Summary of Lecture Notes with Questions - Institute for Particle ...

Durham University,
Ephiphany Term 2011,
Version of July 4, 2011
Flavour Physics and Effective Theories (FPE)
A brief summary of basic concepts and central results will be given.
self-contained script!
This is not a
Dr Thorsten Feldmann
IPPP Durham

Contents
1 Quark Flavour in the Standard Model 1
2 The Weak Effective Hamiltonian 4
2.1 Quantum Corrections to b → csū . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Wilson Coefficients in Perturbation Theory . . . . . . . . . . . . . . . . . 9
2.3 Penguins and Boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Flavour Transitions (on the hadron level) 16
3.1 Example: ¯B → DK decays . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Example: B → πK decays . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Mixing and CP Violation 22
4.1 K 0 - ¯K 0 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 B 0 - ¯B 0 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 Semi-Leptonic Decays 36
5.1 |V cb | from exclusive decays and HQET . . . . . . . . . . . . . . . . . . . 36
5.2 |V cb | from inclusive decays . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.3 |V ub | from inclusive decays and SCET . . . . . . . . . . . . . . . . . . . . 47

1 Quark Flavour in the Standard Model
• Quark multiplets in the SM, and the Yukawa Sector of the Lagrangian
L yuk = −Y ij
D
¯Q i LHD j R − Y ij
U
¯Q i L ˜HU j R + h.c.
coupling left-handed doublets and right-handed singlets with the Higgs field.
• Diagonalization of the Yukawa matrix by bi-unitary transformations
and m u = y u 〈H 0 〉 etc.
Y U = V uL diag(y u , y c , y t ) V †
u R
, Y D = V dL diag(y d , y s , y b ) V †
d R
• Flavour mixing in charged weak currents (weak basis → mass basis)
Ū L gW µ γ µ D L → Ū ′ LV †
u L
gW µ γ µ V DL D ′ L
≡ (V CKM ) ij (Ū ′ L) i gW µ γ µ (D ′ L) j
with unitary Cabbibo-Kobayashi-Maskawa matrix, V CKM = V †
u L
V dL .
• CKM parameter counting:
– 36=18+18 parameters in Y U and Y D
– broken flavour symmetries for quark multiplets:
makes 26 symmetry generators.
U(3) QL × U(3) UR × U(3) DR /U(1) B
– leaves 10 physical parameters: 6 masses, 3 angles, 1 CP-violating phase
• interactions with γ and Z 0 unaffected by unitary rotations: no flavour-changing
neutral currents (FCNCs) in the SM (at tree-level)
• FCNCs can be induced through quantum loops. Example: s → dγ (diagram, see
lecture), which involves the CKM factors V is Vid ∗ with i = u, c, t.
• W ± only couple to left-handed fields: Flavour transitions with right-handed quarks
q R suppressed by m q /m W .
• GIM mechanism (Glashow-Iliopoulos-Maiani): For equal quark masses in FCNC
loops, the contributions would add up to zero, since ∑ i V isVid ∗ = 0 as a consequence
of the CKM-unitarity. For hierarchical quark masses, the heaviest quark usually
dominates (e.g. top-quark in b → sγ).
1

• Standard CKM parametrization (PDG) in terms of angles and phase.
• Wolfenstein parametrization, based on θ 13 ≪ θ 23 ≪ θ 12 ≪ 1
λ ≡ sin θ 12 =
¯ρ + i¯η ≡ − V udV ∗
ub
V cd V ∗
cb
|V us |
√
|Vud | 2 + |V us | 2 ,
≃ sin θ 13e iδ
Aλ 3 .
Aλ2 ≡ sin θ 23 = λ |V cb|
|V us | ,
• Unitary triangle(s) in complex plane: Standard triangle from
0 = V ∗
ub V ud + V ∗
cb V cd + V ∗
tb V td
V ∗
cb V cd
with tip of the triangle at ¯ρ + i¯η. Area of any CKM triangle given by Jarlskog
determinant J/2.
• Sides and angles of the triangle from absolute values and arguments of ratios of
CKM elements. Aim of flavour physics: Overconstrain the CKM triangle to precisely
measure (¯ρ, ¯η) and test the CKM mechanism in the SM.
– |V ij | from semi-leptonic decays (i ≠ t, no top bound states)
– |V tb | ≃ 1 from t → bW +
– |V td | and |V ts | from loop induced rare decays or meson-mixing involving b → d,
b → s or s → d transitions.
– CKM angles from interference effects, leading to CP asymmetries between
particle and anti-particle decay.
Requires precision experiments with kaons and B-mesons and theoretical control
on strong interaction effects.
Questions/Exercises:
• Calculate
V CKM = R 23 (θ 23 ) · diag(1, 1, e iδ ) · R 13 (θ 13 ) · diag(1, 1, e −iδ ) · R 12 (θ 12 )
with R ij =rotations in the i − j plane.
• Disentangle angles and phases in the Yukawa matrices and in the broken
flavour symmetry generators, and find that there are 3 physical angles and
1 physical CP phase left. Repeat the analysis for (2,4,n) generations.
• What is the GIM mechanism
2

