CP Violation in D0 Decays A Signal of New Physics?
Master Thesis in Theoretical Physics
by
Philipp E. Frings
from Wesel-Büderich, Germany
August MMXII
Prepared at the
National Institute for Subatomic Physics (Nikhef)
under the Supervision of
Professor Robert Fleischer
Abstract:
The recent discovery of the Higgs boson represents the last success of a series of successes of the Standard Model (SM). However, there are observations that suggest
that the SM is incomplete, this motivates many physicists to search for new physics
(NP). A recent LHCb measurement deviates significantly from the naive Standard
Model expectation and might represent a sign of NP. The LHCb collaboration measured ∆ACP ≡ ACP (D0 → K + K − ) − ACP (D0 → π + π − ) = [−0.82 ± 0.21 ± 0.11]%,
where ACP (D0 → h+ h− ) with h ∈ {π, K} is a CP asymmetry that quantifies the
rate difference of the decays D0 → h+ h− and D̄0 → h+ h− . It is necessary to evaluate whether this deviation is actually due to new physics or whether it can be
accommodated in the SM. In this Master thesis, we discuss a possible theoretical
explanation of ∆ACP within the SM. First, we derive the SM expression of ∆ACP
and argue that it naively has a value at the 10−4 level. The main uncertainty of
this expectation for ∆ACP is due to unknown (hadronic) matrix elements which
are a consequence of the non-calculable long-distance contributions of the strong
interactions. However, different matrix elements of D decays can be related by the
SU(3) flavor symmetry so that this symmetry can be used to express all charm decay amplitudes by a small number of parameters (topologies). The SU(3) symmetry
holds only approximately and we find that especially for the decays D0 → π + π −
and D0 → K + K − we have to include SU(3)-breaking effects. We investigate three
different kinds of SU(3) breaking. This leads us to a framework in which a large
value for ∆ACP is possible, so that we find that ∆ACP could be accommodated in
the SM. However, we cannot exclude that ∆ACP is generated by NP.
3
Contents
1 Introduction
7
2 The Standard Model
2.1 Discrete Symmetries . . . . . . . . . . . . . . .
2.1.1 Discrete Symmetries in General . . . . .
2.1.2 CPT Invariance . . . . . . . . . . . . . .
2.2 CP Violation in the Standard Model . . . . . .
2.2.1 Introduction to the Standard Model . . .
2.2.2 Quantum Chromodynamics . . . . . . .
2.2.3 Electroweak Weak Symmetry Breaking .
2.2.4 Fermion Masses . . . . . . . . . . . . . .
2.3 The Cabibbo-Kobayashi-Maskawa Matrix . . . .
2.3.1 Degrees of Freedom in the CKM Matrix
2.3.2 Parametrizations of the CKM Matrix . .
2.3.3 Parameters in the Charm System . . . .
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3 Theory of CP Violation
3.1 CP Violation in Decay - Direct CP Violation . . . . . . . .
3.2 Indirect CP violation . . . . . . . . . . . . . . . . . . . . .
3.2.1 Time Evolution of a Neutral Meson System . . . .
3.2.2 The Decay Rate . . . . . . . . . . . . . . . . . . . .
3.2.3 CP Violation in Mixing . . . . . . . . . . . . . . . .
3.2.4 CP Violation in the Interference of Decay With and
3.3 CP Violation in Neutral D Decays . . . . . . . . . . . . . .
3.3.1 Numerical Values in for the Neutral Charm System
3.3.2 The Observable . . . . . . . . . . . . . . . . . . . .
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Mixing
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4 Weak Decays of D Mesons
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4.1 Low-Energy Effective Hamiltonians . . . . . . . . . . . . . . . . . . . . . . 48
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Contents
4.2
4.3
4.1.1 A Simple Example of an OPE . . . . . . . . . . . . . . . .
4.1.2 Low-Energy Effective Hamiltonian for CF and DCS Decays
4.1.3 Low-Energy Effective Hamiltonian for SCS Decays . . . . .
Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Leptonic Decays . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Semileptonic Decays . . . . . . . . . . . . . . . . . . . . .
4.2.3 Factorization of the Tree Amplitude . . . . . . . . . . . . .
The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 The SM Prediction for the Observable . . . . . . . . . . .
4.3.2 The Measurements . . . . . . . . . . . . . . . . . . . . . .
4.3.3 The Tension with the SM . . . . . . . . . . . . . . . . . .
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5 Symmetries of the Decay Amplitudes
5.1 The Topological Approach . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Flavor SU(3) Symmetry . . . . . . . . . . . . . . . . . . . . .
5.1.2 Decomposition of D Meson Decay Amplitudes into Topologies
5.1.3 Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Flavor Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Amplitude Ratios . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Large SU(3) Breaking in Exchange Topologies . . . . . . . . .
5.2.3 Large Penguins . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Conclusion
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A Abbreviations
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B Notation
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Acknowledgments - Danksagung
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Bibliography
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1 Introduction
Physicists investigate the fundamental laws of nature. These laws range from the behavior
of the tiniest known particles to the formation of the structure of the universe. The scales
we know range from 10−18 m, where we find the constituents of the nuclei, to 1026 m, which
is the size of the universe. Considering these large scale differences, it is not surprising
that a theory of everything has yet not been found.
In particle physics, however, the Standard Model (SM) gives explanations for a wide
range of experiments. At particle colliders, such as the Large Hadron Collider (LHC) at
CERN in Geneva, almost all processes observed so far are extremely well described by
the SM. Nevertheless, there are good reasons that suggest that the SM is incomplete.
Namely, the SM faces conceptual problems like the hierarchy problem and does not give a
candidate particle for dark matter. Additionally, the SM does not explain why we find so
little antimatter in our universe. Therefore, it is clear that the SM needs to be extended
or replaced in the future.
Hence, many particle physicists are searching for phenomena which deviate from the
SM, generally called new physics (NP). Experimentally, this can be done in a direct way
by searching for new particles and new decays processes or indirectly by searching for
deviations in established SM processes that are sensitive to NP.
In the past, the decay of long-lived kaons into two pions, observed by Cristenson et al.
in 1964 [1], was such evidence of NP. At that time, it was thought that long-lived kaons
could not decay into two pions since this would violate the charge-parity (CP) symmetry.
Even though parity (P) violation had been found by Wu et al. [2], it was thought that
the combined symmetry CP was a fundamental symmetry of nature, since charge (C)
violation occurs in such a way that it cancels P violation. In 1973, the explanation for
this NP was given by Kobayashi and Maskawa [3]. Their explanation implies that at least
six quarks must exist, so that they predicted three new quarks in addition to the three
quarks, which were known at that time. This remarkable prediction lead to the award of
the Nobel Prize to Kobayashi and Maskawa in the year 2008.
Today, indirect search for NP is, for example, done at the LHCb experiment, which is
one of the four large experiments at the LHC and investigates mainly the decay of particles
7
8
1 Introduction
called B mesons. The LHCb collaboration investigates observables that are sensitive to
NP contributions, which includes CP asymmetries such as
ACP ≡
Γ(P → f ) − Γ(P̄ → f¯)
.
Γ(P → f ) + Γ(P̄ → f¯)
(1.1)
The CP symmetry relates matter and antimatter, ACP quantifies consequently the difference between matter and antimatter for the decay of a particle P to a final state f .
In general, there are three ways in which a CP asymmetry can be generated. One
results from the interference of different decay amplitudes that contribute to a decay,
generally referred to as direct CP violation. Another source of CP violation is linked to
the fact that a neutral particle can transform into its antiparticle, this is generally called
CP violation in mixing. At last, there is the interference of the decay of a particle with
and without mixing that can generate CP violation. The latter two kinds of CP violation
are sometimes referred to as indirect CP violation.
CP asymmetries are sensitive to NP because new particles can easily introduce new CP
violation. Especially because these new particles do not need to be produced directly, it
is sufficient if they contribute to a process at loop-order to generate CP violation. Hence,
if one finds in an experiment that an observable deviates from the theoretical SM expectation, this indicates that NP could have been found. Recently, such a deviation has been
found by the LHCb collaboration in the decay of the D0 meson, which is called a charm
meson since it contains one charm quark D0 = cū.
First theoretical studies on CP asymmetries for the decays of charmed particles were
preformed by Golden and Grinstein who found that relatively large asymmetries are possible [4], and Buccella et al., who concluded that asymmetries in the charm system would be
small [5]. Generally, the expectation is that CP violation is in the range of O(10−4 ) in the
decays of charmed particles. Therefore, it was surprising when the LHCb collaboration
announced the following result [6]:
∆ACP ≡ ACP (D0 → K + K − ) − ACP (D0 → π + π − )
= [−0.82 ± 0.21(stat.) ± 0.11(sys.)]%,
(1.2)
which represents the difference of two time-integrated CP asymmetries and benefits from
systematic uncertainty cancellations. Additionally, the contribution of indirect CP violation largely cancels out in the difference, as discussed in Refs. [7–9]. Thus, ∆ACP mainly
consists of the difference of direct CP asymmetries. A similar CDF measurement supports
9
1 Introduction
this result [10]
∆ACP = [−0.62 ± 0.21(stat.) ± 0.10(sys.)]%.
(1.3)
It is clear that these measurements are in tension with the usual SM expectations, so
that a great number of authors dealt with this subject. In general, we might divide these
studies in the two possible interpretations of these measurements. One possibility is to
discuss NP models which would yield this large value, this has been done, for instance,
in Refs. [11–21]. However, we will not delve into this subject, but reevaluate the SM
prediction and investigate whether the experimental value can be accommodated within
the SM.
In the SM, the decay of charmed particles is described by an effective Hamiltonian
which describes the interplay of the weak and the strong interactions. Because of the
non-perturbative long-distance contributions of quantum chromodynamics (QCD) one
encounters (hadronic) matrix elements that cannot be evaluated in the SM. Nevertheless,
it is possible to relate the hadronic matrix elements of different charm decays by certain
symmetries.
Most studies exploited these symmetries to investigate whether or not it is possible to
accommodate the value of ∆ACP in the SM. The authors of Ref. [22], for example, used
the flavor SU(3) symmetry and analyzed SU(3) breaking based on the earlier work of Refs.
[4] and [23], they concluded that the measurement is compatible with the SM. Based on
the symmetry SU(3) it is also possible to make a topological approach which was used in
[24, 25] and [26] to understand the decay amplitudes which contribute to these decays.
In this topological approach, using the SU(3) symmetry, one can parametrize all decay
amplitudes with a small set of parameters, called topologies, which can be extracted from
some known branching ratios and then used to predict other branching ratios. This allows
a better understanding of the decay amplitudes that generate ∆ACP .
A symmetry described by the subgroup SU(2) of SU(3) is U-spin, it relates the d and
the s quark, U-spin breaking was used by Brod et al. [27], who found that ∆ACP could
be explained within the SM.
It is also possible to ignore the long-distance effects and to evaluate only the size of
the short-distance contributions this is done in Ref. [28], where one finds an analysis of
the size of the amplitudes which generate ∆ACP in the SM, whereas Ref. [29] considers
SM and NP effects for this analysis. In total, most analyses find that the measurement is
marginally or not at all compatible with the SM.
In this work, we will investigate two distinct topics. Firstly, we have a close look at
the indirect CP violation which contributes to the observable ∆ACP . We are going to
1 Introduction
10
investigate the time dependence of the most general CP asymmetry and reduce it to the
relevant expression in the charm system. We will show that the asymmetry mostly consists of direct CP violation, which is generated by the interference of two different decay
amplitudes that contribute to one decay. Therefore, we analyze the decay amplitudes
using the topological approach. We will see that an exact SU(3) symmetry is insufficient
to describe all charm decays, hence, we will investigate how SU(3) breaking can be taken
into account, this will also enable us to explain a large value for ∆ACP .
The observation of large ∆ACP is one of the few intriguing deviations from the SM.
Therefore, it is necessary to investigate whether, against naive expectations, ∆ACP can
be accommodated in the SM in order to see whether ∆ACP is actually a sign of NP. We
will take up this task in this thesis.
Outline
This thesis has four main chapters. In Chapter 2, a general particle physics introduction,
which is appropriate for our means, is given. This mainly includes an explanation of the
three discrete symmetries C, P and T and how CP is broken in the SM. Hence, we introduce the SM in order to show where CP violation can occur. In this part we establish
some conventions and give the numerical values for the CP violating parameters in the
SM, which are used throughout this thesis.
In Chapter 3, we introduce the different kinds of CP violation and we discuss neutral
meson oscillations. This enables us to find the most general time-dependent CP asymmetry, we use then approximations, which are valid in the charm system, to come to an
expression for ∆ACP . We also find that ∆ACP mainly consists of the difference of the
direct CP asymmetries which are generated by the interference of different decay amplitudes that contribute to one decay. Therefore, we discuss the decay amplitudes whose
interference generates ∆ACP in Chapter 4, this is done by using low-energy effective
Hamiltonians. We establish the relevant Hamiltonian Hef f and use it to find the naive
SM expectation for ∆ACP . We find that the SM expression of ∆ACP is in tension with
the measurement and formulate the problem which we want to solve.
In Chapter 5, we introduce the topological approach and explore how we can describe
the branching ratio (BR) data of charm decays if we exploit the SU(3) symmetry which
Hef f exhibits. We will find that an exact SU(3) symmetry does not appropriately describe
the data so that we use SU(3) breaking. We explore three different kinds of SU(3) breaking and find that one sort of SU(3) breaking can generate a sizable ∆ACP and explain
1 Introduction
11
the BR data.
Finally, we conclude in Chapter 6 and give a brief outlook. The conventions adapted
as well as a list of abbreviations can be found in the Appendix.
2 The Standard Model
In particle physics, one intends to describe the behavior of the smallest building blocks of
matter, elementary particles, the SM does this very successfully. It is a quantum field theory that is invariant under three fundamental continuous symmetries SU (3)C ⊗ SU (2)I ⊗
U (1)Y . C refers to the color, I to the weak isospin and Y to the hypercharge symmetry.
The known elementary particles quarks and leptons are described by matter fields and
they interact via massive and massless gauge bosons. The SM symmetry is spontaneously
broken to the symmetry SU (3)C ⊗ U (1)Q , where Q is the electromagnetic charge. Apart
from the continuous symmetries there are three discrete symmetries: Charge-symmetry
C, parity P and time reversal T. A wide range of processes of elementary particles is
invariant under these transformations, nevertheless in some processes the symmetries C,
P and T are explicitly and maximally violated, which causes CP violation. Nevertheless,
the combination CPT is conserved in the SM.
First, we introduce these discrete symmetries and we comment on CPT invariance, then
a layout of the SM will be given in order demonstrate how CP is violated and how the
CKM-mechanism describes CP violation. The phenomenological consequences and manifestations of CP violation will be described in the next chapter. The discussion in this
chapter is mostly text book material and based on the textbooks [30] and [31, 32].
2.1 Discrete Symmetries
In this section, we are going to set up some general properties of the discrete symmetries.
In general, here and below, most of the properties just will be given without showing the
proofs. The continuous symmetries and the SM will be described in the next section.
2.1.1 Discrete Symmetries in General
In the SM, we know three discrete symmetries C, the particle-antiparticle symmetry, P,
parity, and T, time reversal. P and T are known for a long time from classical mechanics. C arose with relativistic quantum mechanics when antiparticles were recognized as
13
2 The Standard Model
14
physical objects.
Parity P
Classically, the parity P is a reflection in the origin, it amounts to a change of all space
coordinates from ~r to −~r, this is also called space inversion. Practically, a right-handed
coordinate system is changed by this symmetry into a left-handed one. Parity can also be
understood as a 180◦ -rotation followed by a simple reflection. Phrased differently, parity
is looking at the mirror image of a physical process, if this mirror image is a possible
physical process then parity is a good symmetry.
Vectors are defined as objects that transform under rotations just like ~r but it is unclear
how the parity operator P acts on a vector. There are two different possibilities a vector
can change its sign under parity, this kind of vector is called a (polar) vector. The vector
can also just remain invariant, this type of vector is called pseudovector. A product of
a vector and a pseudovector changes sign under parity and is called a pseudoscalar in
contrast to an usual scalar which remains invariant under parity.
Time Reversal T
Classically, time reversal is a change of the time coordinate t to -t. The question the T
symmetry raises is whether a physical process, if it happens backwards in time, is still
possible. Macroscopically, this is usually not possible because of thermodynamics but
microscopically T is a good symmetry of the classical equations of motion.
Charge conjugation C
The particle-antiparticle symmetry C has no classical equivalent, since the concept of antiparticles only occurs in relativistic quantum theories. The question C raises is, whether
a physical process still happens in the same way if one replaces all particles by their
antiparticles.
2.1.2 CPT Invariance
It has been proven [33] that a broad class of quantum field theories is invariant under the
combined transformation CPT , where the curly letters represent the respective operators.
The SM belongs to this class, this implies that if a part of this symmetry is broken, as,
15
2 The Standard Model
for example, in the case of CP violation, also the complement symmetry T needs to be
violated. Another important consequence of the CPT theorem is that a particle and an
antiparticle have an exactly equal and opposite charges and magnetic dipole moments.
The mass and total decay width of a particle and its antiparticle are identical
M (P ) = M (P̄ ),
Γ(P → anything) = Γ(P̄ → anything).
(2.1)
Yet, CPT invariance is just a mathematical property of the SM, so that it is not clear
whether CPT is actually conserved in nature. CPT violation is probed by measuring the
difference of particle and antiparticle masses, but up to date no CPT violation has been
found [34].
2.2 CP Violation in the Standard Model
In the previous section, some basic notions about the symmetries C, P and T were given.
In this section, we will first point out the structure of the SM and then see how C, P and
CP violation are implemented in the SM.
2.2.1 Introduction to the Standard Model
There is by now evidence for three different kinds of elementary particles in nature, these
are quarks, leptons and gauge bosons. Quarks and leptons have spin one half and are
thus fermions, the gauge bosons have spin one. All their interactions1 are described by
the SM Lagrangian which is based on the symmetry group SU (3)C ⊗ SU (2)I ⊗ U (1)Y .
Myriad results of experiments have been predicted with excellent precision using this one
formula. Physically, the symmetry groups give rise to three forces, which are mediated
by the gauge bosons. We distinguish:
• The photon γ, which mediates the electromagnetic force and couples to charged
particles only.
• The charged weak bosons W ± and the uncharged weak boson Z, which mediate the
weak interaction, generate CP violation (only W ± ) and couple to all particles.
• Eight gluons which mediate the strong interaction and only couple to quarks and
themselves.
1
All interactions relevant for this thesis. Gravity is not described by the SM but can safely be neglected.
16
2 The Standard Model
• The Higgs boson which generates electroweak symmetry breaking (EWSB) and gives
rise to the particle masses. Therefore, it couples to all particles which have a mass.
The quarks and leptons can be grouped in generations (or families) as is indicated in Table
2.1. We see that there are six quarks, three with a positive charge +2/3 and three with
Generation I Generation II
Up-type Quarks
u
c
Down-type Quarks
d
s
−
Charged Leptons
e
µ−
Neutral Leptons
νe
νµ
Generation III
t
b
τ−
ντ
Table 2.1: Particle content of the SM. The second and third generation are a heavier
copy of the first generation, they have the same quantum numbers as the first
generation so that they behave exactly like the first generation, when it comes
to the interactions. The flavor quantum number which distinguishes the copies
from the first generation does not have an influence on how the interactions
act on these particles.
the negative charge −1/3. Moreover, there are three charged leptons and three uncharged
leptons called neutrinos. Generation II and III are heavier copies of generation I, almost
all existing visible matter is composed by generation I only. To every particle there exists
an antiparticle with opposite charge and opposite charge-like quantum numbers but with
equal mass and spin. Whereas leptons appear as free particles in nature, quarks are
always confined in baryons, combinations of three quarks, or mesons, a pair of a quark
and an antiquark. In this thesis, the following pseudoscalar mesons will occur: The pions
and kaons
Pion
π+
Quark content ud¯
π0
π−
¯ dū
√1 (uū − dd)
2
Kaon
K + K 0 K̄ 0 K −
¯
Quark content us̄ ds̄ ds
ūs
(2.2)
The charm mesons
D meson
D0
Quark content cū
D̄0
c̄u
D+
cd¯
D−
c̄d
Ds+
cs̄
B̄d0
¯
db
Bs0
sb̄
B̄s0
s̄b
Ds−
c̄s
And at last the B mesons
B meson
Bd0
Quark content db̄
We will encounter some more spinless particles with odd parity (η, η ′ , KS0 , KL0 ) but they
have a more complicated valence quark structure so that we will introduce them in the
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2 The Standard Model
context where we need them in Section 5.1.1. There are some more B mesons but they are
not relevant for this thesis, all vector mesons and baryons are widely ignored in this thesis
and are not necessary for the discussion of ∆ACP which we want to conduct. Having
introduced the relevant particles we come to their interactions, where we go back to the
matter fields that describes quarks and leptons. In order to see how they couple to the
three different interactions we need to introduce left and right-handed fields
1
ψR/L = (1 ± γ 5 )ψ.
2
(2.3)
Left-handed particles are weak isospin doublets. Right-handed particles are isospin singlets. In the SM, there do not exist right-handed neutrinos or left-handed antineutrinos.
This gives rise to P and C violation, since the parity-conjugated reactions involving neutrinos are impossible. Furthermore, this leads to charge violation, since processes that
involve neutrinos always involve also a charged leptonic partner. Nevertheless, this C and
P violation always arises simultaneously so that the combined CP-conjugated reactions
are always possible and CP is conserved. For our purposes, we can largely ignore leptons,
since no leptonic CP violation occurs in the SM.
In Table 2.2 is summarized how the quarks couple to the different symmetries. These
quantum numbers are the generalized charges for the respective interactions. For the
spontaneously broken part of the Lagrangian, they effectively add up to the electromagnetic charge.
uL
dL
uR
dR
Symmetry
SU (3)C
Quantum number Color
r,g,b
r,g,b
r,g,b
r,g,b
SU (2)I
T3
+1/2
−1/2
0
0
U (1)Y
Y
+1/6
+1/6
+2/3
−1/3
U (1)Q
Q
+2/3
−1/3
+2/3
−1/3
Table 2.2: Quantum numbers for the first quark generation. The electromagnetic charge
Q is related to the third isospin component and the weak hypercharge by
Q = T3 + Y
After having described the matter content of the SM and the couplings of the different
particles, we are ready to explore the structure of the SM Lagrangian. All the particles
are described by matter fields, the gauge bosons are described with vector fields and the
fermions are described with spinors. The SM Lagrangian can be written down in the
18
2 The Standard Model
following decomposition
LSM = LQCD + LGW S + Lf ermion + LY ukawa .
(2.4)
LQCD describes the strong interactions, that is the dynamics of the gluons and how the
gluons act on the quarks and themselves. LQCD is explained in Section 2.2.2. LGW S
describes the weak interaction and the electromagnetic interaction and how they act on
the fermions. Furthermore, it describes how the Higgs field generates EWSB, this part of
the SM will further be elucidated in Section 2.2.3. The Lagrangian Lf ermion describes the
leptons and quarks. The invariance of the matter fields under the gauge groups gives rise
to the coupling to the gauge fields and describes therefore how the particles interact. At
last, LY ukawa describes how the fermions acquire their mass by the Higgs field taking a
non-trivial ground state. This can be written down in such a way that complex couplings
occur in LGW S this will be shown in Section 2.2.4.
2.2.2 Quantum Chromodynamics
Quantum Chromodynamics (QCD) describes the strong interactions which affects gluons
and quarks. QCD is based on the non-abelian gauge group SU (3)C , where C stands for
color. Color is the charge on which QCD acts, it can be red, green or blue. Since SU (3)C is
non-abelian not only the quarks carry color but also the mediators of the strong force the
gluons this leads to gluons coupling to each other and hence to concepts like confinement
and asymptotic freedom. Asymptotic freedom means that the coupling constant decreases
with higher energy and increases with lower energy, consequently perturbative calculations
are not possible for low energies in QCD. This has far-reaching consequences whenever
hadronic particles are in play, so that the non-perturbativity of QCD will be one of the
main problems of this thesis. The property of confinement leads to the fact that we do
not observe single quarks but only mesons and baryons. An extensive treatment of QCD
can be found in a book by Muta [35]. The Lagrangian of QCD can be written as
1
a
LQCD = − Gaµν Ga,µν + gS Q̄α γ µ Tαβ
Aaµ Qβ .
4
(2.5)
Here Gaµν is the field strength tensor and defined as
Gaµν = ∂µ Aaν − ∂ν Aaµ + gS f abc Abµ Acν .
(2.6)
19
2 The Standard Model
Aaµ is the vector field representing the gluon and gS is the strong coupling constant. Qα is
a column vector of six quark fields which correspond to the six flavors, the quark fields are
color triplets, so that the indices α and β run over the colors. The T a are the generators
of the group SU (3)C . The sum over repeated indices is understood. CP violation can
only occur if there are complex couplings in the Lagrangian, but gS is real if the Aaµ
are Hermitian fields, so that there is no CP violation in QCD. Nevertheless, there is the
possibility of a renormalizable operator that violates CP
ǫµνρσ Gaµν Gaρσ .
(2.7)
This operator is part of the SM but its coupling constant must be < 10−10 , since there is
no experimental evidence for this operator to date, this requires that the coupling of this
operator is fine-tuned. Why this coupling is so small is not understood and this subject
is usually referred to as the strong CP problem. The best limit of searches for strong
CP violation is the measurement of the electric dipole moment of the neutron dn whose
current upper limit is [36]
|dn | < 2.9 × 10−26 e cm.
(2.8)
2.2.3 Electroweak Weak Symmetry Breaking
In this part we will describe how the symmetry SU (2)I ⊗ U (1)Y is broken to U (1)Q . We
do this in order to find how the weak gauge bosons W ± , Z 0 and the photon γ couple to
the fermions in the SM. This framework describes the weak interaction, which acts on
all particles, and the electromagnetic interaction, which acts on charged particles only.
It was established by Glashow, Weinberg and Salam [37–39] and is therefore also known
as the GWS model. Within the GWS model the Higgs mechanism [40–44] explains how
the initially massless gauge bosons acquire mass in a spontaneous way. The photon the
gauge boson of the electromagnetic interaction remains massless. We begin by writing
down the Lagrangian for the gauge fields and the Higgs Boson. Wµa , with a running
over 1,2,3, and Bµ are the gauge fields corresponding to the SU (2)I and U (1)Y symmetry,
⊥
respectively. The Higgs field is a scalar doublet φ = (φ+ , φ0 ) and subject to the potential
V (φ) = −m2 |φ|2 + λ2 |φ|4
LGW S = −
¢
1 ¡ a a,µν
Wµν W
+ Bµν B µν + |Dµ φ|2 + V (φ).
4
(2.9)
20
2 The Standard Model
Where the field strength tensors are defined by
a
Wµν
= ∂µ Wνa − ∂µ Wνa + gǫabc Wµb Wνc ,
Bµν = ∂µ Bν − ∂ν Bµ
(2.10)
and the covariant derivative is
Dµ = ∂µ − ig ′ Bµ − igτ a Wµa ,
(2.11)
and gives rise to the coupling of the Higgs boson to the gauge bosons. If the coupling
2
constants in the
q Higgs potential, m and λ are positive, the Higgs field takes its minimum
2
at |φ| = v ≡ mλ . One expands then the Higgs field around this minimum and three
linear combinations of the gauge bosons get a mass. These linear combinations are
¢
1 ¡
Wµ± = √ Wµ1 ∓ iWµ2
2
¡
¢
1
Zµ = p
gWµ3 − g ′ Bµ
g 2 + g ′2
with mass
with mass
v
mW = g ,
2
p
v
mZ = g 2 + g ′2 .
2
They correspond to the physical W and Z bosons. Of course, there is a fourth linear
combination of fields, this linear combination stays massless and represents the photon
¡ ′ 3
¢
1
g Wµ + gBµ .
Aµ = p
g 2 + g ′2
(2.12)
Since the photon is massless, it still provides a residual U (1)Q gauge symmetry, which
gives rise to the electromagnetic interaction. The linear combinations of the W 1 and W 2 ,
W ± , are defined in such a way that they carry an electromagnetic charge and give rise
to charged currents. The Z boson is neutral and gives rise to neutral currents. The
experimental evidence of neutral currents was the first evidence for the correctness of the
GWS model.
If it comes to the Higgs boson, there has only recently been strong evidence that the
massive excitation of the Higgs field, the Higgs boson, has been found, this is why we
would like to digress into this topic. Already in December 2012 the LHC experiments
CMS [45] and ATLAS [46] published data that pointed to a possible detection. Today
the discovery of a new particle, that has a strong resemblance with the Higgs particle, is
certain. In the decay channels, that have the best mass resolution, the decay of a Higgs
boson to two photons H → γγ and the decay into four leptons (electrons and/or muons),
evidence for the Higgs boson well above 5 sigma level has been found [47, 48]. Figure 2.1
2 The Standard Model
21
Figure 2.1: Left: Higgs exclusion plot taken from [49]. The black solid line indicates
the observed normalized total cross section. The excess of the observed cross
section with respect to the expected cross section at 126GeV hints to a Higgs
boson of that mass. Right: Measured two photon events [50] in contrast
to the expected two photon events, CMS sees an excess at 125GeV of 4.1
standard deviations. Both experiments find independently a 5σ evidence in
their combined data.
illustrates the results with a Higgs exclusion plot provided by the ATLAS collaboration
that shows a resonance at 126 GeV and excludes a Higgs boson for most other accessible
energies and a plot of the CMS collaboration that shows a clear resonance in the two
photon channel around 125.3 GeV. This also shows that the particle found, must be a
boson and cannot have spin one. All the current data points to the detection of a SM
Higgs boson. More properties of the detected particle will be investigated until the end
of 2012 when the LHC will be shut down for a major upgrade.
Let us come back to the analysis of LGW S , we want to find where CP violating terms
can appear. The gauge couplings g and g ′ can always be chosen real, so also in this part
of the SM Lagrangian we cannot get CP violating terms. However, there is also a Yukawa
coupling of the Higgs boson to the fermions which gives rise to the fermion masses after
spontaneous symmetry breaking. We will see that we cannot make all couplings appearing
in the SM real, it will be the couplings of the fermions to the gauge fields Wµ± which we
will choose to be complex as we will see in the following.
We want to rewrite the covariant derivative Eqn. (2.11) in terms of the physical fields.
In order to do this we will first define some new quantities. The weak angle θw is the
22
2 The Standard Model
rotation angle between the basis of (W 3 , B) and (Z 0 , A)
Ã
Z0
A
!
=
Ã
cos θw − sin θw
sin θw cos θw
!Ã
W3
B
!
.
(2.13)
The electric charge entering is a combination of the couplings to SU (2)I and U (1)Y
e= p
gg ′
g 2 + g ′2
.
(2.14)
This gives also rise to the electric charge quantum number
Q = T3 + Y
(2.15)
where T 3 is the the third component of the weak isospin and Y is the hypercharge. At
last, we introduce the lowering and raising operators of the isospin group T ± = T 1 ± iT 2 .
Now we may write the covariant derivative in terms of the physical fields as
¢
g
g ¡
Zµ (T 3 − sin2 θw Q) − ieAµ Q.
Dµ = ∂µ − i √ Wµ+ T + + Wµ− T − − i
cos
θ
2
w
(2.16)
This equation concludes this subsection which dealt with the gauge bosons of SU (2)I ⊗
U (1)Y and how the Higgs field spontaneously breaks this symmetry. In the following
subsection the effect of electroweak symmetry breaking on the fermions is described.
2.2.4 Fermion Masses
Until now, we did not discuss how the fermions interact with the field in the GWS model,
this will be taken up now. We first discuss this for the first quark generation and generalize
to all three quark generations later. We will not consider terms that involve leptons,
such as e, νe , since there is no leptonic CP violation because of the masslessness of the
neutrinos. The left-handed quarks of the first generation are given in an isospin doublet
QL = (uL , dL ) and the right-handed quarks are two isospin singlets uR and dR . The
SU (2)I ⊗ U (1)Y gauge-invariant Lagrangian can be written as
/ L + iūR Du
/ R + id¯R Dd
/ R.
Lf ermion = iQ̄L DQ
(2.17)
/ = γ µ Dµ includes the coupling to the gauge fields as in Eqn. (2.16). We can
where D
decompose Lf ermion in terms of a kinetic Lagrangian and in two Lagrangians that describe
23
2 The Standard Model
the charged and the neutral currents
Lf ermion = Lkin + LN C + LCC
Lkin = iū∂/u + id¯∂/d,
with
(2.18)
(2.19)
LN C = eJµem Aµ + gJµZ Z µ ,
¢
g ¡
LCC = √ Jµ+ W +,µ + Jµ+ W +,µ .
2
(2.20)
(2.21)
In the kinetic Lagrangian, which describes free quark fields, left and right-handed fields
are recombined in one field, but this does not mean that left and right-handed fields
couple to each other here. In LN C J em and J Z appear, they are the quark currents which
couple to the neutral gauge bosons and are of no further interest. The charged current
is Jµ+ = ŪL γµ DL with Jµ− = (Jµ+ )† , one finds that in the charged-current Lagrangian up
quarks couple to down quarks, this is not surprising since the W bosons carry a charge
and charge must be conserved. Therefore, there is no such coupling in LN C . It seems that
also here no CP violation could occur since the coupling constants g and g ′ need to be
real, if Wµa and Aµ are Hermitian. However, there is the Yukawa coupling in the Yukawa
Lagrangian which we may write as
LY ukawa =
X
i,j
¡
¢
(GU )ij Ūi,L , D̄i,L
Ã
φ0
φ−
!
¡
Uj,R + (GD )ij Ūi,L , D̄i,L
¢
Ã
φ+
φ0
!
Dj,R + H.c.,
(2.22)
where we reintroduced the generation structure, the capital letters indicate three component vectors with i,j running over the three possible generations. The vector UL , for
example, means UL = (uL , cL , tL )⊥ . When the Higgs field has a non-zero expectation
value φ0 = v, then the fermions acquire a mass through this term which can be written
as
MU = GU v and MD = GD v.
(2.23)
MU and MD are mass matrices and arbitrary 3 × 3 matrices in the SM, since the Yukawa
(GU )ij and (GD )ij couplings are arbitrary. However, non-diagonal mass matrices are
inconvenient, since the mass eigenstates are associated to the propagating particles. Thus,
in order to obtain the mass eigenstates, we are going to diagonalize the mass matrices
using the following transformations on the quark fields
′
UL/R = ÛL/R UL/R
,
′
DL/R = V̂L/R DL/R
.
(2.24)
24
2 The Standard Model
These rotations in the flavor space rotate the quark vectors in such a way that the mass
matrices become diagonal
MUdiag = ÛL† MU ÛR = Diag(mu , mc , mt ),
MDdiag = V̂L† MD V̂R = Diag(md , ms , mb ). (2.25)
The terms in LN C and LQCD are invariant under this rotation, yet, in LCC more exactly
in J + and J − the basis change introduces the matrix
VCKM = ÛL V̂L† ,
(2.26)
where the label CKM stands for Cabibbo-Kobayashi-Maskawa [3] the founders of this
theory. Since VCKM consists of unitary rotation matrices, it is unitary as well. Since
the rotation matrices were introduced to diagonalize the arbitrary mass matrices, VCKM
contains a priori complex values and thus gives rise to CP violation. For reference we write
down the CP violating part of the SM which is LCC after choosing the mass eigenstates
¢
g ¡
∗
LCC = √ Wµ− ŪL γ µ VCKM DL + Wµ+ D̄L γ µ VCKM
UL .
2
(2.27)
This means that there are two relevant sets of eigenstates for the matter fields in the
SM: The weak interaction eigenstates and the eigenstates which are the eigenstates of
the electromagnetic and strong interaction and which are also the mass eigenstates. This
leads to the feature that the weak interaction does not conserve the flavor quantum
number, which distinguishes the three quark generations. Nevertheless, one can choose
the rotations in Eqn. (2.24) in such a way that either the up or the down states are still
flavor and mass eigenstates. Usually, one chooses the up-states to be the same so that the
following relation holds for the flavor down eigenstates and the mass down eigenstates:
 
