2 CHAPTER 1. THEORY
In particle physics, described by the Standard Model, the electromagnetic and strong
interactions preserve P, C and T, one by one, while the weak interactions violate P and
C separately. However, the composite transformation CPT is a symmetry preserved in
all the universe, that means in each lagrangian field theory experimentally tested.
The parity violation in weak interactions was suggested for the first time in 1956 by
Lee and Yang [2], before the experimental evidences coming from studies of pseudoscalar
quantities (first of all the particle helicity). Historical experiments of great importance
in testing the weak intaction were done by Garwin et al. and Wu et al. in 1957 [4],
Goldhaber et al. in 1958 [5] who measured the neutrino helicity.
In weak interactions C and P are both violated at the same time, so at the end the
symmetry under CP was considered preserved.
A first clear evidence of CP violation comes in 1964 [6] in the K0L decay, who was seen
to be not symmetric in the two C-conjugated decay modes, as was expected consider-
ing the neutral K0L the antiparticle of itself. Therefore the K0L is not eigenstate of CP
transformation.
CP violation has a great theoretical importance:
• the barionic asymmetry (there is much more matter than antimatter in the observed
universe), could only be generated from an initial situation in which the amount
of matter and antimatter were equal, balance that has evolved after CP violating
processed [7];
• elementary particles can have an electric dipole moment, who violates both P and T
symmetries. In the case of T violation, CPT has to be conserved, so a CP violation
is necessary at the end of the game.
In quantum mechanics the transformations P, T and C, are associated to operators
P , T and C respectively. Operators like T are written by exponentiation of transforma-
tion generators. A different approach should be used for P and C operators, who don’t
correspond to continuous transformations, since are transformation associated to finite
quantum numbers. P and C operators can by defined only by not taking into account
the weak interaction, since it is necessary to commute with the hamiltonian operator H,
1.1. CP VIOLATION 3
that generate the time translactions; operators can commute only when their relative
transformations correspond to symmetries not violated in the theory described by the
hamiltonian. Both P and C symmetries are not good symmetries of Nature, their oper-
ators can be defined only switching off the weak interaction, since they do not commute
with the hamiltonian of the weak interaction.
P and C are unitary. They correspond to discrete transformations, therefore they can
be associated to their eigenstates, that means multiplicative quantum numbers (at the
opposite side, for the continuum transformation, quantum numbers are additive).
T is antiunitary. T can have two different eigenvalues, therefore it cannot be associated
to any quantum number. To be very precise is not correct to say that T is conserved
(nothing is conserved), but is just valid the invariance under T.
We can improperly say that CPT is conserved (“CPT theorem” that is kind of preju-
dice). CPT conservation means that a CP violation has to be connected to a T violation.
The violations or conservations of some symmetries are related to the theory (the hamil-
tonian) and not to the single observables.
Strong and weak phases
The presence of complex phases in the transition amplitude is closely related to the CP
violation. Only the phases that are rephasing-invariant may have a physical meaning,
and in particular lead to CP violation. Those phases are in general the relative phases of
the various coherent contributions to a particular transition amplitude.
Three kind of phases may arise in transition amplitudes:
• weak or odd phase: is defined to be one which has opposite signs in the transition
amplitude for a process and in the transition amplitude for its CP-conjugate process;
• strong or even phase: has the same sign in the transition amplitude for its CP-
conjugate process;
• spurious phase: is a conventional relative phase between the amplitude and the
amplitude of the CP-conjugate process.