Further Reading:
• [1, page 13-16]; [2, page 3-11]; [3];
3

2 The Weak Effective Hamiltonian
Example: b → csū decay
• Can be approximated by 4-quark interactions, similar to the Fermi model for muon
decay (µ − → e −¯ν e ν µ )
• In this approximation, the decay amplitude can be obtained from an effective
operator in the interaction Hamiltonian:
with two left-handed currents,
O eff = G F
√ J α
(b→c) g αβ J (u→s)
β
(1)
2
J (b→c)
α = V cb [¯cγ α (1 − γ 5 )b] ,
J (u→s)
β
and the usual definition of the Fermi constant,
= V ∗
us [¯sγ β (1 − γ 5 )u] , (2)
G F
√
2
≡
g2
8m 2 W
≃ 1.166 · 10 −5 GeV −2 .
• We encounter one of the simplest examples for an Effective Theory, with the
following general features:
– operators in the effective interaction Hamiltonian have dimension > 4 (in the
above case dim=6), and the corresponding coupling constants (here G F ) have
negative mass dimension, and the theory – in general – is non-renormalizable,
i.e. only valid up to some cut-off scale (here m W );
– effective operators are contain light (low-energetic) field modes, only
(here light quarks);
– effect of large energy/mass scales (here m W ) absorbed into coupling constants.
4

Questions/Exercises:
• Write down the Feynman diagram and the effective operator for the decay
c → su ¯d in the Fermi model.
2.1 Quantum Corrections to b → csū
A typical QCD loop correction involves the exchange of an additional gluon between the
two hadronic currents:
• The loop momentum q µ can take any values between −∞ and +∞, and therefore
a simple expansion in |q|/m W as in the Fermi model is not possible.
• The idea of “Factorization” within the context of Effective Field Theories is
to systematically separate the cases of
– |q| ≥ m W (“short-distance” dynamics)
– |q| ≪ m W (“long-distance” dynamics)
An intuitive approach (which can be mathematically realized within e.g. dimensional
regularization) is to consider a Feynman integral in the “full theory” (which, itself can
be an effective field theory) and divide it into its different momentum regions by
expanding the integrand according to the scaling with the large mass/energy parameter:
• In our case, the full theory integral for the above Feynman diagram would yield a
generic function
(
I α s , m b
, m )
c
, . . .
m W m b
where we assumed the function I to be dimensionless by factoring out an overall
factor of G F , and therefore it can only depend on dimensionless quantities, i.e.
the strong coupling constant and ratios of masses (and external momenta) of the
particles being involved. 1
1 We assume here, that potential UV divergences have been renormalized by appropriate counter
terms and absorbed into running coupling/mass parameters.
5

• In the infra-red (IR) region of “soft” loop momenta |q µ | ≪ m W , we can still approximate
the W -exchange by a point-like 4-fermion interaction, which correspond
to taking the limit m W → ∞ in the integrand, and the above diagram would reduce
to:
– It represents a diagram within the effective theory, i.e. a contribution to the
vertex correction to the effective Hamiltonian.
– Since one propagator has shrunk to a point, the diagram behaves differently
for large momenta. In particular – in the general case – additional
UV-divergences appear which are not compensated by the counter terms inherited
from the full theory. One therefore has to introduce an additional
UV-regularization (e.g. by an explicit cut-off of order m W , or in terms of
dimensional regularisation).
– The corresponding (dimensionally renormalized) integral will have the generic
form
(
I IR = I IR α s , ln µ , m )
c
, . . . ,
m b m b
i.e. it does not depend on the heavy mass m W , but instead it depends on an
arbitrary UV-renormalization scale µ.
• On the other hand, in the UV-region, we have
|q| m W ≫ m b , m c , . . .
and it can be obtained from the full theory integral by neglecting the masses (as
well as external momenta) of the light degrees of freedom, m b , m c , . . . → 0.
– The UV behaviour is as for the full theory (i.e. regularized by the same counter
terms).
– But now, one encounters new IR divergences, because the light masses have
been set to zero.
6