 
d
Vud Vus Vub
d′
 
 ′  
 s  =  Vcd Vcs Vcb   s  .
b
Vtd Vts Vtb
b′

(2.28)
The indices, as for instance ud, indicate that in Eqn. (2.27) the coupling between an up
quark and a down quark is proportional to Vud .
2 The Standard Model
25
2.3 The Cabibbo-Kobayashi-Maskawa Matrix
In this section, we are going to discuss how the structure of VCKM can be described. We
will show that there is only one complex parameter which governs CP violation in the
SM. We will also see how strong the coupling between different generations is. Moreover,
two different parametrizations will be introduced and we will define the combinations of
CKM matrix entries that are relevant in the charm system and attribute numerical values
to them.
2.3.1 Degrees of Freedom in the CKM Matrix
The CKM matrix in a theory with n generations and therefore n dimensions is a complex
n × n matrix which means that it has 2n2 degrees of freedom. Unitarity U † = U −1 ⇔
U U † = 1 gives n2 constraints on this. Leaving n2 free parameters. For every generation
there are two quark fields whose unphysical phases eliminate a free phase each in VCKM .
However, there remains one overall phase which cannot be used. Consequently, the matrix
has n2 − (2n − 1) = (n − 1)2 parameters. An orthogonal (real) n × n matrix has n2 (n − 1)
real parameters (angles), so that the number of complex parameters is given by the
number of parameters (n − 1)2 minus the number of real angles. The number of real
and complex parameters for different numbers of generations is given in Table 2.3. Table
# Generations
1 2 3 4 n
free parameters in CKM matrix 0 1 4 9 (n − 1)2
real parameters in CKM matrix 0 1 3 6 n(n − 1)/2
complex phases in CKM matrix 0 0 1 3 (n − 1)(n − 2)/2
Table 2.3: Degrees of freedom for VCKM in different dimensions. In the SM, CP violation
is only possible if there are three or more generations.
2.3 implies that only a theory which has three generations or more can have complex
couplings and can consequently have CP violation. In the year 1973, when the KM
mechanism was formulated [3], although only the three lightest quarks were known, CP
violation had been detected, so that this formalism predicted the existence of at least
three more quarks which were found subsequently and supported the KM theory. The
SM has three generations and therefore VCKM can be parametrized with three real angles
and one complex phase which describe quark mixing. This implies that all CP violation
in the SM is related to this single parameter. However, some additional conditions need
26
2 The Standard Model
to be satisfied for CP violation to occur, these are summarized by
J(m2t − m2c )(m2c − m2u )(m2u − m2t )(m2b − m2s )(m2s − m2d )(m2d − m2b ) 6= 0,
(2.29)
where J is the Jarlskog invariant [51] and defined by
∗
J ≡ |ℑ(Vkm
Vlm Vkn Vln∗ )|.
(2.30)
The first condition, J 6= 0, is that the three angles and the phase are non zero. The second
condition which Eqn. (2.29)n requires is that all up-type and all down-type masses are
non degenerate. If the above conditions would not be satisfied in the SM no CP violation
would be possible because the complex phase in VCKM could be rotated away.
2.3.2 Parametrizations of the CKM Matrix
There is an infinite number of possible parametrizations of the CKM matrix, in the
preceding subsection we found that the CKM matrix has four free parameters, three real
rotation angles and one complex phase. It makes sense to parametrize the matrix in terms
of these quantities, Chau and Keung [52] published therefore the following decomposition
which is also advocated by the Particle Data Group (PDG)[34]
VCKM



1
0
0
c12 s12
c13
0 s13 e−iδ13



=  0 c23 s23  
0
1
0
  −s12 c12
iδ13
0
0
−s13 e
0
c13
0 −s23 c23

c12 c13
s12 c13
s13 e−iδ13

=  −s12 c23 − c12 s23 s13 eiδ13 c12 c23 − s12 s23 s13 eiδ13
s23 c13
s12 s23 − c12 c23 s13 eiδ13 −c12 s23 − s12 c23 s13 eiδ13 c23 c13

0

0 
1


 . (2.31)
This parametrization makes the dependence on mixing angles explicit with cij = cos θij
and sij = sin θij , where θij is the angle that mixes the ith and the j th generation. We stress
that it is a pure matter of choice how the CP violating phase is included in VCKM . One
finds that the decomposition in Eqn. (2.31) is not very useful if it comes to applications.
To see this we will give the absolute values of the matrix entries as given by the PDG
VCKM

0.97428 ± 0.00015 0.2253 ± 0.0007 0.00347+0.00016
−0.0011


+0.00012
=  0.2252 ± 0.0007
0.97345+0.00015
0.0410
.
−0.00016
−0.0007
0.00862+0.00026
0.0403+0.0011
0.999152+0.000030
−0.00020
−0.0007
−0.000045

(2.32)
27
2 The Standard Model
These numerical entries are the result of a fit which exploits the unitarity of VCKM , the
uncertainties on this fit are very small, some entries are given with a considerably smaller
uncertainty than they are actually known. For example, the elements Vtd and Vts are not
directly accessible in experiments and consequently are subject to large uncertainties. The
numerical form of the matrix can be used as a motivation to introduce a more convenient
notation, we see from the entries in Eqn. (2.32) that transitions within one generation
are favored. Transitions between the first and the second generation are suppressed by a
factor of approximately 0.23, transitions to the third generation are even more disfavored
this led Wolfenstein to introduce another parametrization which reflects this suppression
scheme [53], introducing four quantities λ, A, ρ and η related to the θij and δ13 by [54]
λ = s12
Aλ2 = s23
Aλ3 (ρ + iη) = s13 eiδ13 ,
(2.33)
we may rewrite VCKM as

1 − 21 λ2 − 81 λ4
λ

2 5 1
(ρ + iη))
1 − 21 λ2 − λ4 ( 18 + 12 A2 )
VCKM = 
 −λ + A ³λ ( 2 −2 ´
Aλ3 (1 − 1 − λ2 (ρ + iη) −Aλ2 + Aλ4 ( 21 − (ρ + iη))
Aλ3 (ρ − iη)
Aλ2
1 − 12 A2 λ4


 + O(λ6 ).