4 CHAPTER 1. THEORY
Given CP conjugated states i and i¯, f and f¯ , g and g¯, where for example
CP|f〉 = eiξf |f¯〉 (1.1)
with arbitrary phase ξf considered equal to 1, CP violation is possible when the transition
amplitudes are the sum of two or more transition amplitudes with different strong or weak
phases, like
〈f |T |i〉 = A1ei(δ1+φ1) + A2ei(δ2+φ2) (1.2)
where the CP-conjugate amplitude is
〈f¯ |T |¯i〉 = A1ei(δ1−φ1+θ) + A2ei(δ2−φ2+θ) (1.3)
where T is the transition matrix, A1 and A2 the modules of the transition amplitudes,
and δ1 6= δ2 (CP strong phases), φ1 6= φ2 (CP weak phases) and θ the common spurious
phase. The asymmetry between the two amplitudes is
|〈f |T |i〉|2 − |〈f¯ |T |¯i〉|2
|〈f |T |i〉|2 + |〈f¯ |T |¯i〉|2 (1.4)
It is possible to have CP violation even in absence of strong phases or amplitudes
interference, for quantities like this
〈f |T |i〉〈g|T |¯i〉 − nfng〈g|T |i〉〈f |T |¯i〉 = 2iA1A2ei(δ1+δ2+θ) sin(φ1 − φ2) (1.5)
where there is a relationship between the two final states f and g, like for physical states
that are a superposition of two CP eigenstates.
In any case, due to CPT invariance, the total decay width of i and i¯ has to be equal∑
f
|〈f |T |i〉|2 =
∑
f
|〈f¯ |T |¯i〉|2 (1.6)
therefore it is necessary to study CP violation using partial decay channels of the particles,
since it is precluded the observation of CP-violating difference between the total decay
width of a particle and its antiparticle.
Neutral meson system
The neutral mesons, interesting for the study of CP violation, decay mostly through weak
interactions. Let’s consider a generic neutral meson (that can be the a D0, K0, B0d or B0s )
P 0 with antiparticle P¯ 0, and common mass m0, for which
1.1. CP VIOLATION 5
• |P 0〉 and |P¯ 0〉 are eigenstates of strong and electromagnetic interactions with mass
m0, so they are flavor eigenstates1;
• as consequence of the weak interaction, described by the non vanishing HW , the two
states oscillate between themselves before decaying.
At a certain time t it is possible to have a state that is a superposition of initial states
and final states |ni〉, where the two |P 0〉 and |P¯ 0〉 may decay
a(t)|P 0〉+ b(t)|P¯ 0〉+
∑
n
ci(t)|ni〉 (1.7)
where ci(t = 0) = 0 are the amplitudes of the final states.
In the Wigner-Weisskopf [8] approximation, taking into account the interaction HW
|ψ(t)〉 ' ψ1(t)|P 0〉+ ψ2(t)|P¯ 0〉 (1.8)
where the wave function satisfies an equation equivalent to the Schro¨dinger equation
i
d
dt
(
ψ1
ψ2
)
=
(
R11 R12
R21 R22
)(
ψ1
ψ2
)
(1.9)
where
R = M− i
2
Γ (1.10)
with
M = 12(R+R†),
Γ = i(R−R†) (1.11)
The matrices M e Γ are hermitian, while the matrix R is anti-hermitian. The weak
interaction is considered like a small perturbation with respect to the strong and electro-
magnetic interactions; in second-order perturbation theory the matrices M e Γ, by sums
over intermediate states n, are
Mij = m0δij + 〈i|HW |j〉+
∑
n
P
〈i|HW |n〉〈n|HW |j〉
m0 − En , (1.12)
Γij = 2pi
∑
n
δ(m0 − En)〈i|HW |n〉〈n|HW |j〉 (1.13)
1States with a defined quark content, that are eigenstate of an effective hamiltonian that describes the strong interaction
6 CHAPTER 1. THEORY
where the operator P projects out the principal part and En are the energies of the states.
From the second order expansion the box Feynman diagrams came out. The mass matrix
has first and second order terms, therefore virtual states connect the real states.
It follows that
d
dt
(|ψ1|2 + |ψ2|2) = −(ψ∗1ψ∗1)Γ
(
ψ1
ψ2
)
(1.14)
where the left-hand side of the equation (1.14) must be negative (since the mesons decay),
hence the Γ is positive definite.
The eigenstates of R are complex, since the matrix is not-hermitian, so they can be
defined as
µH = mH − i2ΓH ,
µL = mL − i2ΓL
(1.15)
Then we define
∆m = mH −mL > 0,
∆Γ = ΓH − ΓL,
∆µ = µH − µL
(1.16)
from which we can obtain
∆m = 2|M12| (1.17)
These eigenstates and eigenvectors correspond to particles with different masses m and
timelifes (or decay width) Γ; hence is possible to label them taking into account one of
that two characteristics: in this case they are labelled with H and L, for the heavy and
for the light respectively, having in mind the mass differences. This decision is suitable
for the B0 − B¯0 system, where the two mass eigenstates have a relevant mass difference
with respect to the lifetimes difference. For the system K0 − K¯0, is better to take into
account the lifetimes, being the filetime of one particle really longer in comparison with
the other one (using thus labels L and S for the long-lived and short-lived respectively).