Therefore, the UV-region of the integral (after renormalization) will have the
generic form
(
I UV = I UV α s , ln µ )
m W
depending now on an IR-regulator scale µ, but being manifestly independent of the
low-energy (long-distance) parameters of the theory. It thus represents a genuine
“short-distance” effect which can be interpreted as a quantum correction to socalled
Wilson Coefficients multiplying the effective operators in the interaction
Hamiltonian.
• Notice that, due to the colour exchange, the operator basis generally has to be
extended.
To summarize, a generic 1-loop Feynman integral in the full theory is reproduced by the
sum of a 1-loop matrix element of an effective operator within the effective theory (the
IR region), and a 1-loop short-distance coefficient (from the UV region) multiplying the
tree-level matrix element of (another) operator in the effective theory:
I
(
α s , m b
m W
, m c
m b
, . . .
)
= 〈O〉 1−loop
(
Effective Operators for b → csū
α s , ln µ m b
, m c
m b
, . . .
)
+ C ′ (
α s , ln µ
m W
)
〈O ′ 〉 tree (3)
Use symmetry considerations (colour, flavour, Lorentz, . . . ) to constrain the operator
basis
• short-distance QCD corrections preserve chirality (Wilson coefficients calculated
with m q → 0)
⇒ only (V − A) ⊗ (V − A) structure involving left-handed quark fields
• there are two independent possibilities to construct a colour singlet from two quark
and two anti-quark fields. Two independent colour structures can be chosen as
O 1 = (¯s ) i
Lγ α u j j
L
(¯c
L γα bL) i ,
O 2 = (¯s ) )
i
Lγ α u i j
L
(¯c
L γα b j L
(4)
• The decays b → csū are thus described by two current-current operators in the
effective Hamiltonian and the corresponding Wilson coefficients
7

H eff = − 4G F
√
2
V cb V ∗
∑
C i (µ) O i + h.c. (5)
us
i=1,2
Here, the Wilson coefficients C i (µ) contain all information on short-distance physics
above the scale µ, whereas the matrix elements of the operator reproduce the IR
behaviour of the theory.
Questions/Exercises:
– How are the different momentum regions of Feynman integrals in the full
SM theory reproduced within the effective theory based on the weak effective
Hamiltonian
– Write down the two current-current operators for the decay c → su ¯d. Explain
the appearance of the two colour structures.
– Why does the electroweak Hamiltonian in the SM does not contain currentcurrent
operators with right-handed quarks
– Consider the toy integral
∫ ∞
0
dk
M − m
(k + M)(k + m)
∗ Calculate the integral exactly, and then determine the leading term for
m/M ≪ 1.
∗ Calculate the UV-region of the integral by setting m → 0 in the integrand.
Use a regulator (k/µ) δ in the integrand with δ → 0 + .
∗ Calculate the IR-region of the integral by considering M → ∞ in the
integrand, and use the same regulator (k/µ) δ with δ → 0 − .
∗ Verify that the sum of the IR and UV region reproduce the full integral
in the limit m/M ≪ 1.
Further Reading:
– [2, chapter 2.1]
8

2.2 Wilson Coefficients in Perturbation Theory
Following the above lines, the 1-loop result for the Wilson coefficients C 1,2 (µ) (renormalized
in the MS scheme) reads
{ } 0
C i (µ) =
1
+ α s(µ)
4π
(ln µ2
m 2 W
+ 11 ) { } 3
6 −1
(6)
As indicated, the result depends on the arbitrary renormalization scale µ. The advantage
of the effective-theory construction precisely lies in the fact that the Wilson coefficients
only depend on the physical short-distance scale m W . Therefore, the natural procedure
is to consider µ ∼ O(m W ), which in this context is called the “Matching Scale”:
• the perturbative expansion of C i (µ ∼ m W ) does not contain coefficients enhanced
by large logarithms
• the Wilson coefficients can be reliably calculated in fixed-order perturbation
theory, since α s (m W )/π ≪ 1.
Anomalous Dimension
In order to compare with experiment, the matrix elements of the effective-theory operators
are needed at low-energy (hadronic) scales. For instance, in B-meson decays
we typically need µ ∼ m b in order that no large logarithms ln µ m b
appear. From the
effective-theory construction, it is clear that only the combination
∑
C i (µ)〈O i 〉(µ)
i
is µ-independent in perturbation theory. The advantage of the effective theory is that it
also allows to calculate the scale dependence of the operator matrix elements, respectively
of the Wilson coefficients, within the effective theory. The scale dependence is quantified
by the Anomalous Dimension Matrix γ ij which itself has an expansion in α s (µ),
∂
∂ ln µ C i(µ) ≡ C j (µ) γ ji (µ) , (7)
γ(µ) = γ(α s (µ)) = α s(µ)
4π γ(1) + . . . (8)
where γ (1,2,...) are numerical coefficient matrices that can be calculated from the renormalization
constants for the 〈O i 〉. In our case
9

γ (1) =
( ) −2 6
6 −2
(9)
with eigenvectors C ± = 1 √
2
(C 2 ± C 1 ) and eigenvalues γ (1)
± = +4, −8. The differential
equation (7) can then formally be solved by separation of variables, leading to
ln C ∫ ln µ
±(µ)
C ± (M) = d ln µ ′ γ ± (µ ′ ) (10)
ln M
Substituting the QCD β-function,
d ln µ ′ = dα s
2β , 2β = −2β 0
4π α2 s + . . . (11)
and keeping only the leading coefficients β 0 and γ (1)
± , one ends up with the solution
( ) (1) −γ αs
±
(µ)
/2β 0
C ± (µ) ≃ C ± (m W )
α s (m W )
(12)
α
Re-expanding the ratio s(µ)
α s(m W
, one realizes that the above formula resums the socalled
“leading logarithms”. Starting at the matching scale m W we can thus use
)
the ”renormalization-group running” of the Wilson coefficients in the effective theory
to obtain “renormalization-group improved” predictions at the low-energy scale
(as compared to the fixed-order result in (6)).
Numerical Values (for our example):
C 1 (m b ) C 2 (m b ) approximation
−0.514 1.026 leading-log
−0.303 1.008 next-to-leading-log
Remark: Possible effect of New Physics contributions:
• change value of Wilson coefficients at matching scale
• change operator basis (right-handed, scalar, or tensor currents)
10