(2.34)
5
Here, we include terms up to order O(λ ), since the CP violating effects in the charm
system only occur at that order. This indicates that CP violating observables are very
small in the charm system. The element Vub does not receive any corrections, Vus , Vcd and
Vtd receive corrections at order O(λ7 ), Vcb receives corrections at order O(λ8 ).
The unitarity alluded to in the
previous subsection gives a nice
geometrical representation of η
VtdVtb∗
∗
and ρ, multiplying the first
VudVub
VcdVcb∗
Vcd Vcb∗
and the third column, Vud Vub∗ +
Vcd Vcb∗ + Vtd Vtb∗ = 0, we may introduce an equation which can be
(1, 0) graphically interpreted as a tri(0, 0)
angle in the complex plane called
Figure 2.2: The unitarity triangle, ρ and η are the co- the unitarity triangle (UT) as in
ordinates of the apex in the complex plain. Fig. 2.2. In this triangle, ρ and
η are the coordinates of the apex
(ρ, η)
28
2 The Standard Model
lud
excluded area has CL > 0.95
η
exc
1.5
ed
γ
at C
L>
γ
1
∆md & ∆ms
0.9
1.0
5
sin 2β
β
0.5
∆md
η
εK
0.0
∆md
∆md
∆ms
0.5
α
εK
β
γ
α
0
V ub
V cb
α
Vub
sin(2β+γ )
-0.5
α
-0.5
εK
-1.0
CKM
fitter
Winter 12
-1.5
-1.0
-0.5
γ
sol. w/ cos 2β < 0
(excl. at CL > 0.95)
0.0
0.5
1.0
1.5
ρ
-1
2.0
-1
-0.5
0
0.5
1
ρ
Figure 2.3: Global fits of the CKMfitter group [55] and the UTfit collaboration [56] to the
unitarity at next-to-leading order.
of the triangle. In fact, there are five more possible UTs, the one we are showing here has
all sides of similar length in contrast to e.g. the UT triangle in the charm system, that
has two sides of length O(λ) and one side of length O(λ5 ). The surface, however, of all six
possible UTs is equal to half the Jarlskog invariant. The above UT is only exact to order
O(λ3 ), for the LHCb experiment this precision does satisfy anymore. Therefore, Buras et
al. [54] have introduced a straightforward extension of the Wolfenstein parametrization
by defining
µ
¶
µ
¶
λ2
λ2
ρ̄ ≡ ρ 1 −
, η̄ 1 −
.
(2.35)
2
2
Using this we can easily rewrite Vtd as
Vtd = Aλ3 (1 − ρ̄ − iη̄),
(2.36)
the unitarity triangle becomes
Aλ3 [(ρ̄ + iη̄) − 1 + (1 − (ρ̄ + iη̄))] + O(λ7 ) = 0
(2.37)
This triangle and the parameters which describe it are is of high interest in flavor physics,
if the respective angles in the triangle do not add up to 180◦ , this is an indication for new
physics, currently no tensions with the SM are apparent as can be seen in Fig. 2.3. The
29
2 The Standard Model
normalized side lengths of the enhanced UT are given by
Rb
Rt
¯
µ
¶ ¯
p
λ2 1 ¯¯ Vub ¯¯
2
2
,
ρ̄ + η̄ = 1 −
≡
2 λ ¯ Vcb ¯
¯ ¯
p
¯ Vtb ¯
1
≡
(1 − ρ̄)2 + η̄ 2 = ¯¯ ¯¯ .
λ Vcb
(2.38)
(2.39)
The PDG gives the following numerical values for the Wolfenstein parameters
λ = 0.2253 ± 0.0007, A = 0.808+0.022
−0.015 ,
+0.023
ρ = 0.135−0.014 ,
η = 0.350 ± 0.013.
(2.40)
The error on λ is of per mil order and negligible the other quantities are at least multiplied
by λ2 , therefore their uncertainties are negligible, too. Consequently, we are going to
ignore the uncertainties of the Wolfenstein parameters in this thesis.
2.3.3 Parameters in the Charm System
The decays of charm mesons are proportional to the elements of first two rows of VCKM ,
since they appear in quark transitions where a c quark decays into lighter quarks. The
UT for this system is obtained by multiplying the complex conjugated first row and the
second row. One obtains
Vcd∗ Vud + Vcs∗ Vus + Vcb∗ Vub = 0.
(2.41)
This can be likewise be written as
λd + λs + λb = 0 with λq ≡ Vcq∗ Vuq ,
q ∈ {d, s, b}.
(2.42)
λq must not be confused with the Cabibbo angle λ, even though they are of similar size.
The numerical values of the λq and their decomposition in the Wolfenstein parametrization
are as follows:
λ5
1
λd = −λ + λ3 + (1 + 4A2 ) − A2 λ5 (ρ − iη) + O(λ7 ) = −0.219 + 0.001i (2.43)
2
8
1 3 λ5
λs = λ − λ − (1 + 4A2 ) + O(λ7 ) = 0.219
(2.44)
2
8
λb = A2 λ5 (ρ − iη) + O(λ11 ) = (5.12 − 13.2i) × 10−5
(2.45)
Where numerical values are given to three significant digits. Since the equality λd ≈ λs ≈
λ holds to good precision we call processes whose amplitude is proportional to λd or λs
2 The Standard Model
30
singly Cabibbo suppressed (SCS). Furthermore, we will often encounter the following
quantities
1
ζ = Vcs∗ Vud = 1 − λ2 − A2 λ4 + O(λ6 ) = 0.948,
2
ξ = Vcd∗ Vus = −λ2 + O(λ6 ) = −0.0508.
(2.46)
Where ζ is of order one and processes proportional to ζ are called Cabibbo favored
(CF), ξ is of order O(λ2 ) if a quantity is multiplied by ξ it is called doubly Cabibbo
suppressed (DCS).
3 Theory of CP Violation
After a chapter which mostly contained basic particle physics content, we now explore CP
violation. One distinguishes direct and indirect CP violation. In the first section, we will
discuss direct CP violation that occurs because of the structure of the decay amplitudes
and which is independent of time. In second section, we will discuss first neutral meson
oscillations which are necessary for the two kinds of indirect CP violation. We then
introduce indirect CP violation and distinguish CP violation in mixing and CP violation
in the interference of decay with and without mixing. At last, we discuss the observable
∆ACP and how the different kinds of CP violation contribute to it. The discussion in this
chapter is mainly based on the lecture notes by Fleischer [57] and the text books [31, 32].
3.1 CP Violation in Decay - Direct CP Violation
CP violation in decay appears on the amplitude level. The general form of a CP asymmetry, given by
Γ(P → f ) − Γ(P̄ → f¯)
ACP (f ) ≡
,
(3.1)
Γ(P → f ) + Γ(P̄ → f¯)
where Γ(P → f ) is the decay rate of the particle P to the final state f . The decay rates
take following form
Γ(P̄ → f¯) = |A(P̄ → f¯)|2 Γ̃f .
Γ(P → f ) = |A(P → f )|2 Γ̃f ,
(3.2)
Where A(P → f ) is the decay amplitude for the decay of a particle P to a final state f
and Γ̃f is a phase space factor. Hence, we obtain the general definition of a direct CP
asymmetry as
¯
¯
¯ Ā(P̄ →f¯) ¯2
1 − ¯ A(P →f ) ¯
ACP (f ) = adir
(3.3)
¯
¯ .
CP (f ) ≡
¯ P̄ →f¯) ¯2
1 + ¯ Ā(
¯
A(P →f )
31
32
3 Theory of CP Violation
This gives the defining condition for direct CP violation as
¯
¯
¯ A(P̄ → f¯ ¯
¯
¯
¯ A(P → f ) ¯ 6= 1 .
(3.4)
Direct CP violation occurs if two different amplitudes contribute to a single decay, e.g.
a tree and a higher order process, such that the interference of these two contributing
amplitudes leads to a different decay rate for the CP-conjugated process. To show this we
assume that the final state is a CP eigenstate such that (CP)A(P̄ → f¯) = ηf A(P̄ → f ),
where ηf = ± for f being an even/odd CP eigenstate. Furthermore, the amplitudes, which
represent the decay of P and its CP-conjugate P̄ to the final state f , can be parametrized
in the following way
A(P → f ) = A1 e−iφ1 eiδ1 + A2 e−iφ2 eiδ2 ,
ηf A(P̄ → f ) = A1 eiφ1 eiδ1 + A2 eiφ2 eiδ2 .
(3.5)
(3.6)
Where A1 and A2 are real CP-conserving amplitudes, δ1 and δ2 are CP-conserving strong
phases and φ1 and φ2 are weak phases which change sign under CP-conjugation. Using
these definitions we obtain for the direct CP asymmetry (3.3)
adir
CP (f ) =
A21
2A1 A2 sin(φ2 − φ1 ) sin(δ2 − δ1 )
.
+ A22 + 2A1 A2 cos(φ2 − φ1 ) cos(δ2 − δ1 )
(3.7)
This equality shows that CP violation in decay can only occur if there are two nonzero amplitudes with non-trivial strong and weak relative phases which contribute to this
decay.
Let us assume that A2 ≪ A1 , we obtain then
adir
CP (f ) ≈ 2
A2
sin(φ2 − φ1 ) sin(δ2 − δ1 ).
A1
(3.8)
Direct CP violation has been first found in the the Kaon system [58, 59] and is now also
established in the B system [60]. Finding direct CP violation was the ultimate prove
that a theory by Wolfenstein [61], called superweak theory, was wrong. The superweak
theory explained CP violation by a fifth very weak force that would mediate processes
that change flavor (strangeness, charm, bottomness, topness) by two, ∆F = 2 processes,
yet direct CP violation is a clear sign for ∆F = 1 CP violating processes.
33
3 Theory of CP Violation
3.2 Indirect CP violation
After this simple example of direct CP violation we come now to indirect CP violation. In
contrast to direct CP violation, which is possible for neutral and charged meson decays,
indirect CP violation is only possible in the decays of neutral mesons, because they can
transform into their antiparticle. This will be explained in the first subsection to show
how indirect CP violation can be generated in the second subsection.
3.2.1 Time Evolution of a Neutral Meson System
The Standard Model of particle physics gives rise to four neutral pseudoscalar mesons that
are not their own antiparticle, these are K 0 , D0 , B 0 and Bs0 . These mesons differ from
their anti-meson by nothing but flavor. Flavor is not conserved by the weak interaction and
it is possible that these mesons change through the weak interaction into their antiparticle.
This is commonly called neutral meson oscillations, in the SM this is not possible via
a tree level process (Glashow-Iliopoulos-Maiani (GIM) mechanism [62]), but must be
mediated by the box diagram, this is shown for a D0 meson in Fig. 3.1. neutral meson
D0
D̄0
W
W
ū
u
d, s, b
c
¯ s̄, b̄
d,
c̄
Figure 3.1: The oscillation of a D0 meson to a D̄0 meson is mediated by the weak interaction. This oscillation is possible since the weak interaction does not conserve
the quantum number charm, which is the only quantum number that distinguishes a D0 form a D̄0 .
oscillations can effectively be described with non-relativistic quantum mechanics, we will
give the formalism which is necessary to describe these oscillations and which will be used
to describe CP violation in mixing and CP violation in the interference of decay with and
without mixing.
We consider a general neutral meson P 0 and its antiparticle P̄ 0 . Since the only quantum
number which distinguishes them is flavor, which is not conserved by the weak interaction,
34
3 Theory of CP Violation
they form a coupled system which is described by the following effective Hamiltonian
H∆F =2 =
Ã
M11 − 2i Γ11 M12 − 2i Γ12
∗
M12
− 2i Γ∗12 M11 − 2i Γ11
!
i
= M − Γ.
2
(3.9)
Since the SM only gives rise to ∆F = 1 processes this Hamiltonian arises from combining
two ∆F = 1 processes, this is done under the assumption that we can limit the quantum
mechanical system to the two neutral mesons and include the final particles in which
they decay only by the entries of Γ [63, 64]. Furthermore, this makes use of the WignerWeisskopf approximation [65, 66] that all interactions take place on time scales much
larger than the time scale of the strong interaction. The Hermitian structure of M and Γ
is due to CPT-invariance and describes the oscillation of the mesons as well as the decay
of the neutral mesons into lighter particles. The off-diagonal entries give the probability
amplitude that the meson transforms to its antimeson. There are two possibilities how
this can happen, through virtual particles (M12 ) and real particles on-shell (Γ12 ), which
are the intermediate states between the two ∆F = 1 processes. The off-diagonal entries
¯ ®
translate into the fact, that |P 0 i and ¯P̄ 0 are not the mass eigenstates of the system.
The eigenstates can be found after computing the eigenvalues of the Hamiltonian
λ1,2
sµ
¶
¶µ
i ∗
i
∗
M12 − Γ12
M12 − Γ12
2
2
µ
¶
i
i
q
≡ M11 − Γ11 ± M12 − Γ12
.
2
2
p
i
= M11 − Γ11 ±
2
(3.10)
(3.11)
Where we introduced
s
∗
M12
− 2i Γ∗12
q
≡±
p
M12 − 2i Γ12
with |p|2 + |q|2 ≡ 1.
(3.12)
There is actually a twofold ambiguity of interchanging the label 1 and 2. We make the
choice to attribute the + to 1 and - to 2, which also amounts to choosing the positive sign
in Eqn. (3.12). Of course, the final result is independent of this choice. We may write
down the mass eigenstates as
¯ ®
¯ ®
|P1 i = p ¯P 0 + q ¯P̄ 0 ,
¯ ®
¯ ®
|P2 i = p ¯P 0 − q ¯P̄ 0 .
(3.13)
(3.14)
35
3 Theory of CP Violation
These are the eigenstates to the eigenvalues λ1 and λ2 respectively, therefore we can
now easily write down the evolution over time of the states |P1 i and |P2 i, as |P1 (t)i =
e−iλ1 t |P1 (0)i and |P2 (t)i = e−iλ2 t |P2 (0)i. Moreover, this readily yields the time evolution
¯ ®
of |P 0 i and ¯P̄ 0 , the eigenstates of the weak interaction, as follows
¯ ® q
¯ ®
¢
1 ¡
|P1 i e−iλ1 t + |P2 i e−iλ2 t = g+ (t) ¯P 0 + g− (t) ¯P̄ 0 ,
2p
p
¯ 0 ®
¯
¯ ®
¡
¢
®
p
1
¯P̄ (t) =
|P1 i e−iλ1 t − |P2 i e−iλ2 t = g− (t) ¯P 0 + g+ (t) ¯P̄ 0 .
2q
q
¯ 0 ®
¯P (t) =
(3.15)
(3.16)
Where we defined g± (t) ≡ 21 (e−iλ1 t ± e−iλ2 t ). In order to evaluate decay rates the following
two expressions will prove useful later on
´
Γ1 +Γ2
1 ³ −Γ1 t
e
+ e−Γ2 t ± 2e− 2 t cos(∆M t)
4
µ
µ
¶
¶
∆Γt
e−Γt
cosh
± cos(∆M t) ,
=
2
2
´
Γ +Γ
1 ³ −Γ1 t
−Γ2 t
− 12 2t
∗
e
−e
+ 2ie
sin(∆M t)
g+ (t)g− (t) =
4
µ
µ
¶
¶
∆Γt
e−Γt
sinh
− i sin(∆M t) .
=
2
2
|g± (t)|2 =
(3.17)
(3.18)
(3.19)
(3.20)
Here, it was convenient to define the masses and the decay rate of P1 and P2 using the
eigenvalues λ1 and λ2
M1 ≡ ℜ(λ1 ),
Γ1 ≡ −2ℑ(λ1 ),
(3.21)
M2 ≡ ℜ(λ2 ),
Γ2 ≡ −2ℑ(λ2 ).
(3.22)
Furthermore, we introduced their averages and differences
Γ1 + Γ2
,
2
∆Γ ≡ Γ2 − Γ1 .
M ≡ M1 + M 2 ,
Γ≡
∆M ≡ M2 − M1 ,
(3.23)
(3.24)
At last we introduce the dimensionless quantities
x≡
∆M
,
Γ
y≡
∆Γ
2Γ
and τ ≡ Γt
(3.25)
36
3 Theory of CP Violation
in order to have the even more compact expressions
e−τ
(cosh(yτ ) ± cos(xτ )),
2
e−τ
∗
g+ (t)g−
(t) =
(sinh(yτ ) − i sin(xτ )).
2
|g± (t)|2 =
(3.26)
(3.27)
Even though the formalism is the same for all neutral mesons, the parameters x and y
are different so that the oscillation behavior and the experimental approaches are quite
different.
3.2.2 The Decay Rate
¯
®
Until now we found the time-dependent behavior of |P 0 (t)i and ¯P̄ 0 (t) , for experiments
one actually needs observables, for example, the decay rate Γ(P 0 (t) → f ). This timedependent observable indicates how many particles P 0 of a number of initial particles are
expected to have decayed into a certain final state f after a certain time t. This observable
does not only contain the time evolution of the initial state but also the probability that
it decays into a given final state. This probability is encoded into the decay amplitudes
¯ ®
Af ≡ hf | H∆F =1 ¯P 0 ,
¯ ®
Āf ≡ hf | H∆F =1 ¯P̄ 0 ,
¯ ®
­ ¯
Af¯ ≡ f¯¯ H∆F =1 ¯P 0 ,
¯ ®
­ ¯
Āf¯ ≡ f¯¯ H∆F =1 ¯P̄ 0 .
(3.28)
(3.29)
In order to write down the decay rates, using Eqns. (3.15) and (3.16) with the expressions
of Eqns. (3.26) and (3.27) and the amplitudes in (3.28) and (3.29), we introduce now the
parameter λf 1
q Āf
λf ≡
.
(3.30)
p Af
Putting all the ingredients together we get
¯ ®
Γ(P 0 (t) → f ) ≡ | hf | H ¯P 0 |2
£
e−τ
|Af |2 (1 + |λf |2 ) cosh(yτ ) + (1 − |λf |2 ) cos(xτ )
=
2
+ 2ℜ(λf ) sinh(yτ ) − 2ℑ(λf ) sin(xτ )]
1
This λf is neither connected to λ1,2 nor to the Cabibbo angle λ.
(3.31)
(3.32)
37
3 Theory of CP Violation
¯ ®
Γ(P̄ 0 (t) → f ) ≡ | hf | H ¯P̄ 0 |2
(3.33)
"Ã
!
Ã
!
¯
¯
¯
¯
¯ 1 ¯2
¯ 1 ¯2
e−τ
2
¯
¯
|Āf |
1 + ¯ ¯ cosh (yτ ) + 1 − ¯¯ ¯¯ cos(xτ )
=
2
λf
λf
µ ¶
µ ¶
¸
1
1
+ 2ℜ
sinh(yτ ) − 2ℑ
sin(xτ )
(3.34)
λf
λf
¯ ¯2
¯ ¯ £
e−τ
2 ¯p¯
|Af | ¯ ¯ (1 + |λf |2 ) cosh(yτ ) − (1 − |λf |2 ) cos(xτ )
=
2
q
+ 2ℜ(λf ) sinh(yτ ) + 2ℑ(λf ) sin(xτ )]
(3.35)
Where the phase space factor has not been considered because it is universal and does not
influence the final result. Furthermore, one can obtain the decay rate to the final states
f¯ just by replacing f with f¯. A good and short way [67, 68] of writing Eqns. (3.32) and
(3.35) is
Γ
Ã
P 0 (t)
P̄ 0 (t)
→f
!
−τ
=e
|Af |
2
"
Cy
C̄y
cosh(yτ ) +
Cx
C̄x
cos(xτ ) +
Sy
S̄y
sinh(yτ ) +
Sx
S̄x
#
sin(xτ ) .
(3.36)
Where the appearing new coefficients are defined in the following way
1
1
Cy ≡ (1 + |λf |2 ) Cx ≡ (1 − |λf |2 ) Sy ≡ ℜλf
2
2
and
¯ ¯2
¯p¯
(C̄y , S̄y ) ≡ ¯¯ ¯¯ (Cy , Sy )
q
Sx ≡ −ℑλf
¯ ¯2
¯p¯
(C̄x , S̄x ) ≡ − ¯¯ ¯¯ (Cx , Sx ).
q
(3.37)
(3.38)
Equation (3.36) can in principle be used to understand the results of experiments, it is
quite compact because we introduced the new parameters but has a complex time dependence and thus does not find direct application. In the Bd0 system, the time dependence
can be simplified, y = 0 as well as |p/q| = 1 are a very good approximation and simplify
the expression considerable. In the charm system, these approximations are not possible,
we therefore will need other approximations as expanding in x and y, we will describe
these approximations in the next section.
Above we found the time-dependent behavior of neutral mesons and their decays, it is
now possible to explain indirect CP violation. We distinguish CP violation in mixing and
CP violation in the interference of decay with and without mixing.
38
3 Theory of CP Violation
3.2.3 CP Violation in Mixing
CP violation in mixing occurs if a particle P 0 cannot decay into a final state f¯ but its
CP-conjugate P̄ 0 can. Consequently, P 0 needs to oscillate to the antiparticle state before
decaying into the given final state f¯. As an example we may take semileptonic decays
P 0 → P̄ 0 → l+ + X − ←P
/ 0,
P̄ 0 → P 0 → l− + X + ←
/ P̄ 0 ,
(3.39)
where we have f = l− X + and f¯ = l− + X + . In the SM the decay amplitudes of these
processes are of the same size
|Af | = |Āf¯|,
(3.40)
whereas the other decay amplitudes are zero by construction
Af¯ = Āf = 0.
(3.41)
We define the corresponding asymmetry in the following way
aSL =
Γ(P 0 → f¯) − Γ(P̄0 → f )
,
Γ(P 0 → f¯) + Γ(P̄0 → f )
(3.42)
where the ’SL’ means semileptonic. Using Eqn. (3.36) with λf = λ−1
= 0 we obtain
f¯
aSL
¯ ¯4
¯ ¯
1 − ¯ pq ¯
=
¯ ¯4
¯ ¯
1 + ¯ pq ¯
(3.43)
which yields the condition for CP violation in mixing
¯ ¯
¯q ¯
¯ ¯ 6= 1 .
¯p¯
(3.44)
This is not a surprising result and can be understood if we consider the definition of q/p
in Eqn. (3.12)
¯ ¯2
i ∗
∗
¯q ¯
¯ ¯ = |M12 − 2 Γ12 | .
(3.45)
¯p¯
|M12 − 2i Γ12 |
As we stated before the off-diagonal entries of the Hamiltonian (3.9) are proportional to
¯ ®
the probability of a particle |P 0 i oscillating to ¯P̄ 0 and vice versa, if these amplitudes
do not have the same size, CP violation in mixing follows. CP violation in mixing is a
39
3 Theory of CP Violation
clear sign for CP violation in ∆F = 2 processes. (Nevertheless, in the SM this ∆F = 2 is
a combination of two ∆F = 1 processes.)
3.2.4 CP Violation in the Interference of Decay With and Without
Mixing
If mixing followed by decay and direct decay interfere this creates an additional form of
CP violation. The final state must be common to P 0 and P̄ 0 . An example for this kind
of CP violation are the decays D0 → K + K − and D̄0 → K + K − and D0 → π + π − and
D̄0 → π + π − , in this case there is no simple form of a CP asymmetry as we will see in
Section 3.3.2, we consequently shift the discussion of this case to next section. However,
we will show in what follows, that even without net direct CP violation and net CP
violation in mixing there is the possibility of CP violation. Let us assume that f is a CP
eigenstate with CP |f i = ηf |f i and that
|Af | = |Āf |,
¯ ¯
¯q ¯
¯ ¯=1
¯p¯
(3.46)
hold. With these assumptions Eqns. (3.32) and (3.35) become
Γ (P (t) → f ) = eτ |Af |2 (cos(yτ ) + ℜ(λf ) sinh(yτ ) − ℑ(λf ) sin(xτ ))
(3.47)
µ ¶
¶
µ
µ ¶
¡
¢
1
1
sinh(yτ ) − ℑ
sin(xτ ) (3.48)
Γ P̄ (t) → f = eτ |Af |2 cos(yτ ) + ℜ
λf
λf
Furthermore, the modulus of λf =
phase of λf as
q Āf
p Af
becomes unity, for this reason we may define the
λf ≡ |λf |eiφ = eiφ
(3.49)
and obtain for Eqn. (3.1)
ACP (f ) = −
sin φ sin(xτ )
.
cos(yτ ) + cos φ sinh(yτ )
(3.50)
Hence, there are two conditions which need to be satisfied so that CP violation in the
interference of decay with and without mixing occurs: The mass eigenstates must have
different mass and the phase of λf must be nontrivial
∆M
x=
6 0 and φ ≡ arg λf = arg
=
Γ
µ
q Āf
p Af
¶
6= 0.
(3.51)
40
3 Theory of CP Violation
The above conditions give CP violation in the interference of ∆F = 1 and ∆F = 2
processes for times τ 6= 0. However, we find that the integrated asymmetry is only non6= 0 is satisfied as well.
zero if y = ∆Γ
2Γ
Ā
In conclusion, one might say that parameter λf = pq Aff contains all information about CP
violation in a certain decay. The absolute value of the amplitude ratio |Āf /Af | encodes
the direct CP violation, the absolute value of |q/p| represents CP violation in mixing.
The phase φ of λf expresses CP violation in the interference of decay with and without
mixing and is an observable on its own. The phases of Āf /Af and of q/p, on the contrary,
are no observables.
This concludes the section on general CP violation, we found that there exist three kinds
of CP violation: Direct CP violation, which is due to CP violation in decay and is also
possible for charged mesons, CP violation in mixing and CP violation in the interference
of decay with and without mixing, which only are possible for neutral mesons, since these
can transform into their antiparticles.
3.3 CP Violation in Neutral D Decays
Having explored CP violation in general we now turn to the first part of our analysis. In
this section we are going to explore the structure of ∆ACP . We are interested which of
the three types of CP violation contribute to ∆ACP . First, we give the mixing parameters
in the charm system, then we turn to a general single time-dependent asymmetry, the
expression of this asymmetry is very complex so that we reduce it to a simpler expression
for the charm system which is subsequently used to find an expression for ∆ACP .
3.3.1 Numerical Values in for the Neutral Charm System
Since we are going to discuss CP violation in charm decays we introduce in this subsection
some experimental values for the quantities related to mixing. We begin by choosing the
¯ ®
following CP properties for |D0 i and ¯D̄0
¯ ® ¯ ®
C ¯D0 = ¯D̄0 ,
¯ ®
¯ ®
P ¯D 0 = − ¯ D 0 .
(3.52)
The final physical results are independent of this choice, this choice would mean that D1
(D2 ) in Eqn. (3.14) was CP odd (even), if CP was conserved, since CP is violated the
41
3 Theory of CP Violation
mass eigenstates are no CP eigenstates, we can only argue that D1 (D2 ) is more CP odd
(even) in the sense that it mostly decays into CP odd (even) states. We cannot predict
