. Social Robotics, 11:709–726.
  * Carvalho et al., 2021 Carvalho, M., Avelino, J., Bernardino, A., Ventura, R. M. M., and Moreno, P. (2021).  Human-robot greeting: tracking human greeting mental states and acting accordingly.  2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 1935–1941.
  * Chuah and Yu, 2021 Chuah, S. H.-W. and Yu, J. (2021).  The future of service: The power of emotion in human-robot interaction.  Journal of Retailing and Consumer Services, 61:102551.
  * Dinges et al., 2021 Dinges, L., Al-Hamadi, A., Hempel, T., and Al Aghbari, Z. (2021).  Using facial action recognition to evaluate user perception in aggravated hrc scenarios.  In 2021 12th International Symposium on Image and Signal Processing and Analysis (ISPA), pages 195–199.
  * Dini et al., 2017 Dini, A., Murko, C., Yahyanejad, S., Augsdörfer, U., Hofbaur, M., and Paletta, L. (2017).  Measurement and prediction of situation awareness in human-robot interaction based on a framework of probabilistic attention.  In 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 4354–4361.
  * Elprama et al., 2016 Elprama, S., El Makrini, I., Vanderborght, B., and Jacobs, A. (2016).  Acceptance of collaborative robots by factory workers: a pilot study on the role of social cues of anthropomorphic robots.
  * Finke et al., 2005 Finke, M., Koay, K. L., Dautenhahn, K., Nehaniv, C., Walters, M., and Saunders, J. (2005).  Hey, i’m over here - how can a robot attract people’s attention?  In ROMAN 2005. IEEE International Workshop on Robot and Human Interactive Communication, 2005., pages 7–12.
  * Fischer et al., 2021 Fischer, K., Naik, L., Langedijk, R. M., Baumann, T., Jelínek, M., and Palinko, O. (2021).  Initiating human-robot interactions using incremental speech adaptation.  In Companion of the 2021 ACM/IEEE International Conference on Human-Robot Interaction, HRI ’21 Companion, page 421–425, New York, NY, USA. Association for Computing Machinery.
  * Foster et al., 2017 Foster, M. E., Gaschler, A., and Giuliani, M. (2017).  Automatically classifying user engagement for dynamic multi-party human–robot interaction.  International Journal of Social Robotics, 9.
  * Hempel et al., 2022 Hempel, T., Abdelrahman, A. A., and Al-Hamadi, A. (2022).  6d rotation representation for unconstrained head pose estimation.  In 2022 IEEE International Conference on Image Processing (ICIP), pages 2496–2500.
  * Howard et al., 2019 Howard, A., Sandler, M., Chu, G., Chen, L.-C., Chen, B., Tan, M., Wang, W., Zhu, Y., Pang, R., Vasudevan, V., Le, Q. V., and Adam, H. (2019).  Searching for mobilenetv3.  In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV).
  * Kato et al., 2015 Kato, Y., Kanda, T., and Ishiguro, H. (2015).  May i help you? - design of human-like polite approaching behavior-.  In 2015 10th ACM/IEEE International Conference on Human-Robot Interaction (HRI), pages 35–42.
  * Kendon, 1990 Kendon, A. (1990).  Conducting Interaction: Patterns of Behavior in Focused Encounters.  Cambridge University Press, Cambridge, U.K.
  * Miller et al., 2021 Miller, L., Kraus, J., Babel, F., and Baumann, M. (2021).  More than a feeling—interrelation of trust layers in human-robot interaction and the role of user dispositions and state anxiety.  Frontiers in Psychology, 12.
  * Mollahosseini et al., 2017 Mollahosseini, A., Hasani, B., and Mahoor, M. H. (2017).  Affectnet: A database for facial expression, valence, and arousal computing in the wild.  IEEE Transactions on Affective Computing, 10:18–31.
  * Müller-Abdelrazeq et al., 2019 Müller-Abdelrazeq, S. L., Schönefeld, K., Haberstroh, M., and Hees, F. (2019).  Interacting with Collaborative Robots—A Study on Attitudes and Acceptance in Industrial Contexts, pages 101–117.  Springer International Publishing, Cham.
  * Mumm and Mutlu, 2011 Mumm, J. and Mutlu, B. (2011).  Human-robot proxemics: Physical and psychological distancing in human-robot interaction.  In 2011 6th ACM/IEEE International Conference on Human-Robot Interaction (HRI), pages 331–338.
  * Naneva et al., 2020 Naneva, S., Gou, M. S., Webb, T. L., and Prescott, T. J. (2020).  A systematic review of attitudes, anxiety, acceptance, and trust towards social robots.  International Journal of Social Robotics, pages 1–23.
  * Oertel et al., 2020 Oertel, C., Castellano, G., Chetouani, M., Nasir, J., Obaid, M., Pelachaud, C., and Peters, C. (2020).  Engagement in human-agent interaction: An overview.  Frontiers in Robotics and AI, 7.
  * Repiso et al., 2018 Repiso, E., Garrell, A., and Sanfeliu, A. (2018).  Robot approaching and engaging people in a human-robot companion framework.  In 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 8200–8205.
  * Satake et al., 2009 Satake, S., Kanda, T., Glas, D. F., Imai, M., Ishiguro, H., and Hagita, N. (2009).  How to approach humans?-strategies for social robots to initiate interaction.  In 2009 4th ACM/IEEE International Conference on Human-Robot Interaction (HRI), pages 109–116.
  * Spezialetti et al., 2020 Spezialetti, M., Placidi, G., and Rossi, S. (2020).  Emotion recognition for human-robot interaction: Recent advances and future perspectives.  Frontiers in Robotics and AI, 7.
  * Syrdal et al., 2007 Syrdal, D. S., Lee Koay, K., Walters, M. L., and Dautenhahn, K. (2007).  A personalized robot companion? - the role of individual differences on spatial preferences in hri scenarios.  In RO-MAN 2007 - The 16th IEEE International Symposium on Robot and Human Interactive Communication, pages 1143–1148.
  * Vinkemeier et al., 2018 Vinkemeier, D., Valstar, M. F., and Gratch, J. (2018).  Predicting folds in poker using action unit detectors and decision trees.  In FG, pages 504–511.
  * Wegrzyn et al., 2017 Wegrzyn, M., Vogt, M., Kireclioglu, B., Schneider, J., and Kissler, J. (2017).  Mapping the emotional face. How individual face parts contribute to successful emotion recognition.  PLOS ONE, 12(5):e0177239.
  * Werner et al., 2017 Werner, P., Handrich, S., and Al-Hamadi, A. (2017).  Facial action unit intensity estimation and feature relevance visualization with random regression forests.  In 2017 Seventh International Conference on Affective Computing and Intelligent Interaction (ACII), volume 2018-Janua, pages 401–406. IEEE.