Questions/Exercises:
• Proof the Fierz identity between colour matrices,
t a ikt a jl = − 1
3N C
δ ik δ jl + 1 2 δ ilδ jk ,
to reproduce the relative weights between the 1-loop corrections to C 1 and
C 2 in (6).
• Verify the eigenvectors and eigenvalues of the anomalous dimension matrix
in (9).
• Work out the details in the derivation of (12).
• Re-expand (12) in fixed-order perturbation theory, using the one-loop approximation
for the ratio α s (µ)/α s (m W ). Compare with (6).
Further Reading:
• [2, chapter 2.2]; [4, chapter 18.2]; [1, page 14-41]
11

2.3 Penguins and Boxes
Example: b → sq¯q decays
We are now considering hadronic b → s transitions with an arbitrary (flavour-singlet) q¯q
pair in the final state (analogously b → dq¯q or s → dq¯q).
• As before, we obtain effective current-current operators, but now there are two
different flavour structures contributing
V ub V ∗
us [ū L γ µ b L ] [¯s L γ µ u L ] ≡ λ u O (u)
2 ,
V cb V ∗
cs [¯c L γ µ b L ] [¯s L γ µ c L ] ≡ λ c O (c)
2 . (13)
By gluon exchange, we also generate the corresponding current-current operators
with differing colour structure
O (u)
1 , O (c)
1 .
• The same final state can also be obtained by so-called penguin diagrams, where
the q¯q-pair is produced from gluon radiation off a loop diagram:
– The produced q¯q pair now can refer to left- or right-handed fields. This gives
rise to four new types of “strong penguin operators” O 3−6 of the form
∑
[¯s L γ µ b L ] [¯q L γ µ q L ] ,
q=u,d,s,c,b
∑
q=u,d,s,c,b
[¯s L γ µ b L ] [¯q R γ µ q R ] , (14)
each coming in two colour variants. The Wilson coefficients for these operators
are suppressed by the strong coupling constant and a loop factor.
– Actually, we also have to allow for operators describing the partonic b → sg
process, because it will generate the same flavour quantum numbers. Using
QCD gauge invariance and exploiting the equations of motion for the quark
12

fields, one realizes that the only relevant operators have to have the form of a
“chromomagnetic penguin” coupling left- and right-handed quark fields,
g s
O g 8 ≡
16π m 2 b ¯s L σ µν G µν b R ,
O g 8 ′ ≡
g s
16π m 2 s ¯s R σ µν G µν b L . (15)
Notice that the operators include an additional quark-mass factor, and therefore
formally are counted as dim-6. The second operator is mostly negligible
since m s ≪ m b .
– The CKM factor for the penguin operators can be determined by exploiting
the GIM mechanism: The result of any penguin diagram gets contributions
from up, charm and top-quarks in the loop. As discussed before, in the
matching calculation we can set m u = m c = 0, and therefore, generically, one
ends up with functions
∑
V ib Vis ∗ f(m 2 i /m 2 W ) = λ u f(0) + λ c f(0) + λ t f(m 2 t /m 2 W )
i=u,c,t
= λ t
(
f(m
2
t /m 2 W ) − f(0) ) , (16)
where in the last step, we have used the CKM unitarity.
The weak effective Hamiltonian for the b → sq¯q transitions (including strong-interaction
effects) therefore reads
H eff = − 4G F
√
2
{ ∑
+ h.c.
i=1,2
(
C i (µ) λ u O (u)
i
(
)
6∑
)}
+ λ c O (c)
i + λ t C i (µ)O i + C8(µ)O g g 8
i=3
(17)
Electroweak penguin operators
Electroweak corrections to the weak effective Hamiltonian can become important for
• precision observables, in particular isospin asymmetries,
• weak decays with leptons or photons in the final state.
To this end, penguin and box diagram with additional photon or Z-exchange have to be
included. This yields:
13