from theory whether D1 or D2 should be heavier this has to be found by experiment. The
Heavy Flavor Averaging Group (HFAG) [69], whose convention we adapted, gives for the
mixing parameters in the D0 − D̄0 system the following values2
x = (0.63+0.19
−0.20 )%,
y = (0.75 ± 0.12)%.
(3.53)
Here we find that the heavier mass eigenstate has also a larger decay width and has thus a
shorter decay time. The ratio x = ∆M
can be seen as the ratio of the average decay time
Γ
τ
and the oscillation period (via off-shell modes) x = decay
the small value of x indicates
τosc
that τdecay ≪ τosc . Since the same is true for on-shell oscillation modes so that y is also
small, this means that a D0 meson is likely to decay before it does oscillate to a D̄0 . This
is the reason why D0 − D̄0 oscillations have only been found in the recent years [70, 71].
The quantity q/p is relevant in mixing as well, its size and phase are
¯ ¯
¯q ¯
¯ ¯ = 0.88+0.18
−0.16
¯p¯
◦
and φc = (−10.1+9.5
−8.9 ) .
(3.54)
We see that these quantities suffer from large uncertainties, which is due to the slow
mixing of the neutral D mesons. Note, that the phase φc is a priori convention dependent
¯ ®
but, since we choose a convention for |D0 i and ¯D̄0 , a can value can be attributed to φc .
Furthermore we are going to use the following numerical data taken from the PDG [34]
for the D System
mD0 = 1864.8 ± 0.1MeV, mD+ = 1869.6 ± 0.2MeV, mDs+ = 1968.5 ± 0.3MeV,
ΓD0 = 1.605 ± 0.006meV, ΓD+ = 0.633 ± 0.004meV, ΓDs+ = 1.316 ± 0.018meV.
(3.55)
The uncertainties of these quantities are < 1%, so that we can safely neglect them in this
thesis.
3.3.2 The Observable
In this section we are going to give the theoretical derivation of a generic CP asymmetry
and see how it reduces under the approximations which are valid in the charm system.
We will consequently derive the theoretical expression of ∆ACP and show that it mainly
2
These values and the values for |p/q| and φ are the values extracted from the data if CP violation is
allowed.
42
3 Theory of CP Violation
consists of the difference of the direct CP asymmetries. We restrict ourselves to the
simplified situation, when f is a CP eigenstate with ηf = ±1 for f being a CP-even/odd
final state. We consequently can write λf in the following parametrization
¯ ¯¯ ¯
¯ q ¯ ¯ Āf ¯
λf ≡ ηf ¯¯ ¯¯ ¯¯ ¯¯ eiφ
p Af
(3.56)
The time-dependent CP asymmetry can then be defined as the normalized rate difference
of a D0 meson decaying into a final state and a D̄0 decaying into the final state
ACP (t) ≡
Γ(D0 (t) → f ) − Γ(D̄0 (t) → f )
.
Γ(D0 (t) → f ) + Γ(D̄0 (t) → f )
(3.57)
We plug in Eqns. (3.36) and get
ACP (t) =
P− Cy cosh(yτ ) + P+ Cx cos(xτ ) + P− Sy sinh(yτ ) + P+ Sx sin(xτ )
,
P+ Cy cosh(yτ ) + P− Cx cos(xτ ) + P+ Sy sinh(yτ ) + P− Sx sin(xτ )
(3.58)
¯ ¯2
¯ ¯
with P± ≡ 1 ± ¯ pq ¯ , one may want to write the asymmetry solely in terms of quantities
which are accessible in experiment. The mixing parameters x and y can be measured in
mixing [72, 73], |p/q| can be related to the semileptonic asymmetry aSL which in principle
can be measured in semileptonic decays3 by
¯ ¯2 √
¯p¯
¯ ¯ = √1 − aSL ,
¯q ¯
1 + aSL
(3.59)
the ratio of decay amplitudes can be related to the direct CP asymmetry by
¯ ¯ s
dir
¯ Āf ¯
¯ ¯ = 1 − aCP
¯ Af ¯
1 + adir
CP
resulting in
(3.60)
´
´
³
³
1
A−a
√
,
P
C
=
,
1 + √1−aA
A
+
+ x
1+A
1−a2 ´
1−a2´
³
³
1
1
,
P− Cx = 1+A
,
P− Cy = 1+A
A − √A−a
1 − √1−aA
2
1−a
³q
³q1−a2
q ´
q ´
√
√
2
2
1−A
1−A
4 1+a
4 1+a
4 1−a
4 1−a
P± Sx = ηf 1+A
sin
φ,
P
S
=
η
cos φ.
±
±
± y
f 1+A
1−a
1+a
1−a
1+a
(3.61)
P+ Cy =
3
1
1+A
However, this has not been done yet because this asymmetry suffers from huge uncertainties due to a
large background signal.
43
3 Theory of CP Violation
Where we set A ≡ adir
CP and a ≡ aSL , this gives the asymmetry
ACP (t) =
¡ √
¢
¡ √
¢
A 1 − a2 − A + a cosh(yτ ) + A 1 − a2 + A − a cos(xτ )
¡√
¢
¡√
¢
(3.62)
1 − a2 + 1 − aA cosh(yτ ) +
1 − a2 − 1 + aA cos(xτ )
√
£¡√
¢
¡√
¢
¤
√
√
1 + a − 1 − a cos φ sinh(yτ ) +
1 + a + 1 − a sin φ sin(xτ )
+ηf 1 − A2
√
£¡√
¢
¤ . (3.63)
¢
¡√
√
√
+ηf 1 − A2
1 + a + 1 − a cos φ sinh(yτ ) +
1 + a − 1 − a sin φ sin(xτ )
Which expresses ACP (t) only in terms of measurable quantities. We see that this is a
inconvenient form with a complex time dependence. In order to obtain a simpler expression we approximate in x and y since they are smaller than one percent as mentioned in
Section 3.3.1. Consequently ACP (t) becomes linearly dependent on time
Cy + Cx + Sy yτ + Sx xτ − C̄y − C̄x − S̄y yτ − S̄x xτ
+ O(x2 , y 2 , xy) (3.64)
Cy + Cx + Sy yτ + Sx xτ + C̄y + C̄x + S̄y yτ + S̄x xτ
h ³
´
τ
dir
= adir
+
x
S
−
S̄
−
a
(S
+
S̄
)
x
x
x
x
CP
CP
Cy + C̄y + Cx + C̄x
³
´i
+ y Sy − S̄y − adir
+ O(x2 , y 2 , xy)
(3.65)
CP (Sy + S̄y )
·
µ¯ ¯ ¯ ¯
µ¯ ¯ ¯ ¯¶¶
¯p¯ ¯q ¯
¯q ¯ ¯p¯
ηf τ
¯ ¯−¯ ¯
¯ ¯ x sin φ ¯¯ ¯¯ + ¯¯ ¯¯ − adir
= adir
CP + ¯¯ A ¯¯
CP
¯p¯ ¯q ¯
¯ Āf ¯
f
q
p
+
¯ Āf ¯ ¯ Af ¯
{z
}
|
ACP (t) =
ηf τ
2
q
2
1−adir
CP
µ¯ ¯ ¯ ¯¶¶¸
µ¯ ¯ ¯ ¯
¯q ¯ ¯p¯
¯p¯ ¯q ¯
dir
+ O(. . . )
+ y cos φ ¯¯ ¯¯ − ¯¯ ¯¯ − aCP ¯¯ ¯¯ + ¯¯ ¯¯
p
q
q
p
(3.66)
where we used the expressions of the C’s and S’s in terms of λf as in Eqn. (3.56)in the
third step. Finally, we find
·
µ¯ ¯ ¯ ¯
µ¯ ¯ ¯ ¯¶¶
q
¯p¯ ¯q ¯
¯q ¯ ¯p¯
ηf τ
dir
dir 2
ACP (t) =
+
1 − aCP x sin φ ¯¯ ¯¯ + ¯¯ ¯¯ − aCP ¯¯ ¯¯ − ¯¯ ¯¯
2
q
p
p
q
µ¯ ¯ ¯ ¯¶¶¸
µ¯ ¯ ¯ ¯
¯p¯ ¯q ¯
¯q ¯ ¯p¯
¯ ¯+¯ ¯
+ O(. . . ).
(3.67)
+y cos φ ¯¯ ¯¯ − ¯¯ ¯¯ − adir
CP
¯q ¯ ¯p¯
p
q
adir
CP
This CP asymmetry does not
¯ ¯have
¯ ¯any approximations except in x and y and contains
¯
¯q¯
¯
p
−4
the contribution y cos φadir
CP (¯ q ¯ + ¯ p ¯) ∼ O(10 ) which has been recently pointed out by
Gersabeck and others [9]. Since the adir
CP is also expected to be of percent order or smaller
in the charm system (See Section 4.3), usually also terms proportional to x, y and adir
CP
are neglected. This leads to the expression which is known from the literature [7, 8]
ACP (t) =
adir
CP
= adir
CP
µ¯ ¯ ¯ ¯¶¸
·
µ¯ ¯ ¯ ¯¶
¯q ¯ ¯p¯
¯p¯ ¯q ¯
τ
x sin φ ¯¯ ¯¯ + ¯¯ ¯¯ + y cos φ ¯¯ ¯¯ − ¯¯ ¯¯ + O(x2 , y 2 , xy)
+
2
q
p
p
q
ind
2 2
(3.68)
+ τ aCP + O(x , y , xy).
44
3 Theory of CP Violation
Where we have defined
aind
CP
1
≡
2
µ
µ¯ ¯ ¯ ¯¶
µ¯ ¯ ¯ ¯¶¶
¯p¯ ¯q ¯
¯q ¯ ¯p¯
x sin φ ¯¯ ¯¯ + ¯¯ ¯¯ + y cos φ ¯¯ ¯¯ − ¯¯ ¯¯
.
q
p
p
q
aind
CP
(3.69)
¯ ¯ ¯ ¯
¯ ¯ ¯ ¯
as the term proportional to ¯ pq ¯ − ¯ pq ¯
Here, we can identify CP violation in mixing in
¯ ¯
¯ ¯
because it vanishes at ¯ pq ¯ = 1. CP violation in the interference of decay with and without
mixing is ascribed to the term proportional to sin φ, since it vanishes if φ = 0. (We
found φ 6= 0 as a necessary condition for CP violation in the interference of decay with
and without mixing in Section 3.2.4.) With Eqn. (3.68) we have a viable theoretical
prediction which can be used for comparison with experiment. Recall that we had defined
τ = tΓ. From hereon, we prefer to write it as τ = t/τD with τD = 1/Γ. We find that a
time-integrated decay asymmetry can be written as
ACP (f ) = adir
CP (f ) + ηf
< tf > ind
aCP .
τD
(3.70)
The direct CP asymmetry is dependent on the final state, since it is dependent on the
decay amplitude which is different for every final state. The indirect CP violation is
universal for all final states to a good approximation (we neglected terms proportional to
ind
adir
CP in Eqn. (3.67)), since aCP is a function of the mixing parameters only. The average
R∞
decay time for the final state f , < tf >= 0 D(t)tdt, where D(t) is the decay time
distribution of the D meson in the detector, enters the expression. In an ideal detector
<tf >
would just be one, yet, in a real detector the decay time is an effective quantity
τD
that is dependent on the properties of the detector. (The detector does not detect all
decays, especially very early decays are not seen by the detector. In an ideal detector
D(t) would just be the usual exponential function.) The effective decay time < tf >
cancels detector dependent asymmetries that would bias the asymmetry. The effective
decay time is also dependent on the final state this is because different decay processes
have smaller/ larger phase spaces and therefore are less/ more likely to occur after a
certain time. For D0 → π + π − and D0 → K + K − we have ηf = +1 the difference ∆ACP
can then be written as
∆ACP = ACP (K + K − ) − ACP (π + π − )
ind ∆ < t >
= ∆adir
.
CP + aCP
τD
(3.71)
(3.72)
45
3 Theory of CP Violation
dir
dir
+ −
dir
+ −
∆adir
CP is defined by ∆aCP = aCP (K K ) − aCP (π π ) similarly ∆ < t > is defined as
∆ < t >≡< tKK > − < tππ >. The value for aind
CP can be taken from the world average
value of [69]
aind
(3.73)
CP = (−0.025 ± 0.231)%.
The values for
∆<t>
τD
can be read of from the following table
Experiment
∆<t>
τD
<t>
τD
Reference
LHCb
CDF
0.10
0.25
2.08
2.58
[6]
[10]
We see that indirect CP violation, if it was at three sigma, would contribute
∆<t>
× aind
3σ = 0.1 × 0.7% = 0.07%
τD
(3.74)
to ∆ACP for ∆<t>
= 0.1 at the LHCb. This cannot explain the large value of ∆ACP .
τD
Because of the small value for aind
CP it is a good approximation to take
∆ACP = ∆adir
CP .
(3.75)
We can conclude that ∆ACP mainly consists of the difference of direct CP asymmetries.
At last, we want to show how Eqn. (3.72) can be used to relate indirect and direct
CP violation, if ∆ < t > is known. The collected world data with uncertainties are
summarized in Fig. 3.2.
46
∆adir
CP
3 Theory of CP Violation
0.02
∆ACP BaBar
∆ACP Belle
∆ACP LHCb
∆ACP CDF Prelim.
AΓ LHCb
AΓ BaBar
AΓ Belle
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
-0.02
-0.02 -0.015 -0.01 -0.005
0
0.005 0.01 0.015 0.02
aind
CP
ind
Figure 3.2: The measurements of ∆ACP can be used to write ∆adir
CP as function of aCP as
in Eqn. (3.72). The plot was made by the HFAG [69]. We see that the best fit
value for ∆adir
CP is almost three standard deviations separated from 0. Some
more information on the measurements of ∆ACP can be found in Section 4.3.2
4 Weak Decays of D Mesons
In the previous chapter, we established that ∆ACP is the difference of the two direct
asymmetries, we also found that direct asymmetries are generated by the interference of
at least two different processes that contribute to one decay. Yet, we did not discuss
the decay processes themselves, we will take up this task in this chapter. In principle,
the decay amplitudes can be derived from the SM Lagrangian, nevertheless, this involves
some complications. Corrections of the strong interaction ’QCD corrections’ make the
analysis difficult because not all QCD corrections can be computed from first principles.
This also hinders us from making precise predictions for ∆ACP as we will see in the third
section of this chapter.
In general, there are three different sorts of decays:
• Leptonic decays, only involving leptonic final particles.
• Semileptonic decays, involving leptonic and hadronic final particles.
• Non-leptonic decays, involving only hadronic particles.
Leptons do not interact with the strong interaction so that the analysis of leptonic and
semileptonic decays is simpler than the analysis of non-leptonic decays. Yet, we are
mainly interested in non-leptonic decays, since the decays D0 → K + K − and D0 → π + π −
are non-leptonic. These decays are described by an effective Hamiltonian that describes
the interplay of the weak and the strong interaction. The perturbative contributions
of the strong interaction are included into the (Wilson) coefficients, whereas the nonperturbative contributions are encoded into four quark operators, which result into undetermined hadronic matrix elements in an amplitude. Nevertheless, we can attribute an
approximate value to some of the matrix elements if we use a factorization ansatz. In
the factorization ansatz, the hadronic matrix elements, that describe the decay into two
particles, are factorized into two independent processes. These two processes both can be
described by non-perturbative quantities, which can be obtained from semileptonic and
leptonic decays.
In the first section, we will introduce the low-energy effective Hamiltonians which describe
47
48
4 Weak Decays of D Mesons
non-leptonic decays, so that we encounter the Hamiltonian that is relevant for the decays
D0 → π + π − and D0 → K + K − . In the second section, we introduce leptonic and semileptonic decays in order to obtain the non-perturbative quantities that are needed for the
factorization scheme. Then we introduce the factorization scheme, that can be used to
attribute approximate values to some of the hadronic matrix elements. In the last section,
we discuss the measurements of ∆ACP that were made and how they deviate from the
SM prediction obtained from the effective Hamiltonian.
4.1 Low-Energy Effective Hamiltonians
In order to describe non-leptonic decays we have to take into account corrections of the
strong interaction to the weak interaction. The strong interaction acts on the quark
constituents of the hadrons and is the most difficult part in the evaluation of decay amplitudes. This section will give us the final Hamiltonian which describes weak decays, the
weak and strong interaction are combined in effective operators, this makes use of the
operator product expansion (OPE). The basic idea of low-energy effective Hamiltonians (LEEH) is to divide in short and long-distance interactions. The short-distance
interactions describe the contributions of the heavy degrees of freedom such as the top
and bottom quark and the W boson. They can be evaluated and will be encoded into
the Wilson coefficients. The long-distance effects, however, cannot be evaluated from
first principles. This discussion is based on the review paper of Buchalla, Buras and
Lautenbacher [74], more details and all derivations can be found in this review.
4.1.1 A Simple Example of an OPE
Since we want to describe D meson decays in which CP violation appears we consider
the charged current Lagrangian LCC in Eqn. (2.27). Let us first describe a restricted
class of non-leptonic D decays, namely those which only have tree level contributions.
The flavor structure of such decays has the generic structure c → quq̄ ′ with q, q ′ ∈ {d, s}
and q 6= q ′ as indicated in Fig. 4.1. From LCC and the propagator of the W boson (in
Feynman-t’Hooft gauge) we derive the following amplitude
A=i
g2 ∗
ηµν
ūL γ ν qL′ .
Vcq Vuq′ q̄L γ µ cL 2
2
2
k − MW
(4.1)
Where MW is the mass of the W boson, g the weak coupling constant, Vcq∗ Vuq′ are CKM
matrix entries and the ψL are the left-handed part of the quark spinors. Because of the
49
4 Weak Decays of D Mesons
′
Vuq′ q
c
u
Vcq∗
q
Figure 4.1: A generic weak decay of a charm quark mediated by a W boson.
restrictions on q and q ′ given above ,this describes CF and DCS decays (See Section 2.3.3
for the definition of CF, SCS and DCS). The mass of the D mesons is around 2 GeV, if
we compare this to the mass of the W boson which is MW = 80.4 GeV we see that an
2
expansion in Mk 2 is justifiable so that the amplitude is consequently given by
W
g2
1
A = i Vcq∗ Vuq′ q̄L γ µ cL
ūL γµ qL′ + O
2
2
−MW
µ
k2
2
MW
¶
.
(4.2)
Introducing the two following notations
g2
GF
√ =
2
8MW
2
and (q̄c)V −A = q̄γµ (1 − γ 5 )c = 2q̄L γµ cL .
(4.3)
we may rewrite the amplitudes as
GF
A = −i √ Vcq∗ Vuq′ (q̄L cL )V −A (ūL qL′ )V −A + O
2
µ
k2
2
MW
¶
.
(4.4)
This amplitude could also have been obtained from the following effective Hamiltonian
GF
Hef f = √ Vcq∗ Vuq′ (q̄c)V −A (ūq ′ )V −A + . . . ,
2
(4.5)
2
where the dots indicate higher orders in k 2 /MW
. Hef f is the first example of how an
LEEH can be obtained by an OPE. This process is also referred to as ’integrating out the
W boson’, since in the path-integral formulation of these steps, the W boson is integrated
out as a dynamical degree of freedom in the generating functional. The path integral
formulation is the more formal derivation of the above effective Hamiltonian but yields
the same result. Since the W boson is no dynamical degree of freedom anymore, we obtain
an effective four point interaction as in Fig. 4.2, which is a modern version of the Fermi
four point interaction. The strength of this effective operator is given by coefficient G√F2
50
4 Weak Decays of D Mesons
c
q′
u
q
Figure 4.2: The weak decay mediated by a W boson is an effective four quark interaction
represented by the black circle.
g
g
W
a)
g
W
b)
W
c)
Figure 4.3: First order QCD corrections to the weak interaction. The QCD corrections of
a) are called factorizable, the QCD corrections of b) and c) are non-factorizable
since they alter the overall color flow.
which encodes the contribution W boson. Higher order corrections, which correspond to
operators of higher dimension can usually be neglected, we will neglect all higher order
corrections in this thesis.
4.1.2 Low-Energy Effective Hamiltonian for CF and DCS Decays
In the preceding subsection, we treated the quarks as free particles, this is unphysical,
quarks are always confined in hadrons and subject to the strong interaction. Therefore,
it is not possible to solely use the weak interaction to compute physical quantities, corrections of the strong interaction have non-negligible effects on the decay processes. Such
leading order QCD corrections to the simple tree process are shown in Fig. 4.3. In
Figs. 4.3a)-4.3c) the gluon alters the color flow of the diagram, this leads to a new four
fermion operator, which could not be generated by the weak interaction alone. The QCD
51
4 Weak Decays of D Mesons
corrections alter the effective Hamiltonian in Eqn. (4.5) to
´
³
′
GF
qq ′
Hef f = √ Vcq∗ Vuq′ C1 (µ)Qqq
(µ)
.
(µ)
+
C
(µ)Q
2
2
1
2
(4.6)
Where µ is a renormalization scale, the Ci are Wilson coefficients and the current-current
quark operators are given by
′
′
Qqq
1 ≡ (q̄α cβ )V −A (ūβ qα )V −A
′
′
and Qqq
2 ≡ (q̄α cα )V −A (ūβ qβ )V −A .
(4.7)
For the derivation of this result we refer to [74], nonetheless, let us explain its main features. A renormalization scale enters this expression because gluon loops can be formally
divergent. These divergences need to be regularized, this can be done, for instance, with
dimensional regularization in D=4-2ǫ dimensions. Regularization is needed to parametrize
the divergences. In a second step, renormalization is used to subtract these divergences
from the theory, so that all expressions become finite. The renormalization scale separates the regions of high/energy (short-distance) and low-energy (long-distance). For
D mesons µ is of the order of a few GeV. The short-distance physics is encoded in the
Wilson coefficients, they contain all physics which arise because of particles heavier than
the renormalization scale µ. This regime can be computed perturbatively because of the
asymptotic freedom of QCD.
As above, the effective four point operators arise from an OPE and higher orders have
been neglected. The physics for energies lower than µ are encoded in the effective operators Q1 and Q2 . Here, QCD corrections give rise to a second operator, which has the
name Q1 for historic reasons, this operator arose because the gluon corrections mix the
quark currents which are color singlets. Actually, usual renormalization is insufficient to
render the theory finite, a concept called operator renormalization is necessary to make
the theory finite, this results into the fact that Q1 and Q2 contribute to each other under
renormalization, this is called operator mixing under renormalization.
C1 and C2 are functions of µ which is an arbitrary scale therefore physical quantities
cannot depend on this scale, the µ-dependence in Ci (µ) must cancel the µ-dependence in
the matrix element < Qi (µ) >.
We did not bother about regularization and renormalization until now and will not any
further since it is beyond the scope of this thesis, nevertheless we see that it is important
to be included since it gives rise to new operators and alters the strength of the initial
operators. In this subsection, we found the LEEH for CF and DCS decays, the LEEH for
SCS decays involves penguin operators and will be explained in the following subsection.
52
4 Weak Decays of D Mesons
4.1.3 Low-Energy Effective Hamiltonian for SCS Decays
In the previous subsection, we gave the LEEH which describes CF and DCS decays. In
SCS decays the LEEH has a more complex form. The quark level transition has the form
c → up̄p with p ∈ {u, d, s} so that penguin diagrams as in Fig. 4.4 do contribute.
In Fig. 4.4, the gluon creates an additional quark pair, this type of diagram is called a
W
g, γ, Z 0
Figure 4.4: Another form of QCD correction is the QCD (electroweak) penguin, where an
additional quark pair is generated by a gluon (Z 0 or γ).
QCD penguin. In contrast, we call a diagram an electroweak penguin if the quark
pair is not created by a gluon but by a Z boson or a photon. Because of their flavor
structure penguin diagrams contribute only to SCS decays. We are going to neglect
the electroweak penguins, since they are suppressed by the coupling constants of the
electroweak interaction. The LEEH for SCS decays is given by
∆C=1
Hef
f
"
#
6
X
GF X
qq
= √
λq (C1 Qqq
Ci Qi + C8g Q8g + H.c.
(4.8)
1 + C2 Q2 ) +
2 q∈{d,s}
i=3
#
" 6
X
GF X
qq
= √
Ci Qi + C8g Q8g + H.c. (4.9)
λq (C1 Qqq
1 + C2 Q2 ) − λb
2 q∈{d,s}
i=3
Where the current-current operators are defined as
Qqq
1 = (q̄α cβ )V −A (ūβ qα )V −A
and Qqq
2 = (q̄c)V −A (ūq)V −A .
(4.10)
On the other hand the penguin operators are given as
Q3 = (ūc)V −A
Q5 = (ūc)V −A
X
p
X
p
(p̄p)V −A ,
(p̄p)V +A ,
Q4 = (ūα cβ )V −A
Q6 = (ūα cβ )V −A
X
p
X
p
(p̄β pα )V −A ,
(4.11)
(p̄β pα )V +A .
(4.12)
53
4 Weak Decays of D Mesons
ντ
Vcs∗
Ds+
W
τ
Figure 4.5: Leptonic decay Ds+ → τ ντ , all non-perturbative physics are related to the Ds+
meson and can be factorized into the decay constant fDs+ .
Since the quark pairs in the penguins couple to a gluon V-A and V+A currents contribute,
yet it is common to write these contributions separately. At last there is the magnetic
penguin operator which arises through the mass of the charm quark
Q8g = −
gs
mc ūσµν (1 + γ 5 )Gµν c.
8π 2
(4.13)