As already considered, under a CP transformation
CP|P 0〉 = eiξ|P¯ 0〉,
CP|P¯ 0〉 = e−iξ|P 0〉 (1.18)
with arbitrary phase ξ. Hence is possible to define the two CP eigenstates as
|P±〉 =
1√
2
(|P 0〉 ± eiξ|P¯ 0〉) (1.19)
1.1. CP VIOLATION 7
We have a CP invariance for a theory described by an hamiltonian for the weak inter-
action HW , if
(CP)HW (CP)† = HW (1.20)
which implies that Γ11 = Γ22 and analogously M11 = M22. Since
M21 = e2iξM12,
Γ21 = e2iξΓ12 (1.21)
we get |R11| = |R22|.
It is convenient to introduce the CP-violating parameters
δ = |R12| − |R21|
|R12|+ |R21|
,
θ = R22 −R11∆µ
(1.22)
The eigenstates R, that are the mass eigenvectors (eigenvectors of the weak interac-
tion), can be defined as follows
|PH〉 = pH |P 0〉+ qH |P¯ 0〉,
|PL〉 = pL|P 0〉 − qL|P¯ 0〉 (1.23)
Unlike |P 0〉 and |P¯ 0〉, the mass eigenstates evolve as a function of time, according to
the equation (1.9).
|PH(t)〉 = e−iµH t|PH〉,
|PL(t)〉 = e−iµLt|PL〉. (1.24)
Due to the weak interaction, described by HW , these states evolve like an exponential
function with defined masses mH and mL, and defined decay width ΓH and ΓL, according
to eq. (1.15). The probability to observe P 0 or P¯ 0 is proportional to e−Γt. Starting with
a flavor eigenstate like P 0 (that can be written in terms of mass eigenstates) produced by
the strong interaction, it evolves during the time, displaying the possibility to observe a
flavor eigenstate of the second kind like P¯ 0.
The phase between |PH〉 and |PL〉 is not defined, and it is present in the product
〈PH |PL〉, which gives
|〈PH |PL〉|2 =
(1 + δ2)|1− θ2| − (1− δ2)(1− |θ|2)
(1 + δ2)|1− θ2|+ (1− δ2)(1 + |θ|2) (1.25)
therefore from CP invariance, that oblige to have the parameters δ = θ = 0, it follows that
〈PH |PL〉 = 0. Imposing a convention for the phases of |PH〉 and |PL〉 like for eq. (1.23),
we have 〈PH |PL〉 = δ, that is real.
8 CHAPTER 1. THEORY
Since CP violation is small, assuming that CPT conserved (so qH/pH = qL/pL = q/p)
gives
|PH〉 = (1 + ²)|P 0〉+ (1− ²)|P¯ 0〉 ' |P+〉,
|PL〉 = (1 + ²)|P 0〉 − (1− ²)|P¯ 0〉 ' |P−〉 (1.26)
where the parameter
² = p− q
p+ q (1.27)
allows to quantify the CP violation. In presence of CP violation the mass eigenstates are
not orthogonal, and are not equal to the CP eigenstates.
Classification of CP violation
If CP is conserved
M∗12 = e2iξM12,
Γ∗12 = e2iξΓ12,
q
p
= ±eiξ
(1.28)
and the CP eigenstates are equal to the mass eigenstates
CP|PH〉 = ±|PH〉,
CP|PL〉 = ∓|PL〉 (1.29)
where the sign ambiguity can be solved only by experiments. The condition to have CP
invariance is
∣
∣
∣
∣
q
p
∣
∣
∣
∣
= 1 (1.30)
The CP transformation acts on a final state f in this straightforward way
CP|f〉 = eiξf |f¯〉,
CP|f¯〉 = e−iξf |f〉 (1.31)
so for the decay amplitudes, taking into account equation (1.18), we have
A¯f = ei(ξf−ξ)Af ,
Af¯ = ei(ξf+ξ)A¯f
(1.32)
hence, avoiding the arbitrary phases ξ and ξf , to have CP invariance we need to satisfy
the conditions
|Af | = |A¯f¯ |,
|Af¯ | = |A¯f |
(1.33)
hence the decay rate of P 0 → f and P¯ 0 → f¯ must be equal.