• electroweak corrections to the matching and running of the Wilson coefficients in
(17),
• new isospin-violating effects, to be accounted for by electroweak penguin operators,
O 7 =
∑
e q [¯s L γ µ b L ] [¯q L γ µ q L ] , O 8 = ∑
]
i
e q
[¯s L γ µ b j j
L
[¯q
L γ µqL] i ,
O 9 =
q=u,d,s,c,b
∑
q=u,d,s,c,b
q=u,d,s,c,b
e q [¯s L γ µ b L ] [¯q R γ µ q R ] , O 10 = ∑
q=u,d,s,c,b
]
i
e q
[¯s L γ µ b j j
L
[¯q
R γ ]
µqR
i ,
(18)
which differ from the strong penguin operators by an additional factor e q refering
to the fractional quark charge, and by the fact that their Wilson coefficients are
suppressed by an additional factor α em /α s .
• a new electromagnetic operator, describing b → sγ,
O γ 7 ≡
where we neglected O γ 7 ′ for m s → 0.
Semi-leptonic operators
e
16π 2 m b ¯s L σ µν b R F µν , (19)
For decays into charged leptons, b → sl + l − , another set of semi-leptonic operators has
to be added,
O 9,V = [¯s L γ µ b L ] ¯lγ µ l ,
O 10,A = [¯s L γ µ b L ] ¯lγ µ γ 5 l , (20)
For decays into neutrinos, b → sl + l − , only the corresponding 4-fermion operator has to
be taken into account, as the neutrinos can only arise from short-distance processes, i.e.
penguin diagrams with Z-boson radiation or box-diagrams with two W -boson exchange.
Operators for neutral meson mixing
In neutral meson mixing, we consider flavour transitions of the form
b ¯d ↔ d¯b (|∆B| = 2, ∆S = 0) ,
b¯s ↔ s¯b (|∆B| = |∆S| = 2) ,
s ¯d ↔ d¯s (|∆S| = 2, ∆B = 0) , (21)
This requires box diagrams with two W -exchanges again involving only left-handed
quarks,
14

H |∆B|=2,∆S=0
eff
= G2 F
4π 2 m2 W (VtbV ∗
td ) 2 C(m t /m W , µ) [¯bL ] ]
γ µ d L
[¯bL γ µ d L + h.c. (22)
Because the field content of the two currents in brackets is identical:
• there is only one independent colour structure,
• the CKM factor is the simple square (not the absolute square) of the CKM elements.
Questions/Exercises:
• What would be the CKM factor of the penguin operators in s → dq¯q and in
c → uq¯q decays
• What are the crucial differences between electroweak and strong penguin
operators
• Draw an example for an effective-theory diagram contributing to b → sγ and
involving one of the strong penguin operators O 3−6 (or the operator O g 8).
• Draw examples or box and penguin diagrams contributing to b → sl + l − and
b → sν¯ν.
Further Reading:
• [1, part two (p. 43ff)]
15

3 Flavour Transitions (on the hadron level)
The analysis of quark-flavour transitions is complicated by the fact that quark are confined
in hadronic bound states. As an example, we will study the different flavour
topologies that can contribute to 2-body non-leptonic decays.
3.1 Example: ¯B → DK decays
As an example, we will study the quark-level transition b → csū (see previous section)
within the non-leptonic decay modes B → DK. Depending on the spectator quark in
the B-meson, we may consider the following hadronic decays: 2
(a) ¯B0 d → D + K − with (b ¯d) → (c ¯d)(sū) , (23)
(b)
(c)
¯B0 d → D 0 ¯K0
B − → D 0 ¯K−
with (b ¯d) → (cū)(s ¯d) , (24)
with (bū) → (cū)(sū) , (25)
(c) ¯B0 s → D + s ¯K − with (b¯s) → (c¯s)(sū) . (26)
Let us consider the “flavour topologies” for each decay mode in turn
(a) ¯B 0 d → D+ K − : Here the ¯d-quark simply acts as a spectator quark in the B → D
transition, while the kaon flavour stems from the quarks emitted from the weak
decay.
This case is sometimes refered to as a “Tree Topology” (class-I).
(b) ¯B 0 d → D0 ¯K0 : Here the ¯d-quark acts as a spectator quark in the B → K transition,
while the D-meson flavour stems from the quarks emitted from the weak decay.
2 We may have also considered B → DK decays induced by b → us¯c decays.
16

This case is sometimes refered to as a “Tree Topology” (class-II).
(c) B − → D 0 K − : Here the ū-quark can act as a spectator quark in either the B → D
or the B → K transition.
Therefore, this decay is induced by “Tree Topologies” of class-I and class-II.
(d) ¯B s
0 → D s + K − : This can be induced by a class-I tree topology, but also by an
“Annihilation Topology” (class-III), where both quarks from the initial-state
meson participate in the weak decay.
Each decay topology gives rise to qualitatively different hadronic bound state effects:
17