Evaluated at the charm quark mass µ ≈ mc the Wilson coefficients take the following
numerical values at next-to-leading order (NLO) [28]
C1 = −0.41, C2 = 1.21,
C3 = 0.02,
C4 = −0.04,
C5 = 0.01,
C6 = −0.05, C8g = −0.06.
(4.14)
With the above Hamiltonian we can describe all decays of a D meson into two light
pseudoscalars. The clear problem of the LEEH is that the matrix elements < Qi >
cannot be computed. Yet, sometimes they can be extracted from experiment, how this
can be done is treated in the following section.
4.2 Factorization
In this section, we want to introduce factorization which is a means to attribute an
approximate value to some of the hadronic matrix elements. In order to do this we need
to define some non-perturbative quantities that can be extracted from semileptonic and
leptonic decays which we are going to introduce in below.
4.2.1 Leptonic Decays
We follow the steps of Fleischer [57] to discuss as an example the leptonic decay of
Ds+ → τ ντ . This decay is depicted in Fig. 4.5. The decay amplitude for this process can
54
4 Weak Decays of D Mesons
be written as
A=i
¯
­ ¯ ν
®
ηµν
g2 ∗
¯s̄γ (1 − γ 5 )c¯ Ds+
Vcs [ūντ γ ν (1 − γ5 )vτ ] 2
0
2
8
k − MW
(4.15)
Where MW is the mass of the W boson, g the weak coupling constant and Vcs∗ is a CKM
matrix entry. Again the mass of the D mesons is around 2 GeV, so that an expansion in
k2
with k 2 ∼ m2D is justifiable, the amplitude is consequently given by
M2
W
¯
­ ¯
®
GF
A = −i √ Vcs∗ [ūντ γ ν (1 − γ5 )vτ ] 0 ¯s̄γµ (1 − γ 5 )c¯ Ds+ + O
2
µ
k2
2
MW
¶
.
(4.16)
Ignoring the higher order terms we see that all QCD dynamics are enclosed in
¯
­ ¯
®
0 ¯s̄γµ (1 − γ 5 )c¯ Ds+
(4.17)
¯
®
­ ¯
0 ¯s̄γµ γ 5 c¯ Ds+ (p) = ifDs+ pµ .
(4.18)
which describes the decay of a Ds+ meson. Since it is a pseudoscalar particle the part
h0 |s̄γµ c| Ds+ i of the matrix element must be zero h0 |s̄γµ c| Ds+ i = 0, so that we can define
We call fDs+ the decay constant of Ds+ , it contains all contributions of the strong interaction and is therefore a non-perturbative quantity which cannot be computed from
first principles. However, there are QCD sum rules approaches [75] and lattice approaches [76] that give theoretical expectations for the decay constants. The branching ratio (BR) of this decay BR(Ds+ → τ ντ ) = (5.6 ± 0.4)% can then be used to find1
fDs+ = (260.0 ± 5.4)MeV, since the other quantities are all quite well known. The definition of the decay constant is general and holds as well for the other mesons as the pions
or the kaons. We are going to use in this thesis (in MeV)
fπ = 130.4 ± 0.2,
fK = 156.1 ± 0.9,
fD = 206.7 ± 8.9,
fDs = 260.0 ± 5.4.
(4.19)
In later computations the errors are neglected.
4.2.2 Semileptonic Decays
For semileptonic decays we may take the example of D0 → K − l+ νl . In semileptonic
decays we will encounter the form factors, this a non-perturbative quantity that is used
1
See the note ’Decay constants of charged pseudoscalar mesons’ of the PDG [34]
55
4 Weak Decays of D Mesons
c
Vcs∗
W
νl
l
s
D0
K+
u
Figure 4.6: The decay D0 → K − l+ νl , the leptons to not couple to the strong interaction
so that all non-perturbative physics can be encoded into F D→K (m2l ).
to describe the transition from D meson to a kaon, as in the preceding subsection the
physics of the leptons can be separated from the physics of the quarks because the strong
interaction does not couple to leptons. The Feynman diagram of the decay is shown in
Fig. 4.6. The amplitude of the decay D0 → K − l+ νl can be written as
¯
¯
­
®
GF
A = −i √ Vcs∗ [ūνl γ ν (1 − γ5 )vl ] K − ¯s̄γµ (1 − γ 5 )c¯ Ds+ + O
2
µ
k2
2
MW
¶
.
(4.20)
Where as above we used that k 2 ∼ m2D ≪ m2W . All contributions of the strong interaction are encoded in hK − |s̄γµ (1 − γ 5 )c| Ds+ i, since K − and Ds+ are pseudoscalars we have
hK − |s̄γµ γ 5 c| Ds+ i = 0 so that we parametrize
­
®
K − (p) |s̄γµ c| Ds+ (P ) = f+DK (q 2 )Pµ + f−DK (q 2 )qµ .
(4.21)
The matrix element is a Lorentz vector, because of Lorentz invariance it can only be
proportional to combinations of Lorentz vectors in this decay that is e.g. qµ and Pµ . qµ is
the momentum transferred by the W boson, so that q 2 = m2l with m2e and m2µ negligible,
semileptonic BRs are thus only dependent on f+DK (q 2 ). Another way of writing the matrix
element that will prove useful later is
­
®
m2 − m2
m2 − m2
K − (p) |s̄γµ c| Ds+ (P ) = (Pµ − D 2 K qµ )f+DK (q 2 ) + D 2 K qµ f0DK (q 2 )
q
q
(4.22)
with f0DK (q 2 ) ≡ f+DK (q 2 ) + f−DK (q 2 )q 2 /(m2D − m2K ). Since f+DK is a non-perturbative
quantity its q 2 dependence is an unknown function. There are a number of models which
approximate this dependence. The most recent measurements of the form factors in the
D system have been made by the CLEO collaboration [77], they used the modified pole
model [78], this assumes that the q 2 dependence can be described by one main pole and
one effective pole that effectively describes all other poles. The parametrization goes as
56
4 Weak Decays of D Mesons
follows
f+ (q 2 ) = ³
1−
q2
f+ (0)
´³
´
2
1 − α mq2
m2pole
(4.23)
pole
Where mpole is the mass of D∗ /Ds∗ for pions/kaons (The definition of the form factor for
D → π transitions is completely analogous). In this thesis we are going to use their values
as [77]
f+K (0)
αK
0.739
0.30 2112MeV
f+π (0)
mDs∗
0.666
απ
mD ∗
0.21 2007MeV
.
The form factor f0 (q 2 ) is not accessible in semileptonic decays and theoretical models are
necessary to determine it, it can been shown [79] that to a good approximation f0 (q 2 )
can be identified with f+ (q 2 ) so that for further computations we are going to use this
approximation. We have now introduced all non-perturbative quantities in the charm
system, which describe leptonic and semileptonic decays. We will make use of them in
the following subsection where we introduce factorization.
4.2.3 Factorization of the Tree Amplitude
The tree amplitude of a decay such as D0 → K − π + can be factorized. That is, we can
use experimental data to attribute a value to an amplitude which cannot be computed
from first principles because of the non-perturbative character of QCD. Factorization
only takes into account QCD corrections as in Fig. 4.3a) since they can be factorized in
non-perturbative quantities, such as fP or f0DP (q 2 ) (with P being a kaon or pion), which
can be extracted from experiment. The QCD corrections as in Fig. 4.3b),c) cannot be
included in this way this is why this factorized form of the matrix element gives only an
approximate value. In order to see how factorization works let us write down the tree
amplitude of D0 → K − π + . This CF weak decay is mediated by the effective Hamiltonian
in Eqn. (4.6) with q = s and q ′ = d. We obtain
­
∆F =1
K − π + |Hef
|D0
f
®
®
GF £ ­
= √ ζ a1 K − π + |(ūd)(V −A) (s̄c)(V −A) |D0
(4.24)
2
­
®¤
a
a
+ 2C1 K − π + |(ūα Tαβ
dβ )(V −A) (s̄γ Tγδ
cδ )(V −A) |D0
(4.25)
where we made use of the SU (NC ) color algebra
a
a
Tαβ
Tγδ
1
=
2
µ
δαδ δβγ
1
δαβ δγδ
−
NC
¶
(4.26)
57
4 Weak Decays of D Mesons
to rewrite the operator Q1 and where a1 is defined by a1 =
matrix element is then given by
1
C
NC 1
+ C2 . The factorized
® ¯¯
K − π + |(ūd)(V −A) (s̄c)(V −A) |D0 ¯
f act.
®
­ + µ
®­ −
5
= π |ūγ (1 − γ )d|0 K |s̄γµ (1 − γ 5 )c|D0
­
= ifπ f0DK (m2π )(m2D − m2K ).
(4.27)
(4.28)
(4.29)
The second contribution to the amplitude proportional to C1 is just zero in the factorization approach
­ − +
® ¯¯
a
a
K π |(ūα Tαβ
dβ )(V −A) (s̄γ Tγδ
cδ )(V −A) |D0 ¯
= 0.
(4.30)
f act.
The factorization ansatz is based on the assumption that the quarks that form the pion
depart very fast from the interaction point where they are generated. If they are far
away from the other quarks, it is expected that the decay constant represents the main
contribution of the hadronization process. Even though it is not sure that factorization
also works for other, heavier particles, we will use factorization not only for decays to
pions. The factorization scheme above is only valid for decays that involve a charged final
particle. The decay D0 → π 0 K̄ 0 does not exhibit this structure but there is a second kind
of factorization which can be used for his decay, similarly to above we obtain
­
with a2 = C1 +
¯ ® ¯¯
¯
π 0 K̄ 0 ¯ Hef f ¯D0 ¯
1
C.
NC 2
f act.
GF
= i √ ζa2 fπ f0DK (m2π )(m2D0 − m2K )
2
(4.31)
4 Weak Decays of D Mesons
58
4.3 The Problem
We now found how we can describe decay amplitudes for D mesons, in this section we
will see that this treatment leads to a tension between the SM and the measurements.
We first will derive the SM prediction for ∆ACP and then give an outline of the different
measurements and see that they are in tension with the SM. We will then use the experimental data to see how large the subleading amplitude would need to be to accommodate
the measured value.
4.3.1 The SM Prediction for the Observable
We want to find the SM prediction for ∆ACP . As we saw in Section 3.3.2 the observable
∆ACP can be identified with the difference of the two direct CP asymmetries,
+ −
dir
+ −
∆ACP = adir
CP (K K ) − aCP (π π )
(4.32)
in a very good approximation. The single direct asymmetries are solely depending on the
amplitudes contributing to the respective decay. Hence we need to write down the decays
processes that contribute to the decays D0 → π + π − and D0 → K + K − . We draw the
diagrams that contribute to these decays in the full theory in Fig. 4.7. So that we might
write down the amplitude as
P
p
λd (Tππ + Eππ ) + p λp (Pππ
+ P Apππ )
s
b
− P Asππ ),
λd (Tππ + Eππ − ∆Pππ − ∆P Aππ ) + λb (Pππ
+ P Abππ − Pππ
P
p
λs (TKK + EKK ) + p λp (PKK
+ P ApKK )
b
d
λs (TKK + EKK + ∆PKK + ∆P AKK ) + λb (PKK
+ P AbKK − PKK
− P AdKK ).
(4.33)
With p running over {d, s, b}, in the second equalities we defined ∆PP P ≡ PPs P − PPd P
and ∆P AP P ≡ P AsP P − P AdP P for P ∈ {π, K} and made use of the unitarity of the
CKM matrix. These are the leading amplitudes that contribute to the decays. We find
Aππ =
=
AKK =
=
59
4 Weak Decays of D Mesons
u
Vud (Vus )
u
π − (K −)
D0
π − (K −)
D0
W
d(s)
Vcd∗ (Vcs∗ )
c
d(s)
c
u
Vcd∗ (Vcs∗ )
u
u
W+
π + (K +)
Vud (Vus)
d(s)
π + (K +)
u
π − (K −)
d(s)
u
π − (K −)
D0
c
c
Vcq∗
Vcq∗
d(s)
d(s)
q ∈ {d, s, b}
D0
q ∈ {d, s, b}
d(s)
Vuq
π + (K +)
W+
u
u
Vuq
d(s)
u
π + (K +)
Figure 4.7: Amplitudes that contribute to the decays D0 → K + K − and D0 → π + π − : Tree
amplitude, exchange amplitude, penguin amplitude and penguin annihilation
amplitude. In the latter three one or more gluons or weak gauge bosons create
a quark pair, these are not displayed because they can be attached anywhere in
the diagram. These are the two lowest order diagrams which are proportional
to λd /λs or λb .
expressions for this in the effective theory if we use Hef f in Eqn. (4.9)
Aππ = hπ + π − | Hef f |D0 i
¡
¢
P6
dd
0
+ −
0
= G√F2 λd hπ + π − | C1 Qdd
C
Q
+
C
Q
|D
i
i
i
8g
8g
1 + C2 Q2 |D i − λb hπ π |
i=3
AKK = hK + K − | Hef f |D0 i
¡
¢
P6
dd
0
+ −
0
= G√F2 λs hK + K − | C1 Qdd
C
Q
+
C
Q
|D
i
+
C
Q
|D
i
−
λ
hK
K
|
2
b
i
i
8g
8g
1
2
i=3
(4.34)
In both cases we might rewrite the amplitude in general as
AP P = λq TP P + λb PP P ,
ĀP P = λ∗q TP P + λ∗b PP P .
(4.35)
(4.36)
With P = K (π) and q = s (d). In the second equation we give also the CP-conjugated
amplitude which is the decay amplitude of D̄0 to the final state PP. With this parametriza-
60
4 Weak Decays of D Mesons
tion the direct CP asymmetry in Eqn. (3.3) can be written as
õ ¶ !
¶
2
P
λb
PP
adir
(P
P
)
=
−2ℑ
ℑ
+
O
CP
TP P
λq
Ã
¯
µ ¶2 !
µ ¶¯
λb
λb ¯¯ PP P ¯¯
sin δP P + O
= −2ℑ
¯
¯
λq
TP P
λq
µ
λb
λq
¶
µ
(4.37)
(4.38)
¯ ¯
¯ ¯
where we used ¯ λλqb ¯ ∼ 10−3 as an expansion parameter and defined the relative phase
δP P ≡ arg PTPPPP . We prefer to keep the somewhat unusual ℑ(λb /λq ) since it is just a
numerical factor, its value can be taken from Eqn. (2.45) and is with good precision
µ ¶
λb
= −6 × 10−4
ℑ
λs
µ ¶
λb
and ℑ
= 6 × 10−4 .
λd
(4.39)
So that we obtain in total
∆ACP
¯
¯
¯
¶ ·¯
¸
¯ PKK ¯
¯ Pππ ¯
λb
−4
¯
¯
¯
¯
= − 2ℑ
¯ TKK ¯ sin δKK + ¯ Tππ ¯ sin δππ ∼ 10 .
λs
{z
}
| {z } |
µ
∼10−3
(4.40)
C
∼ C i ∼10−1
2
The main suppression results from the ratio of CKM elements further suppression of
∆ACP is expected because of the scaling of the Wilson coefficients in Eqn. (4.34). We
now have got an SM expectation for ∆ACP , yet we cannot evaluate the hadronic matrix
elements in Eqn. (4.34), so that we do not have a precise value. We expect that the ratio
of hadronic matrix elements is O(1) but because of the long-distance effects no value can
be attributed to this ratio.
4.3.2 The Measurements
In the previous subsection we derived the SM expectation for ∆ACP , in this subsection
we are going to describe which different measurements have been made and why they are
interesting so that we can put them into relation to the expression for ∆ACP we found
in Eqn. (4.40) in the next subsection. The LHCb collaboration announced that they
had found CP violation in the charm system which is, given the current sensitivity of
experiments at percent level, thought to be a clear sign of NP [7, 8]. They measured
the difference of two time-integrated asymmetries which has the convenient property that
it profits from cancellations of systematic uncertainties. Additionally, the two single
asymmetries are expected to be of similar size but of opposite sign so that they effectively
61
4 Weak Decays of D Mesons
add up in the difference. The following value was found2 [6]
0
+ −
0
+ −
∆Adir
CP ≡ ACP (D → K K ) − ACP (D → π π ) = [−0.82 ± 0.21 ± 0.11]%.
(4.41)
This measurement deviates 3.5 Gaussian standard deviations from zero, older measurements for this quantity stem from BaBar [80] and Belle [81] and show no significant CP
violation. Soon after the publication of the LHCb result, CDF published a measurement
of the single asymmetries [82]
ACP (π + π − ) = (+0.22 ± 0.24 ± 0.11)%,
(4.42)
ACP (K + K − ) = (−0.24 ± 0.22 ± 0.09)%,
(4.43)
∆ACP = (−0.46 ± 0.31 ± 0.12)%
(4.44)
so that the difference
does not show any significance. Nonetheless, the LHCb measurement benefits from uncertainty cancellations so that it can be regarded as more reliable than the CDF result
for ∆ACP which is just the mathematical combination of the two measurements and does
not exhibit this convenient feature. Additionally, the CDF collaboration supported the
LHCb measurement, after they had reanalyzed their data and had computed the asymmetry directly with a value of [10]
∆ACP = [−0.62 ± 0.21 ± 0.10]%.
(4.45)
The HFAG gives a combination of all existing relevant results as [69]
∆Adir
CP = (−0.656 ± 0.154)%.
(4.46)
This result is the current world average so that we are going to use it as the value for
∆ACP throughout this thesis. It is, against expectations, of percent order and serves
as a good reason to investigate the phenomenology of charm decays in general and this
asymmetry difference in particular.
Before we confront the measurement of ∆ACP with the expression in Eqn. (4.40), we
want to point out that there is actually another charm decay which shows CP violation
2
All uncertainties on experimental results, here and below, are given in the order: (value)±(statistical
uncertainty) ± (systematic uncertainty).
62
4 Weak Decays of D Mesons
it is the decay D+ → KS0 π + where the following CP asymmetry [83]
Γ(D+ → KS0 π + ) − Γ(D− → KS0 π − )
ACP (D →
≡
= (−0.363 ± 0.094 ± 0.067)%,
Γ(D+ → KS0 π + ) + Γ(D− → KS0 π − )
(4.47)
shows CP violation with a deviation of 3.2 standard deviations from zero. Yet, this is is
no sign of CP violation in charm transitions since the above value is completely consistent
with the CP violation predicted from K 0 − K̄ 0 mixing which leads to the composition of
ACP (D+ → KS0 π + ) into
+
KS0 π + )
0
K
ACP (D+ → KS0 π + ) = A∆C
CP + ACP .
(4.48)
This leads to a CP asymmetry for the charm transition of [83]
+
A∆C
CP (D ) = (−0.018 ± 0.094 ± 0.068)%
(4.49)
which is completely consistent with zero.
4.3.3 The Tension with the SM
We already pointed out that the asymmetry is significantly higher than expected. If
∆ACP is generated by the SM penguins, then these must be larger than expected. We
can actually put their size into relation with the size of the trees TKK and Tππ . The
absolute size of the denominators TP P in ∆ACP can be found from the branching ratios
(BRs) by the Eqn. (5.16), since we know that all processes which are proportional to λq
contribute to the BRs whereas PP P which is proportional to λb is negligible for the BRs,
because |λb /λd | ∼ 10−3 is small. We find
|Tππ | = |Tππ + Eππ − ∆Pππ − ∆P Aππ | = 2.1 keV
(4.50)
|TKK | = |TKK + EKK + ∆PKK + ∆P AKK | = 3.8 keV
(4.51)
We can now use this numerical information to estimate the minimum size of the CP
asymmetry generating penguin. In order to do this we assume flavor SU(3) symmetry
(An introduction to SU(3) flavor symmetry will be given in Section 5.1.1) to hold for the
penguins so that we find for the penguin amplitudes P = PKK = Pππ , additionally this
results into the same relative phase so that we define δ ≡ δKK = δππ , it follows that the
63
4 Weak Decays of D Mesons
observable ∆ACP has the form
∆ACP
µ
λb
= −(0.656 ± 0.154)% = −2ℑ
λs
¶µ
1
1
+
|Tππ | |TKK |
¶
|P| sin δ.
(4.52)
This requires that
ℑP = −(7.3 ± 1.7) keV
⇔
|ℑP/T | = 2.5 ± 0.6,
(4.53)
where we give the values which correspond to the one sigma bounds of ∆ACP , the lower
bound corresponds to a smaller absolute value of ∆ACP . For T we took the average value
of both trees T ≡ (Tππ + TKK )/2, we see that the world average requires a penguin that
is about 2.5 times larger than the size of the average tree amplitude. The approximations
we have taken do not alter this conclusion for |P/T |, they could only increase the value
that is necessary for ∆ACP . The approximation in Eqn. (4.37) increases the absolute
value of adir
CP (P P ). Not taking the same relative phase between TP P and PP P would as
well decrease the value of ∆ACP .
Furthermore, the condition above requires that the penguin amplitude has a relative
strong phase of π2 the more the phase deviates the larger the penguin would need to be
this is illustrated in Fig. 4.8.
In conclusion this means that to accommodate the world average of ∆ACP = (−0.656±
0.154)%, a value as in Eqn. (4.53) or higher is undoubtedly necessary. The expectation
from the short-distance physics is that the penguins are suppressed with respect to T,
any enhancement must be due to long-distance physics. The challenge is whether we can
find anyhow evidence that large penguin amplitudes are possible or even favored.
64
4 Weak Decays of D Mesons
ÈPTÈ
6
5
4
3
2
1
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
∆
Figure 4.8: The normalized penguin |P/T | = |PKK /T | = |Pππ /T | as a function of the
relative phase to Tππ and TKK . The stunning result is that, assuming the
world average, the penguin must be at least 2.5× as large as the ’tree’ T =
(Tππ + TKK )/2. The upper dashed line indicates the upper one sigma bound
of ∆ACP the lower dashed line the lower one. (The graph in the range (π, 2π)
looks identical.)
5 Symmetries of the Decay Amplitudes
With the measurement presented in Section 4.3.2 we face a clear tension between the
SM expectation and the measurement. How indirect CP violation contributes to ∆ACP
was discussed in the Section 3.3.2, indirect CP violation contributes to the observable,
but the main contribution to the observable stems from direct CP violation. Thus, we
are going to investigate in the subsequent sections how the understanding of the decay
amplitudes can be improved, since direct CP violation is generated by the interference
of different amplitudes that contribute to one decay. Since we found that we cannot determine the hadronic matrix elements, we will use the SU(3) symmetry of Hef f to relate
different hadronic matrix elements and parametrize them with a small set of parameters.
This is called the topological approach. Since ∆ACP is the first asymmetry that has been
measured with a value that has a significant deviation from zero, we will mainly discuss
the generation of BRs, which are largely known, to understand the structure of the decay
amplitudes.
In the first section, we introduce the SU(3) flavor symmetry and the topological approach
which is based on the SU(3) symmetry. In the second section we investigate SU(3) breaking and find a way to generate a large value for ∆ACP .
5.1 The Topological Approach
The evaluation of the effective quark operators in Eqns. (4.10 - 4.13) as well as the evaluation of the full theory from first principles is impossible because of the non-perturbativity
of the strong interaction at low energies. There are different frameworks to attribute a
value to these operators, in the B system a model which uses the fact that the b quark
is heavy (mb ∼ 5 GeV) can be used to make viable predictions. For the pions the chiral
approximation is sensible where the light quarks u and d are taken massless. However, in
the D system the mass of the charm quark mc ∼ 1.3 GeV is such that it can neither be
approximated with zero nor is it sufficiently large to use a heavy quark approximation.
Thus, neither the framework used for B mesons nor the framework used for the pions
65
66
5 Symmetries of the Decay Amplitudes
can be used for D mesons, they can only be used to attain crude estimates in the charm
system. Yet, there is a model independent approach based on a flavor SU (3) symmetry
which can be used for non-leptonic meson decays to estimate and relate decay amplitudes
called the topological approach (or quark diagram approach).
In this section, we will introduce the flavor SU (3) symmetry and the topological approach. We conclude with attributing values to the topologies and pointing out some
puzzling aspects of D meson BRs.
5.1.1 Flavor SU(3) Symmetry
For the masses of the three lightest quarks u, d and s holds the following equality
mu ≈ md ≈ ms ≪ ΛQCD
(5.1)
assuming the masses as equal, this introduces a new symmetry in Hef f called flavor SU(3)
(or SU (3)F ) symmetry.1 The quarks form consequently a triplet that transforms under
SU(3)
 