1.1. CP VIOLATION 9
At the end from equations 1.28 and 1.32 follows that
arg
(
p2
q2
Af A¯∗fAf¯ A¯∗f¯
)
= 0 (1.34)
There are tree species of CP violation:
• indirect CP violation: CP violation in the mixing between the flavor eigenstates,
when 1.30 does not hold;
• direct CP violation: CP violation in the decay amplitudes, when 1.33 does not hold;
• interference CP violation: CP violation in the phase mismatch between the mixing
parameters (p, q) and the decay amplitudes, when 1.34 does not hold.
To summarize, CP violation arises always from an interference between phases, phases of
the elements M12 and Γ12 (indirect), phases of two decay amplitudes (direct), or phase of
p/q and the phases of the decay amplitudes (interference).
1.1.2 CP violation in the Standard Model
The charged current term of the electro-weak lagrangian (mediated by the W± boson),
written for the mass eigenstates of just one quark family, is
LqW =
g√
2
(W+µ u¯LγµV dL +W−µ d¯LγµV †uL) (1.35)
where V is an element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [11] [12]. For
three quark families
V =
Vud Vus Vub
Vcd Vcs Vcb
Vtd Vts Vtb
(1.36)
It is a unitary matrix2, with 4 independent parameters and one phase that, as observed
for the first time by Kobayashi and Maskawa, generates CP violation [12]. There is not
an analog matrix for the neutral currents, since flavor changing neutral currents do not
exists at the tree level.
There is CP violation in the Standard Model, if, and only if, any of the rephasing-
invariant functions of the CKM matrix is not real.
2V †V = V V † = 1.
10 CHAPTER 1. THEORY
Unitarity Triangle
Taking into account the unitarity conditions of the CKM matrix, the following relations
hold
VudV ∗us + VcdV ∗cs + VtdV ∗ts = 0,
VusV ∗ub + VcsV ∗cb + VtsV ∗tb = 0,
VudV ∗ub + VcdV ∗cb + VtdV ∗tb = 0.
(1.37)
these are sums of three complex quantities, that can be represented in the complex plane
as sides of a triangle; the lenghts of the sides are invariant, so the triangle does not modify
its shape under a rephasing of all the phases.
From experimental measurements of Vij, the triangles coming out from the first two
equations of 1.37 have a side much shorter than other two, these triangles are connected
to CP violation in the K and Bs system respectively.
The most interesting triangle, related with the physics of the Bd meson, is the third
one built from the orthogonality condition between the first and the second columns of
the matrix. It is the so called “Unitarity Triangle”[13].
Choosing a convention for the phases and rescaling the triangle by dividing each side
by |VcdV ∗cb|, we obtain the triangle of Figure 1.1. The inner angles of the triangle are
Figure 1.1: Unitarity triangle rescaled.
1.2. THE B+ → KSpi+pi0 DECAY 11
α = arg
(
− VtdV
∗
tb
VudV ∗ub
)
,
β = arg
(
−VcdV
∗
cb
VtdV ∗tb
)
,
γ = arg
(
−VudV
∗
cb
VcdV ∗cb
)
.
(1.38)
Wolfenstein parametrization
The first parametrization of the CKM matrix was put forward by Kobayashi and Maskawa
using the three Euler angles, writing the matrix like a product of three rotations.
In 1983 it was realized that the bottom quark b decays predominantly to the charm
quark c, so |Vcb| À |Vub|; then it was noticed by Wolfenstein that |Vcb| ∼ |V 2us|, and [14] a
parametrization in which unitarity only holds approximately was introduced, writing
V =
1− λ2/2 λ Aλ3(ρ− iη)
−λ 1− λ2/2 Aλ2
Aλ3(1− ρ− iη) −Aλ2 1
+O(λ4) (1.39)
involving the parameters λ, A, ρ and η, where λ = sin θC ' 0.22 is small and serves as
an expansion parameter, that is a function of the Cabibbo angle θC , A ' 0.82 and η
rapresent the CP violation phase. We have CP violation if η 6= 0, that means a triangle
area not equal to zero.