Class-I Topology
In the “naive“ picture, we would neglect interactions between the emitted quarks forming
the kaon and the quarks involved in the B → D transition. Starting from the hadronic
matrix element of the weak effective Hamiltonian,
〈D + K − |H eff | ¯B 0 d〉 ∼ 4G F
√
2
V cb V ∗
cs C i (µ) 〈D + K − |O i | ¯B 0 d〉
we obtain the so-called ”naive factorization approximation“,
〈D + K − |O 1,2 | ¯B 0 d〉 ∼ 〈D + |J (b→c) |B 0 d〉
} {{ } × 〈K− |J (u→s) |0〉
} {{ } .
B → D form factor
kaon decay constant
which only involves to (relatively) simple hadronic objects:
• The B → D transition form factor F B→D (q 2 = m 2 K ).
• The kaon decay constant f K .
In order to determine the individual contributions from the two operators O 1,2 , we have
to project onto colour singlet currents:
• The operator O 2 is already given as the product of two colour-singlet currents
J (b→c) and J (u→s) .
• For the operator O 1 we make use of the Fierz identity (see above)
(¯c i Lγ µ b j L )(¯sj L γµ u i L) = 1 3 J (b→c) J (u→s) + octet × octet
Consequently, for the class-I topologies we obtain
∑
C i (µ) 〈D + K − |O i | ¯B d〉 0 ∼
i=1,2
(
C 2 (µ) + 1 )
3 C 1(µ) · F B→D (m 2 K) · f K
However, we immediately see that the above formula cannot be correct, as the transition
form factors and the decay constants do not compensate the scale-dependence of the
Wilson coefficients.
18

The correct scale-dependence of the hadronic matrix elements actually comes from the
”non-factorizing” gluon cross-talk between the quarks in the kaon and the quarks
in the B → D transitions. For the naive factorization hypothesis to yield a good approximation
to the real case, we would have to show that the non-factorizing effects
are
• suppressed by α s (m b ) in perturbation theory
(i.e. dominated by short-distances of order δx ∼ 1/m b ),
• or suppressed by Λ QCD /m b (sub-leading non-perturbative effects).
An intuitive explanation involves the so-called colour-transparency argument:
• The light kaon (in the B-meson rest frame) is rather energetic (E ∼ 2 GeV
∼ O(m b /2)). Therefore the non-factorizing gluons effectively “see” a rather small
colour dipole of a size δx ∼ 1/E. The coupling constant of the gluons to this dipole
is thus given by α s (E) ≪ 1.
• Couplings to higher multipoles in the hadronic transitions will be suppressed by
Λ QCD /E.
A diagrammatic proof that the colour-transparency argument for a certain class of nonleptonic
B decays is correct has been given in [5]. Moreover, the perturbative corrections
can be systematically calculated, giving rise to the so-called “QCD(-improved) Factorization
Approach”. The formula for the hadronic matrix elements is modified as
∑
C i (µ) 〈D + K − |O i | ¯B d〉
0
i=1,2
∼ F B→D · f K ·
∫ 1
0
(
du C 2 (µ) + 1 3 C 1(µ) + α )
s(µ)
4π t(u, µ) φ K(u, µ) + . . .
• The function t(u, µ) can be calculated within QCD perturbation theory and depends
on the energy/momentum fractions u and ū = 1 − u for a quark–antiquark
state in the kaon.
• Therefore, the final result for the transition amplitude is obtained by a convolution
integral, involving a new hadronic input function, φ K (u, µ), characterizing the
probability amplitude for the energy/momentum fraction u in the kaon, with
∫ 1
0
du φ K (u, µ) = 1 .
• The calculation of Λ QCD /m b corrections within the QCDF approach, on the other
hand, is difficult and yields an irreducible source of theoretical uncertainties.
19

Class-II Topology
• Naively, we would expect a similar factorization into a B → K form factor and the
D-meson decay constant.
• Concerning the colour structure, the role of O 1 and O 2 is now reversed, and the
amplitude would be proportional to
C 1 + 1 3 C 2
which happens to be a rather small number (see Table in the previous chapter).
The class-II decay are therefore also refered to as “colour-suppressed”.
• Finally, the colour transparency argument does not apply for the class-II topologies,
because the ratio of E/m D in the decay is moderate and therefore the emitted D-
meson corresponds to a rather large colour dipole.
Fortunately, in this case, the transition of a B-meson at rest into an energetic kaon
itself is a sub-leading effect, since the spectator quark has to be accelerated from
essentially at rest to a constituent of a fast kaon. Therefore the class-II topologies
for B → DK are suppressed by 1/E K with respect to the class-I topology.
Annihilation Topology
Again, to produce the additional fast (back-to-back) s¯s pair from the vacuum, one receives
a suppression of the decay amplitude with 1/E ∼ 1/m b , and the annihilation
topology is sub-leading compared to class-I decays.
3.2 Example: B → πK decays
We briefly comment on the modifications – compared to the previous subsection – that
arise when both final-state mesons are light:
• Both final-state mesons are energetic, E ∼ m b /2, and therefore class-I and class-II
decays can be treated on the same footing, and the colour transparency argument
applies in both cases.
• The perturbative calculation of the non-factorizable gluon exchange diagrams, now
depends on the momentum fraction of all the various quarks involved in the decay.
As a consequence, one has to specify the momentum distributions φ π (u) and φ K (u)
as well as the distribution φ B (ω), where ω is the energy of the spectator quark in
the B-meson.
• As the fundamental quark transitions involve b → sq¯q, one has to extend the
analysis to include hadronic matrix elements of the penguin operators (O 3−6,8 g ,...).
The give rise to flavour topologies like the following,
20