u
 
(5.2)
ψ =  d .
s
The antiquarks form an antitriplet which can be written as
¡
¢
¯ s̄ .
ψ̄ = ū, d,
(5.3)
Mesons consist of an antiquark and a quark, so that the light pseudoscalar mesons form
an octet and a singlet, 3 ⊗ 3̄ = 8 ⊕ 1, the octet consists of the following particles
¯
π + = ud,
π0 =
√1 (uū
2
¯
− dd),
π − = dū,
¯ K − = sū
K + = us̄, K 0 = ds̄,
K̄ 0 = sd,
η8 = √16 (uū + dd¯ − 2ss̄)
The singlet is
1
1
η0 = √ (uū + dd¯ + ss̄).
3
(5.4)
(5.5)
Strictly speaking also corrections of quantum electrodynamics must be neglected. In the rest of this
thesis, whenever we refer to SU(3), we refer to this flavor symmetry and not the QCD gauge group.
67
5 Symmetries of the Decay Amplitudes
The states η8 and η0 are not the mass eigenstates, they are subject to octet singlet mixing,
the real particles are η and η ′ , they are defined as
η ≡ η8 cos θη − η0 sin θη ,
(5.6)
η ′ ≡ η8 sin θη + η0 cos θη .
(5.7)
The mixing angle θη is around 20◦ [84] (and reference [7] therein) so that a good approximation is to take θη = 19.5◦ = arcsin(1/3), η and η ′ can be consequently written
as
1
η = √ (uū + dd¯ − ss̄),
3
1
η ′ = √ (uū + dd¯ + 2ss̄).
6
(5.8)
(5.9)
Yet, we are going to use a parametrization in the quark flavor basis. Using
1
¯ and ηs = ss̄
ηq = √ (uū + dd)
2
(5.10)
we introduce the angle φ in2
Ã
η
η′
!
=
Ã
cos φ − sin φ
sin φ cos φ
!Ã
ηq
ηs
!
(5.11)
where the mixing angle φ = 40.4 ± 0.6◦ has been determined by the KLOE collaboration
[85]. The central value and its small error are not undisputed [86] and φ might even
range between 30◦ and 45◦ , but this goes beyond the discussion in this thesis, so that
we use the given value. Additionally, we ignore that there is a gluonic component which
does contribute to this mixing. We include the η − η ′ mixing to be able to use more
experimental information but it is not our main goal to describe η and η ′ so that the
use of the angle φ = 40.4 ± 0.6◦ is sufficient for this thesis. Note that Eqn. (5.9) and
Eqn. (5.11) deviate in their coefficients at most by 10% so that the differences that are
obtained because of the different parametrizations should be negligible.
In fact, η − η ′ mixing implies that SU(3) is broken, since under SU(3) all 9 mesons would
be massless and mixing would not occur. This means that by using the mixing angle φ
we take into account a certain kind of SU(3) breaking. We encounter with η − η ′ mixing
one of the limits of the SU(3) assumption, which we just made. However, it is necessary
2
This φ is not related to the φ which is used as the phase of λf .
68
5 Symmetries of the Decay Amplitudes
that we include η − η ′ mixing, since we could not make use of any data that involves η or
η ′ otherwise. The neutral kaons K 0 and K̄ 0 are subject to mixing as explained in Section
3.2.1. In experiments we only find the mass eigenstates as in Eqn. (3.14)
¯ ®
¯ ®
|KS i ≡ pK ¯K 0 + qK ¯K̄ 0 ,
¯ ®
¯ ®
|KL i ≡ pK ¯K 0 − qK ¯K̄ 0 .
(5.13)
¯ D+ = cs̄.
D0 = cū, D+ = cd,
s
(5.14)
(5.12)
The kaon mass eigenstates short S and long L are distinguished by their mean lifetime
rather than by their mass, since the masses are very similar and their mean lifetimes are
very different. If CP was conserved KS would be CP even and KL would be CP odd and
pK = qK = √12 , this is a very good approximation for our purposes, so that we adopt
pK = qK = √12 in what follows.
The charm quark is a SU(3) singlet. The charmed mesons with c = 1 form thus an
antitriplet
The charmed meson with c = −1 form a triplet
D̄0 = c̄u,
D− = c̄d,
Ds− = c̄s.
(5.15)
The group SU(3) has three SU(2) subgroups, consequently there are three flavor SU(2)
symmetries. The most renown subgroup is the isospin symmetry in which the u and
the d quark transform into each other. In analogy there exists U-spin, for the symmetry
which connects the d and the s quark and V-spin which connects u and s quarks. These
symmetries are broken by the quark masses. Isospin is the symmetry which is the least
broken. The masses of the u and s quark differ the most, so that the V-spin symmetry is
expected to broken the most.
5.1.2 Decomposition of D Meson Decay Amplitudes into Topologies
Based on the SU(3) symmetry introduced above, we can decompose all charmed meson
decays to light pseudoscalars, generally denoted by D → P P , according to the structure
of the weak interaction and the flavor flows into topologies. Corrections of the strong
interaction are included in these topologies up to all orders, therefore they must not be
confused with Feynman diagrams. For D meson decays we find six relevant topologies.
These topologies can be used to parametrize all decay amplitudes A(D → P P ). We can
5 Symmetries of the Decay Amplitudes
69
then extract the size and the (strong) phase of the topologies by relating |A(D → P P )|2
to the experimental BRs BR(D → P P )exp . If we have extracted the topologies, we
can estimate the size of other BRs. In view of the effective operators, which we cannot
evaluate, the topological approach represents our best handle on the decay amplitudes
since it gives us the most reliable information.
Historically the topological approach has been introduced by Chau et al. in the D system
[87–89] and by Gronau et al. [90–92] in the B system. Even though we will apply the
topological approach only to D → P P decays, it is useful for all non-leptonic decays
including decays to vector mesons.
The six relevant decay topologies are tree diagrams T, color-suppressed tree diagrams
C, weak exchange diagrams E, weak annihilation diagrams A, gluonic penguins P and
penguin annihilation diagrams PA, their structure and characteristics are given in what
follows.
Tree Amplitudes T and C
A generic weak decay of a D meson c → quq̄ ′ as in Fig. 4.1 mediated by a single W ±
boson gives rise to two possible topologies: The spectator quark, i.e. the other quark in
the initial D meson that does not interact via the W boson, either can recombine with q
to form a final meson or with u to form a final meson, this is depicted in Fig. 5.1. If u and
q̄ ′ combine to a final meson, they form a charged color singlet which is independent of the
color flow in the rest of the diagram. This is called a tree topology (T) or external W
emission, in contrast to the color-suppressed tree (C) or internal W emission, which
is naively suppressed by a factor of 1/3 with respect to T because the color flow in the
two final mesons is fixed by the initial D meson. The tree topologies are expected to have
Figure 5.1: Tree topology T (left) and color suppressed tree topology C (right). Topologies
include all quantum corrections so that , for example, even though no gluons
are displayed, any gluon correction is included.
the largest contribution to the full amplitude.
5 Symmetries of the Decay Amplitudes
70
Annihilation Amplitudes A and E
In the case of a charged (uncharged) initial meson a weak annihilation (exchange)
topology A (E) is possible. Both initial quarks interact via the W boson, so that there
is no spectator quark. A generic exchange topology E and a generic annihilation topology
A are depicted in Fig. 5.2. E and A are suppressed with respect to T because a quark
Figure 5.2: Exchange topology E (left) and annihilation topology A (right).
pair needs to be generated from the vacuum whose quarks recombine subsequently with
the other two quarks. If we say that the quark pair needs to be generated from the vacuum we mean, that the strong interaction creates this quark pair. In principle, it is also
possible that strong interaction creates one of the final mesons and the two quarks of the
W vertices create the other quark but this process which belongs to the class of singlet
diagrams is strongly suppressed.
Penguin Amplitudes P and PA
Another possible diagram is that the quark which is generated at the first vertex interacts
also with the second vertex. These diagrams are called penguin diagrams for historical
reasons. Since the intermediate quark can be a d, s or b, there are actually six penguin
topologies, which we need to distinguish. Because of their flavor structure penguin topologies in D meson decays only occur in SCS decays, their generic structure is given in Fig.
5.3. In contrast to P, a PA topology is only possible if the initial quark is uncharged. P
and PA are expected to be small since the Wilson coefficients of the penguin operators
in Eqn. (4.9) are small. However, they may be quite important as we will discuss in the
next section.
Other diagrams as electroweak penguins, where the quark pair in the left diagram of Fig.
5.3 is generated by a neutral electroweak gauge boson (Z 0 or γ) or diagrams including
color singlets generated from the vacuum are neglected throughout this thesis. A priori,
they are small because of αQED or αQCD suppression with respect to the six diagrams
5 Symmetries of the Decay Amplitudes
V∗cp
71
Vup
p ∈ {d,s,b}
Figure 5.3: Penguin topology P (left) and penguin annihilation topology PA (right). All
possible gluon combinations that can create the final state are included in the
value of a topology.
introduced above. Investigations on their size have shown that they are actually negligible
[84]. Therefore, we limit the selection of topologies to the ones above.
5.1.3 Fits
After having introduced the topologies, the strategy is as follows: We will parametrize
all charm decay amplitudes D → P P in the language of the topological approach. Then,
since we cannot compute the value of the topologies from first principles, we will extract
their values from the data and use them to predict other BRs. If this works, we know that
SU(3) is a good symmetry to describe charm decays. Of course, we do not expect that
the symmetry holds exactly because the non-zero masses break it. Yet, if the breaking is
not to large, the SU(3) description should work quite well.
In the Tables 5.1-5.2 we give the parametrizations of D → P P decays in terms of the six
above topologies ordered by Cabibbo suppression. The decay amplitudes for D̄0 , D− and
Ds− to the CP-conjugated final states can be obtained by complex conjugating the CKM
factors, ζ, λd , λs and ξ, in the amplitude decomposition, the parameters were introduced
in Section 2.3.3. The η − η ′ mixing is taken into account by using the mixing angle φ.
If it comes to the kaons, we list K 0 and K̄ 0 in the final states. Yet, since the physical
particles are KS0 and KL0 and not K 0 or K̄ 0 , we have to approximate Γ(D → X K̄ 0 ) =
2Γ(D → XKS0 ). This assumption can in principle be spoiled for CF and DCS by their
mutual interference, nevertheless we expect that the CF decays are only changed little by
this effect. The DCS decays are not measured, so that the inclusion of interference effects
for these decays is not necessary yet.
We note that the penguin topologies Pd and Ps and P Ad and P As are equal under
SU(3) symmetry so that they often have been neglected, taken into account that they are
multiplied by λd and λs respectively, which are of equal size and of opposite sign. The
72
5 Symmetries of the Decay Amplitudes
Meson
D0
D+
Ds+
Final
State
π+K −
π 0 K̄ 0
K̄ 0 η
K̄ 0 η ′
π + K̄ 0
π+π0
π+η
π+η′
K + K̄ 0
Amplitude Representation
ζ(T + E)
√ζ (C − E)
2
φ)
√ (C + E) − sin φE)
ζ( cos
2
ζ( sin√2φ) (C + E) + cos φE)
ζ(T + C)
0
√
ζ(− sin φT +
√ 2 cos φA)
ζ(cos φT + 2 sin φA)
ζ(C + A)
pCM
[MeV]
861
860
772
565
863
975
902
743
850
BRexp
BRf it
[%]
[%]
3.87 ± 0.05 3.88
2.38 ± 0.10 2.34
0.90 ± 0.08 0.96
1.88 ± 0.10 1.90
2.93 ± 0.09 2.94
<0.034
0
1.83 ± 0.15 1.79
3.94 ± 0.33 3.79
2.96 ± 0.16 2.94
Table 5.1: Listing of CF D → P P decays with ζ = Vcs∗ Vud . Representation of the amplitude in the topological approach, center-of-mass momentum, experimental [34]
and fitted BRs given by the CF fit. We used the η − η ′ mixing as introduced
in Section 5.1.1.
same is true for (λd + λs )E in D0 → K 0 K̄ 0 . However, as will be shown later SU(3) seems
to be broken so that it is preferable to write down all contributing topologies here. The
penguins with an internal b quark are multiplied by λb . Numerically |λb | ≪ |λs | ≈ |λd |
holds, so that they are irrelevant for the BRs, nevertheless they are important for ∆ACP
so that we included them here. The experimental values BRexp are taken from the live
version of the PDG [34]. BRs and amplitudes are related by the following equation
BR(D → P1 P2 ) =
Γ(D → P1 P2 )
1 pCM
|AP1 P2 |2 .
=
2
ΓD
ΓD 8πmD
(5.16)
Where ΓD and mD are the total decay width and the mass of the meson D as they were
given in Section 3.3.1, and pCM is the absolute value of the momentum of P1 and P2 in
the center-of-mass frame given by
pCM =
s
m4D + m4P1 + m4P2 − 2m2D m2P1 − 2m2D m2P2 − 2m2P1 m2P2
4m2D
(5.17)
where mP1 and mP2 are the masses of the final pseudoscalar particles. The BRs for DCS
decays are of order 10−4 so that some could not be measured to date.
Let us now use the topological approach for CF decays of D mesons, we want to see
how large the topologies are and whether the SU(3) symmetry gives a good description
of the data. We are going to extract the sizes and strong phases of the topologies, no
penguin topologies can contribute in CF decays, so that there remain four topologies
73
5 Symmetries of the Decay Amplitudes
Meson
D0
D+
Ds+
Final
State
π−K +
π0K 0
K 0η
K 0η′
π+K 0
π0K +
K +η
K +η′
K +K 0
Amplitude Representation
ξ(T + E)
√ξ (C − E)
2
φ)
√ (C + E) − sin φE)
ξ( cos
2
ξ( sin√2φ) (C + E) + cos φE)
ξ(C + A)
√ξ (T − A)
2
√ φ (T + A) − sin φA)
ξ( cos
2
√ φ (T + A) + cos φA)
ξ( sin
2
ξ(T + C)
pCM
[MeV]
861
860
772
565
863
864
776
571
850
BRexp
BRf it
[10−4 ]
[10−4 ]
1.47 ± 0.07 1.11
0.67
0.28
0.54
1.98
1.83 ± 0.26 1.57
1.08 ± 0.17 0.97
1.76 ± 0.22 0.89
0.37
Table 5.2: Listing of DCS D → P P decays with ξ = Vcd∗ Vus . Representation of the amplitude in the topological approach, center-of-mass momentum, experimental
[34] and fitted BRs given by the CF fit. The BRs for the decays to η ′ s is taken
from a recent measurement of the Belle collaboration [93].
which possibly contribute. Since only relative strong phases matter, we can choose the
tree topology to be real. Hence, there are eight measured BRs and 7 free parameters. A
χ2 minimization fit, gives the values for the topologies (in eV)
◦
T = (3112 ± 18),
C = ((2589 ± 42e±i(207±1) )),
+14 ◦
◦
E = ((1538 ± 53)e±i(121±1) ), A = ((367 ± 47)e±i(30−29 ) ).
(5.18)
The topologies need to have the dimension energy as can be seen from Eqn. (5.16). The
numerical precision in these amplitudes is, of course, unphysical and a purely mathematical result we may rewrite this by normalizing to the tree amplitude
◦
T = 3.1 keV,
C/T = (0.83 ± 0.01)e±i(207±1) ,
+14 ◦
◦
E/T = (0.49 ± 0.02)e±i(121±1) , A/T = (0.12 ± 0.02)e±i(30−29 ) .
(5.19)
As expected T is the dominating topology but C and E are considerably larger than
expected from the short-distance behavior of these topologies. The value for χ2 = 1.28
indicates that this is a good fit, so that the topological approach which assumes SU(3)
is a viable approach to describe the decay amplitudes. The errors on the quantities
are obtained by varying the χ2 function to χ2 = 2.82 keeping all but one parameter
fixed. One should not overestimate the physical meaning of these uncertainties, since the
experimental BRs have mostly errors in the range of 5% to 10%, the errors in the fit
have more a mathematical meaning than a physical one. They tell us how strongly the
5 Symmetries of the Decay Amplitudes
74
topology values are mathematically determined by the fit. The twofold ambiguity of the
phases is due to the fact that the absolute value of an amplitude only involves the cosine
of the phase which is insensitive to the sign of the phase.
The values agree very well with those obtained by Cheng and Chiang [94]. They also
agree well with the result obtained by Bhattacharya and Rosner [84], although there are
larger deviations which are probably due to the fact that they use the parametrization
as in Eqn. (5.9) for the description of η − η ′ mixing. The relatively large values for
C and E are due to long-distance effects of the strong interaction which are difficult to
estimate. The amplitude for Ds+ → π + π 0 is zero because of isospin symmetry under which
all contributions to this decay mode cancel. The fit values given for SCS and DCS decays
are computed by using the values of Eqn. (5.18) and by setting P = P A = 0. We see
that there are some large deviations from the experimental values, the BRs with a final
η (′) have still huge errors, so that we will not worry about them to much. For D0 → π + π −
and D0 → K + K − the fit values and the experimental values do not coincide at all, even
though they are equal under U-Spin. This is a long standing puzzle, which was already
discovered some thirty years ago [87]. Another puzzling result is that BR(D0 → K 0 K̄ 0 ) is
of the same order as the other BRs, whereas we would expect to it to be largely suppressed
by ∼ |λb /λd |2 . This is a clear sign that SU(3) symmetry must be broken or that some
NP does contribute. If we do not understand the BRs of the decays D0 → K + K − and
D0 → π + π − then it is not surprising that we do not understand ∆ACP , we will improve
our understanding of the BRs in the following section by introducing SU(3) breaking this
will also help us to better understand ∆ACP .
75
5 Symmetries of the Decay Amplitudes
Meson
D0
Final
State
π+π−
π0π0
π0η
π0η′
ηη
ηη ′
D+
K +K −
K 0 K̄ 0
π+π0
π+η
π+η′
Ds+
K + K̄ 0
π+K 0
π0K +
K +η
K +η′
Amplitude Representation
pCM
[MeV]
λd (T + E) + λp (Pp + P Ap )
922
λp
λ
d
√ (C − E) + √ (Pp + P Ap )
923
2
2
λ
−λd E cos φ− √s2 C sin φ+λp Pp cos φ
846
λs
√
−λd E sin φ+ 2 C cos φ+λp Pp sin φ
678
λd
C
2
√
(C + E) cos φ − λs ( 2 sin 2φ −
755
√2
λp
2
2
2E sin φ) + √2 (Pp + P Ap ) cos φ
λd
(C + E) sin 2φ + λs ( √12 C cos 2φ −
537
2
E sin 2φ) + λp (Pp + P Ap ) sin 2φ
λs (T + E) + λp (Pp + P Ap )
791
(λd + λs )E + λp 2P Ap
789
λd
√
(T + C)
925
2
λd
√
848
(T + C + 2A) cos φ − λs C sin φ +
√2
2λp Pp cos φ
λ
d
√ (T + C + 2A) sin φ + λs C cos φ +
681
√2
2λp Pp sin φ
λs T + λd A + λp Pp
793
λd T + λs A + λp Pp
916
√1 (−λd C + λs A + λp Pp )
917
2
cos
φ
835
(λd C + λs A + λp Pp ) √2 − (λs (T +
C + A) + λp Pp ) sin φ
√ φ + (λs (T +
(λd C + λs A + λp Pp ) sin
646
2
C + A) + λp Pp ) cos φ
BRexp
BRf it
−3
[10 ]
[10−3 ]
1.400 ± 0.026 2.21
0.80 ± 0.05
1.34
0.68 ± 0.07
0.75
0.89 ± 0.14
0.73
1.67 ± 0.20
1.41
1.05 ± 0.26
1.19
3.96 ± 0.08
0.346 ± 0.058
1.19 ± 0.06
3.53 ± 0.21
1.90
0
0.84
1.47
4.67 ± 0.29
3.59
5.66 ± 0.32
2.42 ± 0.16
0.62 ± 0.21
1.75 ± 0.35
5.35
2.68
0.85
0.76
1.8 ± 0.6
1.04
Table 5.3: SCS D → P P , we give the amplitude representations in topological approach,
center-of-mass momenta, experimental [34] and fitted BR given by the CF fit.
We used the η − η ′ mixing as introduced in Section 5.1.1 and used λp = Vcp∗ Vup
where p runs over {d, s, b} for the penguin topologies with an implicit sum over
p understood.
5 Symmetries of the Decay Amplitudes
76
5.2 Flavor Symmetry Breaking
How well the topological approach describes the BR data of D → P P decays under the
assumption of SU(3) symmetry was explained in the previous section. It was found that
the topological approach describes very well the CF data, yet this is not too surprising
since a fit of 8 data points to 7 parameters is used. The description of BRs of SCS
and DCS decays, however, is less successful. It seems that the exact SU(3) symmetry
is insufficient to get a good description of the BRs, this also implies that SU(3) is not
sufficient to describe ∆ACP , since the understanding of the BRs is intimately connected
with the understanding of ∆ACP . Hence, we are going to include SU(3) breaking which
can be done in different ways. We will investigate the following three:
Naive Factorization Naive factorization as introduced in Section 4.2.3 can be used to
correct the SU(3) amplitudes with factorizable quantities. The main question is how
well factorization works in the charm system, that is how large the non-factorizable
corrections are.
Breaking in Exchange Topologies The BR(D → K 0 K̄ 0 ) is zero under SU(3) the relatively large value as in Tables 5.3 could be caused by exchange topologies, that
exhibit large SU(3) breaking.
Large SU(3) breaking Penguins The contributions of penguin topologies to the BRs is
negligible under SU(3), if the penguin topologies are large their contributions can
be relevant because of SU(3) breaking. Large penguin topologies are necessary if
∆ACP is generated by SM processes.
Actually, we will show in the section on large penguin topologies which size of SU(3)
breaking would be necessary to generate ∆ACP , the sections on naive factorization and
the exchange diagrams, on the contrary, are important to estimate SU(3) breaking effects
that are independent of the penguin processes.
We would like to stress that we are not going to investigate large SU(3) breaking in T and
C (SU(3) breaking other than what can be included by naive factorization) since they are
the largest topologies and therefore large SU(3) breaking in these topologies would mean
that even a fit to CF data would most likely not work. Since the fit to CF data works we
think that there is no large SU(3) breaking in these two topologies. The topology A is
small, so that we do not expect that large SU (3) breaking would alter the contribution
of this topology significantly.
77
5 Symmetries of the Decay Amplitudes
5.2.1 Amplitude Ratios
In what follows we are going to analyze amplitude ratios in order to see how naive factorization can improve the SU(3) predictions. In the fit in Eqn. (5.18) we found that T, C
and E are large and of the same order of magnitude, A is an order of magnitude smaller.
We are going to take into account these four topologies for these tests. In the given 24
D → P P decays only linear combinations of these topologies contribute, it is therefore
not possible to probe naive factorization for single topologies. In order to be able to use
naive factorization we must therefore not only make assumptions on how T factorizes,
where factorization is expected to work quite well, we also have to make assumptions on
the factorization behavior of the other three topologies. We correct T and C
T (D → P1 P2 )
TP1 P2
fP2 f0DP1 (m2P2 ) (m2D − m2P1 )
,
=
=
T (D0 → K − π + )
TK − π+
fπ f0DK (m2π ) (m2D0 − m2K )
fP1 f0DP2 (m2P1 ) (m2D − m2P2 )
C(D → P1 P2 )
CP1 P2
,
=
=
Cπ0 K̄ 0
fπ f0DK (m2π ) (m2D0 − m2K )
C(D0 → π 0 K̄ 0 )
(5.20)
(5.21)
as it was shown in Section 4.2.3 (also see [95]). We will use fπ (fK ) and mπ+ (mK + ) as the
decay constant and mass for all pions (kaons). For the form factor we do not distinguish
the D mesons and use f0Dπ (f0DK ) for transitions to all pions (kaons). The numerical
values can be found in Section 4.2. E and A cannot be factorized.
In what follows we will compare amplitudes, therefore let us introduce the definition of
the experimental amplitude
|A(D → P1 P2 )exp | ≡ mD
s
8πΓD BR(D → P1 P2 )exp
p P1 P2
(5.22)
Where mD and ΓD are the mass and the decay width of the decaying D meson and pP1 P2
is the respective center-of-mass momentum as in Eqn. (5.17).
As a first example we compare the decays D0 → K − π + and D0 → K + π − . Let us correct
the amplitude ratio under SU(3) symmetry by CKM factors so that obtain
RSU (3)
¯
¯ ¯
¯ ¯¯
¯ ξ ¯ ¯ A(D0 → K − π + ) ¯ ¯ Tπ+ K − + E ¯
¯ = 1.
¯=¯
= ¯¯ ¯¯ ¯¯
ζ A(D0 → K + π − ) ¯ ¯ TK + π− + E ¯
(5.23)
5 Symmetries of the Decay Amplitudes
78
This is the expectation for the ratio of the experimental amplitudes. However, the experimental amplitudes result in
Rexp
¯ ¯¯
¯
¯ ξ ¯ ¯ A(D0 → K − π + )exp ¯
¯
¯
¯ = 0.869 ± 0.008.
¯
= ¯ ¯¯
ζ A(D0 → K + π − )exp ¯
(5.24)
We see that SU(3) is broken by
1−
fK
Rexp
= 13% ≈ 1 −
= 16.5%
RSU (3)
fπ
(5.25)
which is the nominal SU(3) breaking size. We will now take into account corrections
through naive factorization to rescale the ratio
RSU (3)
¯
¯
¯ Tπ+ K − + E ¯
¯.
= ¯¯
TK + π− + E ¯
(5.26)
We obtained the size of E and Tπ+ K − = T in the fit (5.18), the tree topology in the DCS
amplitude gets the following correction
TK + π− =
fK f Dπ (m2K )(m2D − m2π )
T = 1.24T.
fπ f0DK (m2π )(m2D − m2K )
(5.27)
The exchange diagrams cannot be factorized. With this value which has been improved
by naive factorization we obtain
RSU B
¯
¯
¯ T +E ¯
¯
¯ = 0.97
=¯
1.24T + E ¯
(5.28)
This is closer to the experimental ratio Rexp = 0.869 ± 0.008 but not in total agreement.
This must be due to non-factorizable corrections.
We did not give an uncertainty on RSU B since the uncertainties of the form factor and
the decay constant are small and the uncertainties of the fit parameters can be estimated
to be small as well. On T the uncertainty is about 1%, on C about 2% and for E about
5% these small values will propagate to RSU B so that not the complete deviation from
the experimental value can be explained with these uncertainties. Therefore, the above
analysis is an indication that naive factorization not sufficient to include SU(3) breaking.
To the above decays no penguin topologies can contribute, because they are CF/DCS
decays. Therefore, the SU(3) breaking must come from non-factorizable contributions in
the tree amplitude or in the exchange amplitude.
5 Symmetries of the Decay Amplitudes
79
Let us now compare the decays D+ → π + K̄ 0 (CF) and D+ → π + π 0 (SCS). The second
decay does not receive contributions from penguin topologies, even though it is SCS. We
√
correct the SU(3) expectation by CKM factors and a factor of 2 so that we expect
RSU (3)
¯√ ¯ ¯
¯ 2ζ ¯ ¯ A(D+ → π + K̄ 0 ) ¯¯
¯
¯¯
¯ = 1.
=¯
¯
¯ λd ¯ ¯ A(D+ → π + π 0 ) ¯
(5.29)
The experimental amplitudes result in
Rexp
¯√ ¯ ¯
¯ 2ζ ¯ ¯ A(D+ → π + K̄ 0 ) ¯¯
¯¯
¯
exp ¯
= 0.841 ± 0.018,
=¯
¯¯
+
+
0
¯ λd ¯ A(D → π π )exp ¯
(5.30)
so that SU(3) is broken by
1−
Rexp
= 16%.
RSU (3)
(5.31)
Again we try to adjust the SU(3) breaking with naive factorization, we find Tπ+ K̄ 0 = T ,
Cπ+ K̄ 0 = C and
f0Dπ (m2π )(m2D − m2π )
T = 0.96T,
f0DK (m2π )(m2D − m2K )
fπ f0Dπ (m2π )
=
C = 0.78C,
fK f0Dπ (m2K )
Tπ+ π0 =
(5.32)
Cπ+ π0
(5.33)
this results to
RSU B
¯
¯
¯
¯
T
+
C
¯ = 0.949.
= ¯¯
0.96T + 0.78C ¯
(5.34)
Again we give no uncertainties for the reasons given above. The result is closer to the