In the Wolfenstein parametrization the CKM matrix elements satisfy these relations
VudV ∗ub
|VcdVcb|
= ρ+ iη,
VcdV ∗cb
|VcdVcb|
= −1,
VtdV ∗tb
|VcdVcb|
= 1− ρ− iη.
(1.40)
1.2 The B+ → KSpi+pi0 decay
1.2.1 Experimental and theoretical status
The B+ → KSpi+pi0 decay3 proceeds via quasi-2-body channels, B+ → K∗+pi0,
B+ → K∗0pi+ and B+ → ρ+K0S, or via the non resonant 3-body decay. The K∗ and
ρ resonances decay into Kpi and pipi final states respectively. Many resonances overlap in
3unless otherwise stated, charge conjugate modes are implied
12 CHAPTER 1. THEORY
b
u
+B
+W u
us
u
*+K
0pi b
u
+B
+W s
u
g
u
u
0pi
*+K
b
u
+B
+W u
u
s
u
*+K
0pi
b
u
+B +W su
g u
u
0pi
*+K
Figure 1.2: Trees (left column) and penguins (right column) Feynman diagrams, where the first row are
the colour suppressed and the second row the colour favoured diagrams.
Mode PDG avg. [39] BABAR ref. Belle ref. CLEO ref. New avg.
K∗(892)0pi+ 10.9± 1.8 10.8± 0.6+1.1−1.3 [15] 9.7± 0.6+0.8−0.9 [16] 7.6+3.5−3.0± 1.6 [17] 10.0± 0.8
K∗(892)+pi0 6.9± 2.4 6.9± 2.0± 1.3 [18] 7.1+11.4−7.1 ± 1.0 [17] 6.9± 2.3
K∗0 (1430)0pi+ 47± 5 32.0± 1.2+10.8−6.0 [15] 51.6± 1.7+7.0−7.4 [16] 45.2+6.2−6.3
K∗2 (1430)0pi+ < 6.9 5.6± 1.2+1.8−0.8 [15] < 6.9 [19] 5.6+2.2−1.4
K∗(1410)0pi+ < 45 < 45 [19] < 45
K∗0 (1680)0pi+ < 12 < 15 [20] < 12 [19] < 12
K0pi+pi0 (N.R.) < 66 < 66 [21] < 66
K0Sρ+ 8.0± 1.5 8.0+1.4−1.3 ± 0.6 [22] < 48 [23] 8.0+1.5−1.4
Table 1.1: Compilation of the B0 → K0Spi+pi0 results. Snapshot of December 2008. B+ Branching
Fractions (decays with kaons) (×106). (UL 90% CL).
phase space, therefore is required an amplitude (Dalitz plot) analysis of the 3-body final
states (see section 1.3). Measurements existing up to now are included in table 1.1. While
all final states can be reached via colour allowed penguin diagram and annihilation dia-
grams, the B+ → K∗+pi0 can also proceed through color allowed and color suppressed tree
and penguin graphs (Figure 1.2) with CKM factors λ4 and λ2 respectively. The gluonic
penguin processes are favored by color and CKM. The electroweak penguin transitions
might be sizeable as well.
The most important decays channel that contribute to the final 3-body state are listed
in table 1.2.1 together with the branching fraction prediction; table 1.1 shows the corre-
spondingly measured branching fractions.