We thus encounter many contributions, involving many hadronic input parameters, and
different kind of suppression factors (CKM elements, Wilson coefficients, colour factors,
α s , 1/m b , . . . ). The analysis of systematic theoretical uncertainties is thus very complicated
and difficult. As a general rule, we may conclude:
• The colour-allowed amplitudes (class-I) are generally well-predicted and theoretical
estimates are in line with experimental measurements.
• The colour-suppressed amplitudes (class-II) have large theoretical errors because
of numerical cancellations.
• The annihilation topologies cannot be predicted reliably (but are expected to be
sub-leading).
Questions/Exercises:
• Draw the possible flavour topologies for B → DK decays, but now with the
decay currents for b → us¯c.
• Which decay operators and topologies appear in B → Dπ decays
• What is naive factorization, and how can it be improved
• Why are the class-II and class-III topologies in B → DK decays suppressed
with respect to class-I
Further Reading:
• [6]
21

4 Mixing and CP Violation
The ∆S, B = 2 operators specified in chapter 2, induce mixing between neutral mesons
and their anti-mesons.
4.1 K 0 - ¯K 0 Mixing
• The neutral kaons can be specified by their flavour content
K 0 ∼ (¯sd) ,
¯K0 ∼ (s ¯d)
These flavour eigenstates are not the physical (mass) eigenstates.
• Chosing a phase convention for CP-transformations,
we can construct CP eigenstates,
CP |K 0 〉 = −| ¯K 0 〉 , CP | ¯K 0 〉 = −|K 0 〉 ,
|K 1 〉 ≡ 1 √
2
(
|K 0 〉 − | ¯K 0 〉 ) , CP |K 1 〉 = +|K 1 〉 ,
|K 2 〉 ≡ 1 √
2
(
|K 0 〉 + | ¯K 0 〉 ) , CP |K 2 〉 = −|K 2 〉 . (27)
They would be the physical eigenstates, if CP were conserved.
• The neutral kaons can decay into pions, with |ππ〉 being CP-even, and |πππ〉 being
CP-odd. If CP were a good symmetry, we expect
K 1 → 2π
large phase space → large width → short life-time
K 2 → 3π small phase space → smaller width → larger life-time (28)
• In reality, we observe small corrections from CP-violation in weak interactions, i.e.
small admixtures of the opposite CP states in physical mass eigenstates and decay
amplitudes. We classify the physical kaon eigenstates by their lifetime, and define
K S = K 1 + ¯ɛK
√ 2
= 1 (1 + ¯ɛ)K 0 − (1 − ¯ɛ)
√ ¯K 0
√ ,
1 + |¯ɛ|
2 2 1 + |¯ɛ|
2
K L = K 2 + ¯ɛK
√ 1
= 1 (1 + ¯ɛ)K 0 + (1 − ¯ɛ)
√ ¯K 0
√ , (29)
1 + |¯ɛ|
2 2 1 + |¯ɛ|
2
where ¯ɛ is a (convention-dependent) small complex parameter.
22

• The experimentally measured mass difference amounts to
∆M K = M(K L ) − M(K S ) ∼ 3.5 · 10 −15 GeV
and the width difference in the kaon system satisfies to a good approximation
Theoretical Description
∆Γ K ≈ −2∆M K .
The masses and decay widths of the neutral kaons describe the time-evolution of the
K 0 − ¯K 0 system in quantum mechanics,
i dψ
( ) |K
dt = Ĥ ψ(t) , ψ(t) ∼ 0 (t)〉
| ¯K 0 (30)
(t)〉
where the Hamilton operator contains a hermitian and an anti-hermitian part, related
to virtual and real intermediate states, respectively,
Ĥ = ˆM − i ˆΓ
2
(31)
Here ˆM and ˆΓ are hermitian matrices,
M 12 = M12, ∗ Γ 21 = Γ ∗ 12 .
The CPT theorem further relates particle and anti-particle properties, such that
M 11 = M 22 ≡ M , Γ 11 = Γ 22 ≡ Γ ,
and the Hamiltonian can be written as
( )
M − iΓ/2 M12 − iΓ
Ĥ =
12 /2
M12 ∗ − iΓ ∗ . (32)
12/2 M − iΓ/2
The eigenvalues of the matrix yield the mass and width differences:
∆M = 2 Re Q , ∆Γ = −4 Im Q ,
with Q = √ (M 12 − iΓ 12 /2)(M ∗ 12 − iΓ ∗ 12/2) . (33)
23