experimental value but they are still not in complete agreement. So that we find until to
now that naive factorization cannot accurately take into account SU(3) breaking.
Let us have a look at the decays in which we are most interested D0 → π + π − and
D0 → K + K − . Their amplitudes should be equal
RSU (3)
¯
¯¯
¯
¯ A(D0 → π + π − ) ¯ ¯ T + E ¯
¯¯
¯ = 1.
= ¯¯
A(D0 → K + K − ) ¯ ¯ T + E ¯
(5.35)
The experimental amplitudes give
Rexp
¯
¯
¯ A(D0 → π + π − )exp ¯
¯ = 0.551 ± 0.007,
= ¯¯
A(D0 → K + K − )exp ¯
(5.36)
5 Symmetries of the Decay Amplitudes
80
so that SU(3) is broken by
1−
Rexp
= 45%
RSU (3)
(5.37)
This is a lot higher than in the previous cases. It seems that these decay modes do not
only produce a large direct CP asymmetry but they also exhibit large SU(3) breaking,
this was already noted long before [87] and different explanations have been recently given
Refs. [26, 94, 96].
If we include naive factorization the following expressions result
f0Dπ (m2π )(m2D − m2π )
T = 0.96T,
f0DK (m2π )(m2D − m2K )
fK f0DK (m2K )
=
T = 1.28T,
fπ f0DK (m2π )
Tπ+ π− =
TK + K −
which finally gives
RSU B
¯
¯
¯ 0.96T + E ¯
¯ = 0.96.
¯
=¯
1.28T + E ¯
(5.38)
(5.39)
(5.40)
This barely changes the SU(3) expectation, so that naive factorization cannot accommodate the experimental values and is insufficient to describe SU(3) breaking. This large
SU(3) breaking is likely to be connected to the large value of ∆ACP and we will show
another way how to deal with this in Section 5.2.3.
Let us first continue our analysis of how naive factorization alters the expectations for
amplitude ratios. A comparison of all possible relations is listed in Table 5.4. We find
that the SU(3) breaking for most decays is larger than 0.20 and, hence, relatively large.
The ratio of entry (4) is the most prominent case, as already pointed out. In ratio (9)
we find almost no SU(3) breaking so that the naive factorization actually increases the
SU(3) breaking. In ratio (10) and (11) naive factorization works very well. This could
be because the amplitude is parametrized by (T-A), whereas A ∼ T /10 so that the nonfactorizable structure of A does not greatly disturb the naive factorization corrections.
Yet, for all D0 decays except (3) SU(3) breaking is large and taking into account naive
factorization does lead to an accurate agreement. The reason could be the relatively large
unfactorizable E∼ T/2 topology in the parametrization of D0 decays. Additionally, we
might have to take into account the penguin topologies, since they contribute to all D0
decays (especially P A, that contributes only to D0 decays and not to D+ and Ds+ decays
where naive factorization works relatively good). Another indication for this reasoning is
81
5 Symmetries of the Decay Amplitudes
#
1
2
3
4
5
6
7
8
9
10
11
First Transition
D0 → π + π −
D0 → π + K −
D0 → π + K −
D0 → π + π −
D0 → π + π −
D0 → π − K +
D+ → π + K̄ 0
D0 → π 0 π 0
Ds+ → π + K 0
D+ → π 0 K +
D+ → π 0 K +
Second Transition
D0 → π + K −
D0 → K + K −
D0 → π − K +
D0 → K + K −
D0 → π − K +
D0 → K + K −
D+ → π 0 π +
D0 → π 0 K̄ 0
D+ → K + K̄ 0
D+ → K + K̄ 0
Ds+ → π + K 0
Rexp
1 − Rexp
0.796 ± 0.010
0.204
0.692 ± 0.006
0.308
0.869 ± 0.008
0.131
0.551 ± 0.007
0.449
0.692 ± 0.009
0.308
0.796 ± 0.027
0.204
0.841 ± 0.018
0.159
0.766 ± 0.034
0.234
1.01 ± 0.05
−0.01
0.743 ± 0.075
0.257
0.735 ± 0.032 0.2651
exp
RSU B 1 − RRSU
B
0.996
0.201
0.967
0.284
0.972
0.106
0.963
0.428
0.969
0.286
0.994
0.199
0.949
0.106
0.965
0.206
0.901 −0.122
0.680 −0.091
0.758
0.030
Table 5.4: SU(3) breaking tests. Under SU(3) the expected value for the experimental
ratio Rexp is RSU (3) = 1, deviations quantify the SU(3) breaking which is given
by 1-Rexp /RSU (3) = 1−Rexp . The ratio RSU B is the expected value if corrections
through naive factorization are taken into account in the topological approach,
exp
1 − RRSU
therefore gives the SU(3) breaking if one includes naive factorization.
B
We give no uncertainties on RSU B for the reasons given below Eqn (5.28).
a sum rule which has been proposed by Brod et al. [27]
|A(D0 → π + π − )exp |/λd | + |A(D0 → K + K − )exp /λs |
− 1 = (4.2 ± 0.9)%.
|A(D0 → π + K − )exp. /ζ| + |A(D0 → K + π − )exp /ξ|
(5.41)
Which shows no great deviation from 0, so that this points to penguins that approximately cancel out in the sum.
We can conclude from this section that the two ratios which do not involve penguins
exhibit nominal SU(3) breaking ∼ 15%. Nonetheless, for the D0 decay amplitudes, which
could involve penguins, we find large SU(3) breaking. The indication of large penguins is
further supported by the sum rule Σsum−rule which has a small value. We found furthermore, that including naive factorization does not significantly correct for SU(3) breaking
in the case of D0 decays in contrast to the decays of charged D meson where naive factorization works relatively good.
In total, one has to keep in mind that, since we used the fit values from Eqn. (5.18), this is
also dependent on the values of these topologies. The above relations could be completed
by the BRs of the following DCS decays BR(D0 → π 0 K 0 )exp , BR(D+ → π + K 0 )exp and
BR(Ds+ → K + K 0 )exp which are not measured yet. If they would be known, we would
have a clearer picture of SU(3) breaking in D meson decays.
Σsum−rule =
82
5 Symmetries of the Decay Amplitudes
5.2.2 Large SU(3) Breaking in Exchange Topologies
The large value of BR(D0 → K 0 K̄ 0 ) gives rise to another interpretation of the data. It
might be that SU(3) breaking is large in the topology E, so that no sizable penguins are
needed to explain the deviation of the SCS BRs from the data. In particular, this would
mean that BR(D0 → K 0 K̄ 0 ) is generated by SU(3) breaking in the exchange amplitudes
we therefore distinguish Ed and Es , these diagrams are plotted in Fig. 5.4. Large SU(3)
breaking in the exchange amplitudes might also alter the prediction of BR(D0 → π + π − )
and BR(D0 → K + K − ), yet no direct influence is expected because in both cases Ed does
contribute. We will investigate this problem here and see what can be concluded.
In order to determine sizes for these topologies that are compatible with the rest of the
s
c
d
c
K0
K̄ 0
W
D0
d
W
D0
s
K̄ 0
K0
u
u
s
d
Figure 5.4: SU(3) breaking in exchange amplitudes, we distinguish Ed (left) and Es
(right), where nomenclature is due to the quark-antiquark pair that is generated from the vacuum. These topologies could exhibit large SU(3) breaking in
order to generate BR(D0 → K 0 K̄ 0 ) = (0.346 ± 0.058)10−3 which is expected
to be zero under SU(3) symmetry.
data we will use a fit. In the CF decays the following amplitudes are modified
¶
cos(φ)
√ (C + Ed ) − sin φEs ,
A(D → K̄ η) = ζ
2
µ
¶
sin(φ)
0
0 ′
√ (C + Ed ) + cos φEs .
A(D → K̄ η ) = ζ
2
0
0
µ
(5.42)
(5.43)
Introducing a new topology, amounts to increasing the number of fit parameters from 7
to 9. Therefore, we have to include more BRs to have an over constrained fit. We will
include the DCS decays D0 → π + K − and Ds+ → K + K̄ 0 .
We first try a fit with T, C, E and A alone we obtain (in keV)
◦
T = 3.2
C = 2.6ei272
◦
◦
E = 1.5ei102 A = 0.39ei113
(5.44)
83
5 Symmetries of the Decay Amplitudes
The χ2 = 26 is quite large, this is because we included DCS decays. Especially, the
decay D0 → K + π + contributes with a value of 20 to this. Nevertheless, the values of the
topologies are close to the values obtained before in Eqn. (5.18).
If we include now the Es topology we obtain (in keV)
◦
T = 3.3
C = 2.5ei206
Ed = 1.9ei124
◦
◦
Es = 1.2ei2 A = 0.33ei360
◦
(5.45)
The χ2 = 25 is again very large and does not improve significantly using these values for
Ed and Es we obtain for BR(D0 → K 0 K̄ 0 ).
BR(D0 → K 0 K̄ 0 ) =
pK 0 K̄ 0
|λd (Es − Ed )|2 = 0.144 × 10−3
2
8πmD ΓD
(5.46)
We see that this fit favors only a BR that is less than half of the experimental value
BR(D0 → K 0 K̄ 0 )exp = 0.346±0.058. Hence, we consider it as unlikely that the BR(D0 →
K 0 K̄ 0 ) is generated by SU(3) breaking in exchange diagrams alone. Again we find that
we need to add another SU(3) breaking component. The total amplitude necessary to
generate the BR would be
|∆E| ≡ |Ed − Es | = 1.1 keV.
(5.47)
We conclude that SU(3) breaking in exchange amplitudes is not likely to be the source of
the total value of BR(D0 → K 0 K̄ 0 ).
5.2.3 Large Penguins
We see that the SCS decays are not well described by the CF fit, we would like to
improve on this. One reason is certainly nominal SU(3) breaking but we saw in Section
5.2.1 that compensating for this effect by naive factorization is not successful in all cases.
Additionally, it is difficult to include these effects for all decay processes, since the decay
constants and form factors for η and η ′ are only poorly known. Besides, the uncertainties
of most of the BRs are quite large around 5%-10%. Only the decays D0 → K + K − and
D0 → π + π − have smaller errors of about 2%. Ds+ → K + η and Ds+ → K + η ′ have errors
of about 30%, these large errors are of the size of nominal SU(3) breaking. Therefore, we
might cannot even distinguish between experimental uncertainty and corrections which
come from SU (3) breaking.
There is another obvious step, in what follows we will assume that there is SU(3) breaking
5 Symmetries of the Decay Amplitudes
84
in the penguin amplitudes, as has been done as well in Refs. [26, 27]. This is motivated
by an observation in Section 5.2.1 where in the BR relation (4) the deviation from the
SU(3) expectation was especially large. If we include penguin topologies we see that they
add to the leading amplitudes with respective negative sign. What we mean by this is
clearer if we write down the amplitude parametrization of the 2 involved decays.
A(D0 → π + π − ) = λd (T + E) + λp (Pp + P Ap ) = λd (T + E − ∆P − ∆P A) + O(λb )
A(D0 → K + K − ) = λs (T + E) + λp (Pp + P Ap ) = λd (T + E + ∆P + ∆P A) + O(λb )
where we defined
∆P ≡ Ps − Pd
and ∆P A ≡ P As − P Ad .
(5.48)
The contributions O(λb ) are certainly irrelevant for the value of the BRs. This amplitude
structure in addition to the large SU(3) breaking observed in this ratio motivates us to
investigate large penguins, we can extract absolute values by comparing the experimental
and fit ratio:
¯
¯ ¯
¯
¯
¯
¯ A(D0 → π + π − )exp ¯
¯ A(D0 → π + π − ) ¯ ¯ T + E − ∆P − ∆P A ¯
¯ ¯
¯
¯ = 0.551 = ¯
Rexp = ¯¯
¯ A(D0 → K + K − ) ¯ = ¯ T + E + ∆P + ∆P A ¯
A(D0 → K + K − )exp ¯
(5.49)
With the values of the topologies in (5.18) we obtain (in keV)
∆P + ∆P A = 0.7 + 0.4i ∨ ∆P + ∆P A = 8.0 + 4.5i
(5.50)
We can discard the large solution because it is quite unlikely that the difference of two
penguin diagrams is larger than the tree topology (T ∼ 3 keV). Nevertheless, this ratio
is consistent with a penguin difference that is larger than expected. A single penguin Pd
is expected to have a tenth of the size of a tree topology, now we find that the SU(3)
breaking difference of two penguins is larger than the expectation for a single penguin. If
SU(3) is not too large broken, this means that the difference ∆P is smaller than the two
individual penguin topologies, the above results imply that the single penguins are very
large.
We will come back to this point a bit later we should first find SU(3) breaking penguin
topologies that are consistent with the whole SCS data. We do this by making a global
χ2 fit to SCS data using six topologies: T, C, E and A together with ∆P and ∆P A. The
amplitude decomposition can be found in Table 5.5. We obtain the following values for
85
5 Symmetries of the Decay Amplitudes
the topologies (in keV)
◦
◦
T = 3.4
C = 2.9e±i149
E = 0.8e±i227
◦
◦
◦
A = 0.7e±i9 ∆P = 0.3e±i314 ∆P A = 0.6e±i358 .
◦
(5.51)
◦
T = 3.4
C/T = 0.85e±i149
E/T = 0.24e±i227
◦
◦
◦
A/T = 0.20e±i9 ∆P/T = 0.07e±i314 ∆P A/T = 0.18e±i358 .
Meson
D0
Final
State
π+π−
π0π0
π0η
π0η′
ηη
ηη ′
D+
K +K −
K 0 K̄ 0
π+π0
π+η
π+η′
Ds+
K + K̄ 0
π+K 0
π0K +
K +η
K +η′
Amplitude Representation
pCM
[MeV]
λd (T + E − ∆P − ∆P A)
922
λd
√
(−C
+
E
−
∆P
−
∆P
A)
923
2
−λd (E + ∆P ) cos φ − √λs2 C sin φ
846
λs
√
678
−λd (E + ∆P ) sin φ + 2 C cos φ
λd
2
√
(C + E − ∆P − ∆P A) cos φ −
755
2
√
2
C
λs ( 2 sin 2φ − 2E sin φ)
λd
(C + E − ∆P − ∆P A) sin 2φ +
537
2
1
√
λs ( 2 C cos 2φ − E sin 2φ)
λs (T + E + ∆P + ∆P A)
791
2λs ∆P A
789
λd
√
(T + C)
925
2
λ
d
√ (T + C + 2A − 2∆P ) cos φ −
848
2
λs C sin φ+
λd
√
(T + C + 2A − 2∆P ) sin φ +
681
2
λs C cos φ
λs (T − A + ∆P )
793
λd (T − A − ∆P )
916
−λ
√ d (C + A + ∆P )
917
2
cos
φ
835
λd (C − A − ∆P ) √2 − λs (T + C +
A + ∆P ) sin φ
√ φ + λs (T + C +
λd (C − A − ∆P ) sin
646
2
A + ∆P ) cos φ
(5.52)
BRexp
BRf it
−3
[10 ]
[10−3 ]
1.400 ± 0.026 1.41
0.80 ± 0.05
0.78
0.68 ± 0.07
0.70
0.89 ± 0.14
0.61
1.67 ± 0.20
2.24
1.05 ± 0.26
1.23
3.96 ± 0.08
0.346 ± 0.058
1.19 ± 0.06
3.53 ± 0.21
3.88
0.44
1.27
3.49
4.67 ± 0.29
4.69
5.66 ± 0.32
2.42 ± 0.16
0.62 ± 0.21
1.75 ± 0.35
5.90
2.27
0.81
1.13
1.8 ± 0.6
2.11
Table 5.5: SCS D → P P processes we give the amplitude representations in topological
approach, center-of-mass momenta, experimental [34] and fitted BR. The fit is
a fit to the SCS BRs including SU(3) breaking penguins. We omitted terms
O(λb ) since they do not contribute to the BRs, these terms can be found in
Table 5.3.
In general the size ordering of the topologies is preserved. T is the largest and A is
larger than the penguins, but the penguins are sizable. Especially, PA seems to be quite
86
5 Symmetries of the Decay Amplitudes
large and has almost no phase with respect to T, this can explain why the SU(3) breaking
in D0 decays is larger (PA only contributes to D0 decays) and why naive factorization
cannot improve on the breaking of SU(3) breaking. PA+P deviates from what we found
in Eqn. (5.50), since we find other values for T and C in Eqn. (5.51) than what we used
to determine (5.50).
We obtain a total χ2 = 22.3 for a number of 16-11 degrees of freedom (dof). Of course,
χ2 /dof = 4.5 is a large value but this is because the BRs of D0 → π + π − and D0 → K + K −
have quite small uncertainties, these uncertainties increase the χ2 . There is more SU(3)
breaking than the SU(3) breaking encoded in ∆P and ∆P A, so that this SU(3) breaking
combined with the small uncertainties leads to a large χ2 /dof. The values of the BRs are
given in Table 5.5 we find an overall good agreement.
BRs which involve η or η ′ final mesons are less well described. There are several possible
reasons for this. It could be that the η − η ′ mixing angle as another value than assumed
or that SU(3) breaking is simply larger for η and η ′ since they are heavier than kaons and
pions.
Despite the large χ2 this fit gives us an insight on how large the penguins are: We find
that the difference of penguins is of the same order of magnitude as A that means that
the single penguins that generate ∆ACP are likely to be large. This can be shown if we
conduct an analysis of the SU(3) breaking. We assumed SU(3) breaking with
∆P = Ps − Pd .
(5.53)
We might rewrite this in terms of a breaking scale δ and an SU(3) breaking phase φδ .
Assume P to be SU(3) invariant penguin topology, Pd and Ps deviate from this value
because the d and s quark masses are different so that we might write
δ
Pd = P (1 − )e−iφδ /2
2
δ
and Ps = P (1 + )e+iφδ /2 .
2
(5.54)
Consequently, we obtain
∆P = (δ cos(φδ /2) + 2i sin(φδ /2))P
(5.55)
87
5 Symmetries of the Decay Amplitudes
we therefore find
µ
∆P
1−
=
δ cos(φδ /2) + 2i sin(φδ /2)
µ
∆P
=
1+
δ cos(φδ /2) + 2i sin(φδ /2)
Pd
Ps
¶
δ
e−iφδ /2
2
¶
δ
e+iφδ /2 .
2
(5.56)
(5.57)
If we had information on the breaking scales we would immediately know the size of the
individual penguins that generate the CP asymmetry. Analogously we can obtain for PA
P Ad
P As
µ
∆P A
=
1−
δ cos(φδ /2) + 2i sin(φδ /2)
µ
∆P A
1+
=
δ cos(φδ /2) + 2i sin(φδ /2)
¶
δ
e−iφδ /2
2
¶
δ
e+iφδ /2 .
2
(5.58)
(5.59)
In principle, the SU(3) breaking parameters are not the same for the PA and P topologies,
but since these are unknown parameters that are likely to be of similar size, we might as
well just take them as equal. There are, of course, also penguins Pb that contribute to the
CP asymmetry yet we cannot say anything about these, so that we might ignore them
for now.
Depending on the value of δ and φδ we find that P Ad , P As , Pd and Ps can have a considerable size. We illustrate this in Fig. 5.5, where we plotted the |Ps + P As |/T as a function
of δ for different values of φδ . We find that the smaller δ is, the larger |Ps + P As | must
be, this is because we ∆P and ∆P A are inversely proportional to δ.
To illustrate how ∆ACP is connected to this breaking formalism, we insert the expressions of P and P A into the prediction of the asymmetry in Eqn. (4.40) and use the
parametrization in Eqn. (4.33). With all penguins with index b neglected and Pπs = Ps ,
P Asπ = P As , PKd = Pd and P AdK = P Ad .
∆ACP
µ
¶
Ps + P As
Pd + P Ad
= −2ℑ
(5.60)
ℑ
+
T + E − ∆P − ∆P A T + E + ∆P + ∆P A
ÃÃ
!
µ ¶
(1 + 2δ )e+iφδ /2
(1 − 2δ )e−iφδ /2
λb
= −2ℑ
ℑ
+
×
λs
T + E − ∆P − ∆P A T + E + ∆P + ∆P A
¶
∆P + ∆P A
×
(5.61)
δ cos(φδ /2) + 2i sin(φδ /2)
λb
λs
¶
µ
This formula can be used to illustrate the size of the asymmetry depending on the size of
the SU(3) breaking parameters assuming the fit values in Eqn. (5.51) and different values
88
5 Symmetries of the Decay Amplitudes
ÈPs +PAs ȐT
7
6
5
4
3
2
1
0.00
0.05
0.10
0.15
∆
0.20
Figure 5.5: The normalized absolute size of the the penguins |Ps + P As |/T (The graph
for |Pd + P As |/T looks almost identical so that we do not show it here.) as
a function of the SU(3) breaking parameter δ and φδ fixed at (from top to
bottom) φδ = 2◦ , 4◦ , 8◦ , 12◦ .
for ∆ACP . This is shown in Fig. 5.6, we see that the ranges 0 ≤ δ ≤ 0.08 and 0◦ ≤ φδ ≤ 9◦
are sufficient to generate the CP asymmetry ∆ACP . Even the LHCb measurement can
be possible, but also the lower 1σ bound of the world average is in good agreement with
the SU(3) breaking. Nevertheless, the SU(3) breaking δ ∼ 15% as we had found it before
would be to large to accommodate ∆ACP . So that we agree with the results in Ref. [27],
what distinguishes our approach is that we also take into account SU(3) breaking in the
phase of the penguins
We can conclude that the information, which we can extract from the BR data and which
is the only information which is available, shows that there can be large SU(3) breaking
penguin topologies ∆P and ∆P A. These large penguin differences point to large penguin
topologies Pd , Ps , P Ad and P As and can generate a large value ∆ACP . The outcome of this
topological approach is in contrast to the short-distance expectation for ∆ACP in Section
4.3. Furthermore, it is surprising that the loop-suppressed penguins are of tree size as
can be seen in Fig 5.5. Nevertheless, this is not the first time that long-distance physics
lead to an unexpected result, the long-known ∆I = 1/2 rule [97] is another example.
Therefore, we may conclude that ∆ACP can very well be an effect of long-distance QCD
effects and must not be a sign of NP.
89
5 Symmetries of the Decay Amplitudes
10
Φ∆ in °
8
6
4
2
0
0.00 0.02 0.04 0.06 0.08 0.10 0.12
∆
Figure 5.6: Relation between the SU(3) breaking parameter δ and the SU(3) breaking
phase φδ to generate ∆ACP . The inner, black curve is the condition for the
LHCb value ∆ACP = −0.82%(≈ upper 1σ bound of the world average), the
red curve is the world average ∆ACP = −0.656% and the blue curve represents
the lower one sigma bound of the world average ∆ACP = −0.5%.
6 Conclusion
In this thesis we have investigated the theoretical understanding of the asymmetry
∆ACP = ACP (K + K − ) − ACP (π + π − )
(6.1)
which has been measured to be significantly larger than the naive SM expectation by
the CDF collaboration and by the LHCb collaboration. We investigated whether ∆ACP
can be accommodated in the SM and found that a large value for ∆ACP is not unlikely.
Although a value as large as the world average ∆ACP = −(0.656 ± 0.154)% is clearly
disfavored by the calculable short-distance contributions, the long-distance contributions
of QCD are not understood, so that we cannot exclude that ∆ACP is generated in the
SM.
In order to come to this conclusion, we investigated the influence of indirect CP violation
on ∆ACP and found it to be negligible in Section 3.3.2. We give the most general asymmetry in terms of observable quantities, which is not in the literature.
Since we found that the observable ∆ACP can be described by the difference of the
direct asymmetries with good approximation, we investigated the decay amplitude structure. First, we introduced the effective Hamiltonian Hef f which describes charm decays
and identified the non-calculable hadronic matrix elements as the major uncertainty. Since
the masses of the u, d and s quark are such that we can assume them as light and equal,
Hef f exhibits a SU(3) flavor symmetry which we used to introduce the topological approach in Section 5.1. The topological approach relates the hadronic matrix elements and
describes all charm decays to two pseudoscalar particles (D → P P ) with a small set of
parameters.
The topological approach describes the branching ratios of the Cabibbo favored decays quite well, however, especially the singly Cabibbo suppressed decays, such as D0 →
K + K − and D0 → π + π − , are not well described. This led us to introduce SU(3) breaking.
We first investigated how naive factorization can compensate for SU(3) breaking, in some
decays naive factorization could compensate for all SU(3) breaking, in other decays no
91
6 Conclusion
92
good improvement could be reached.
Since the branching ratio BR(D0 → K 0 K̄ 0 ) is significantly larger than the SU(3) expectation, we had a glimpse on large SU(3) breaking in exchange amplitudes. Nonetheless,
having SU(3) breaking only in exchange amplitudes, in order to generate the deviations
from the SU(3) symmetry, is not favored by our fit.
We find a good agreement with the branching ratio data if we include the SU(3) breaking penguin topologies ∆P and ∆P A, which would be zero under exact SU(3) symmetry.
These can largely explain the branching ratios and could explain a large asymmetry within
the SM, if SU(3) breaking behaves as we expect. Nevertheless, we cannot make a final
verdict of the theoretical understanding of this asymmetry, since SU(3) breaking may
be different from what we expect. Additionally, the large errors might still hide some
P
NP contributions which can have the absolute size of ∆AN
CP ∼ 0.1% and could not be
disentangled from the SM contribution. The theoretical uncertainties are all due to the
long-distance effects of QCD.
What eventually remains is, that ∆ACP can be accommodated by the SM (but probably
with a smaller value).
Outlook
There are new measurements of ∆ACP by the LHCb collaboration whose publication is in
preparation. It will be interesting to see whether their very large value of ∆ACP remains
with the new data. Future colliders like SuperKEKB, which should start data taking in
the year 2015, and SuperB will give further measurements of ∆ACP .
It will also be interesting to see if the single asymmetries ACP (K + K − ) and ACP (π + π − )
will be measured with a significant value, i.e. in the percent range, but this will still take
some time due to the systematic uncertainties in the experiment. Additionally, the single
asymmetries still contain the contribution of the indirect CP violation.
The determination of the unknown branching ratios, BR(D0 → π 0 K 0 )exp , BR(D+ →
π + K 0 )exp and BR(Ds+ → K + K 0 )exp , will help to better understand naive factorization
and SU(3) breaking and will therefore help to strengthen or weaken the reasoning connected to large penguins. The explanation of ∆ACP will in general require more data,
since we do not expect that a way to compute long-distance effects of QCD will be found
in the near future.
The detection of other large CP asymmetries in charm decays could also lead to more
insights. In general, we only expect CP violation in SCS decays, any CP violation in CF
or DCS decays is completely unexpected.
A Abbreviations
BR
C
CF
CKM
DCS
dof
Eqn.
Fig.
GWS
H.c.
HFAG
HQET
LEEH
MFV
NP
OPE
P
PP
QCD
Ref.
SM
SCS
T
UT
Branching Ratio
Charge symmetry
Cabibbo Favored
Cabibbo-Kobayashi-Maskawa (Matrix)
Doubly Cabibbo Suppressed
Degree of Freedom
Equation
Figure
Glashow-Weinberg-Salam (model)
Hermitian Conjugate
Heavy Flavor Averaging Group
Heavy Quark Effective Theory
Low-Energy Effective Hamiltonian
Minimal Flavor Violation
New Physics
Operator Product Expansion
Parity Symmetry
Particle Physics
Quantum Chromodynamics
Reference
Standard Model
Singly Cabibbo Suppressed
Time symmetry
Unitarity Triangle
An appended and small ’s’ indicates the plural, independent of the actual plural form of
the expression abbreviated.
93
Usual names of variables
A
C
E
P
PA
T
λ = 0.2253
λq = Vcq∗ Vuq
Annihilation topology
Color suppressed tree topology
Exchange topology
Penguin topology
Penguin annihilation topology
Tree topology
Cabibbo parameter
Usual index conventions
p
q
α, β
µ, ν, ρ, σ
quark flavor u,d,s
quark flavor d,s
Color indices
Minkowski spacetime indices
94
B Notation
As common in high energy physics we use the notation introduced by Bjorken and Drell.
In the natural units we set the speed of light as well as the reduced Plank quantum to
one
c = ~ = 1.
(B.1)
For the metric of the Minkowski space we take the following metric
ηµν = diag(1, −1, −1, −1)
(B.2)
which we use to lower and raise the indices of four vectors
xµ = (x0 , −x1 , −x2 , −x3 ) = ηµν xν
with xµ = (x0 , x1 , x2 , x3 )⊥ .
(B.3)
Here, the Einstein summation convention over repeated indices is implied. Usual four
vectors besides xµ are
pµ = (E, p~)⊥ , pµ = (E, −~p)
∂ µ = ∂x∂ µ
∂µ = ∂x∂ µ
For the Dirac matrices γ µ in four dimensions, which obey the Clifford algebra
{γ µ , γ ν } = 2η µν ,
(B.4)
we choose to use the Dirac representation
γ0 =
Ã
I2 0
0 −I2
!
γi =
,
Ã
0 σi
−σ i 0
!
.
(B.5)
Here, we made use of the identity matrix in two dimensions and the Pauli matrices
I2 =
Ã
1 0
0 1
!
,
σ1 =
Ã
0 1
1 0
!
,
σ2 =
95
Ã
0 −i
i 0
!
,
σ3 =
Ã
1 0
0 −1
!
.
(B.6)
96
B Notation
At last we introduce γ 5
γ 5 ≡ iγ 0 γ 1 γ 2 γ 3 =
Ã
0 I2
I2 0
!
.
(B.7)
The Gell-Mann matrices can be used to define a representation of the generators of SU(3)
T a = 12 λa , they are given by