The charmless decays B → Kpipi are dominated by b → s penguin transition. Under
1.2. THE B+ → KSpi+pi0 DECAY 13
Mode Model ref. B.F. prediction
ρ(770)±K0S QCDF [24] 10.27±1.96
global fit [25] 6.08±0.79
HMChPT [26] 1.3 +3.0−0.9
K∗(892)±pi0 QCDF [24] 5.25±0.83
global fit [25] 7.00±4.49
HMChPT [26] 1.5+0.3−0.3
FSI [27] 12.4+1.5−0.8
K∗(892)0pi± QCDF [24] 8.90±1.59
global fit [25] 10.64±0.82
HMChPT [26] 1.5+0.4−0.3
FSI [27] 22.5+2.8−0.9
K∗(1430)±pi0 HMChPT [26] 5.5+1.6−1.4
K∗(1430)0pi± HMChPT [26] 5.2+1.6−1.4
K0pi±pi0 HMChPT [26] 10.0+7.1−3.7
Table 1.2: Theoretical predictions (×106) for various models of the signal model of the Dalitz analysis,
together with a global fit. The models listed are: QCD factorization (QCDF), heavy meson chiral
perturbation theory (HMChPT) and final-state interaction (FSI).
the factorization approach [26], the decay amplitude consists of three distinct factorizable
terms: the current-induced process, the transition process and the annihilation process.
Recently, Belle has measured the direct CP violations B− → Kpi decay [ref] that for
the charge B is
ACP (B− → K−pi0) = Γ(B
− → K−pi0)− Γ(B+ → K+pi0)
Γ(B− → K−pi0) + Γ(B+ → K+pi0) = +0.07± 0.03± 0.01, (1.41)
and the average of the current experimental data of BABAR , Belle, CLEO and CDF by
the Heavy Flavor Averanging Group (HFAG) [28] is
ACP (B− → K−pi0) = 0.050± 0.025 (1.42)
A difference is observed between direct CP violations in charged and neutral modes, that
by the HFAG average is
∆A = ACP (B− → K−pi0)− ACP (B¯0 → K−pi+) = 0.147± 0.028 (1.43)
at 5σ level; however, recent calculations based on the QCD factorization approach
(QCDF), the perturbative QCD approach (pQCD) and the soft-collinear effective theory
14 CHAPTER 1. THEORY
(SCET), predicted that ACP (B− → K−pi0) and ACP (B¯0 → K−pi+) are close to each other.
The mismatch between theory and experiment is maybe due to the limited understanding
of the strong dynamics in B decays, but equally possibly due to new physics effects.
Even recent theoretical estimations within the QCDF framework give ACP (B− →
K−pi0) = −0.109± 0.008 [24], very close to ACP (B¯0 → K−pi+), but still in sharp contrast
to experimental data. So it is very hard to accommodate the measured large difference ∆A
in the SM with the available approaches for hadron-dynamics in B decays, even varying
the value of the effective gluon mass that enter in the models. This could be an indication
of new sources of CP violation beyond the SM. Using a set of FCNC effective NP operators
(b → suu¯ and b → sdd¯) the results are more consistent with the experimental data for
ACP (B− → K−pi0).
1.2.2 Constraints on γ angle from B → Kpipi modes
The current methods to measure γ rely on the interference between the colour-allowed
B− → D0K− and the colour-suppressed B− → D¯0K− decay modes resulting in direct
CP violation. They are theoretically very clean, as only tree amplitudes are involved,
but their sensitivity to γ is governed by the rather small relative magnitude of the two
amplitudes, denoted rB: 0.05 . rB . 0.3, depending on the D meson decay channel. As
a consequence, γ is the most poorly determined angle of the unitarity triangle, (78± 12)◦
using only direct determinations (c.f. β = (22.0± 0.8)◦ with the full fit) [29]. Therefore,
any independent determinations of the angle γ should be exploited in order to reduce the
statistical uncertainty. Although a first proposal on using the charmless three-body decays
B → Kpipi to extract the unitarity triangle angle γ via isospin relations was made in 2002
[30], the more recent ideas in [31] [32] are both far more accurate in their estimations of
the theoretical uncertainties of their methods, and more convenient experimentally.
The paper by Ciuchini, Pierini and Silvestrini [32] exploits the use of the phase-
extraction capabilities of the Dalitz plot analysis technique, similarly to what Lipkin-
Nir-Quinn-Synder proposed to measure α in B0 → ρpi → pi+pi−pi0 [33]. They start by
relating the ratio of the amplitudes for the decays B+ → K∗+pi0 and B+ → K∗0pi+ and
their CP conjugates to γ through isospin, and then cleverly take advantage of the Dalitz
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