0 1 0


λ1 =  1 0 0  ,
0 0 0


0 −i 0


λ2 =  i 0 0  ,
0 0 0



1 0 0


λ3 =  0 −1 0  ,
0 0 0





0 0 1
0 0 −i





λ4 =  0 0 0  , λ5 =  0 0 0  , λ6 = 
i 0 0
1 0 0



1 0 0
0 0 0
1 


8
7
λ =  0 0 −i  , λ = √  0 1 0
3
0 0 −2
0 i 0
They obey the following relation

0 0 0

0 0 1 ,
0 1 0


.
£ a b¤
T , T = if abc T c
(B.8)
(B.9)
(B.10)
(B.11)
where the f abc are called structure constants.
For SU(2) the generators are given by τ a = 12 σ a , i.e. in terms of the Pauli matrices.
They obey [τ a , τ b ] = iǫabc τ c Here, we introduced the total antisymmetric tensor of third
grade ǫabc .


 1 if abc is an even permutation of 123,
abc
(B.12)
ǫ =
−1 if abc is an odd permutation of 123,


0
all other cases.
ǫabc gives therefore the structure constants of SU(2). The definition for ǫµνρσ is analogously.
All experimental values are given in the order
value ± statistic uncertainty ± systematic uncertainty
(B.13)
Acknowledgments - Danksagung
I am very thankful that the Theory Group of Nikhef accepted me as a Master student and
welcomed me cordially. Especially, I want to thank my supervisor Robert Fleischer who
introduced me into the scientific world and guided me in preparing this thesis. He advised
me well and supported enthusiastically my search for a PhD position. I am gracious that
Piet Mulders accepted to be the second corrector of this thesis. Furthermore, I want
to thank Rob who helped me over and over again with questions. I shared the office
with Robbert and Jory and enjoyed our daily discussions and the good atmosphere a lot.
Finally, I would like thank all other members of the Theory Group for the good scientific
environment.
Ich danke Isabel für die Unterstüzung die sie mir gegeben hat und mit der ich zwei schöne
Jahre in Amsterdam verbracht habe. Meinen Eltern danke ich, die mich von klein an nach
Herzen unterstützt haben und immer hinter mir stehen, ohne sie wäre mein Studium nicht
möglich gewesen.
97
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