The eigenvectors determine ¯ɛ,
√
1 − ¯ɛ
1 + ¯ɛ = M12 ∗ − iΓ ∗ 12/2
M 12 − iΓ 12 /2
≡ r exp(iκ) . (34)
The deviation of the parameter r from unity (or ¯ɛ from zero) is a measure for CP violation
in kaon mixing,
r = 1 +
2|Γ 12 | 2
4|M 12 | 2 + |Γ 12 | 2 Im M 12
Γ 12
≈ 1 + Im M 12
Γ 12
. (35)
CP asymmetry in semi-leptonic kaon decays
We consider the CP asymmetry for semi-leptonic K L decays into π ± ,
a SL (K L ) = Γ(K L → π − e + ν e ) − Γ(K L → π + e −¯ν e )
Γ(K L → π − e + ν e ) − Γ(K L → π + e −¯ν e )
(36)
Realizing that K 0 ( ¯K 0 ) can only decay into π − (π + ), we obtain
Γ(K L → π − e + ν e ) ∝ ∣ ∣ (1 + ¯ɛ) A(K 0 → π − e + ν e ) ∣ ∣ 2 ,
Γ(K L → π + e −¯ν e ) ∝ ∣ ∣ (1 − ¯ɛ) A( ¯K0 → π + e −¯ν e ) ∣ ∣ 2 (37)
Since the strong interactions are CP-conserving (and there are no interference effects),
the absolute values for the amplitudes cancel in the ratio, and one obtains
a SL (K L ) = |1 + ¯ɛ|2 − |1 − ¯ɛ| 2 2Re ¯ɛ
=
|1 + ¯ɛ| 2 + |1 − ¯ɛ|
2
1 + |¯ɛ| = 1 − r2
2 1 + r 2
≈ 2Re ¯ɛ ≈ Im Γ 12
M 12
(38)
24

This kind of CP asymmetry only depends on the K 0 - ¯K 0 mixing parameters and is thus
dubbed
“CP-Violation in Mixing”
Experimentally one finds Re ¯ɛ ∼ 10 −3 .
Direct and indirect CP violation in kaon decays
CP violation can also manifest itself in the “wrong” decays K L → ππ (or similarly for
K S → 3π).
K L ∼ K 2 + ¯ɛK 1
↓
direct CP-viol.
↓
ππ
↓
indirect CP-viol.
↓
ππ
with “indirect CP violation“ from the small K 1 admixture in K L , and ”direct CP
violation” through the CP-violating effects in the weak decay K 2 → ππ itself.
• As a measure for the indirect CP violation, one defines the parameter
ɛ ≡ A(K L → (ππ) I=0 )
A(K S → (ππ) I=0 )
(39)
which compares the amplitudes for decays into the isospin-zero (I = 0) projection
of the two pions in the final state (the intereference effects with the I = 2 projection
will be relevant for the direct CP violation discussed below). The parameter ɛ can
be expressed as
ɛ = ¯ɛ + iξ ≈ exp(iπ/4) √
2 ∆MK
(Im M 12 + 2ξ Re M 12 ) (40)
25

where the parameter ξ = Im A 0 /Re A 0 measures the weak phase of the I = 0 decay
amplitude A 0 as defined below. Notice that ɛ is phase-convention independent, and
Re ɛ = Re ¯ɛ.
The decay K → (ππ) I=0 proceeds dominantly via s → udū and thus receives a CKM
factor V us Vud ∗ which is real in the standard parametrization. Therefore, ξ ≈ 0, and the
parameter ɛ is basically sensitive to Im M 12 .
• For direct CP-violation, we need the interference of two independent decay amplitudes.
In K → ππ decays these are given by the two alternative isospin projections,
3 √
3
A(K + → π + π 0 ) =
2 A 2 e iδ 2
,
A(K 0 → π + π − ) =
√
2
3 A 0 e iδ 0
+
√
1
3 A 2 e iδ 2
,
A(K 0 → π 0 π 0 ) =
√
2
3 A 0 e iδ 0
− 2
√
1
3 A 2 e iδ 2
. (41)
The pre-factors are obtained from the SU(2) I Clebsch-Gordan coefficients, and
δ 0,2 denote the so-called “strong phases” (which arise as a consequence of nonfactorizable
QCD rescattering effects between the final-state quarks). The weak
phases are still contained in the two independent amplitudes A 0 and A 2 , such that
the CP-conjugate decays will proceed via A ∗ 0,2 (while δ 0,2 will not change sign).
As a measure for indirect CP violation one then considers a particular combination
of decay amplitudes where the contribution from the indirect CP violation drops
out,
ɛ ′ ≡ √ 1 ( A(KL → (ππ) I=2 )
2 A(K S → (ππ) I=0 ) − A(K S → (ππ) I=2 )
A(K S → (ππ) I=0 )
)
A(K L → (ππ) I=0 )
A(K S → (ππ) I=0 )
= 1 √
2
Im A 2
A 0
exp(iΦ ɛ ′) , with Φ ɛ ′ = π 2 + δ 2 − δ 0 (42)
– Im A 2
A 0
comes from the different CKM factors of the operators in H eff contributing
to the I = 0 or I = 2 final state.
– The phase difference δ 2 − δ 0 comes from strong-interaction effects, and can be
independently measured in ππ scattering, leading to Φ ɛ ′ ≈ π/4.
3 The combination of two isospin-1 pion triplets yields 3 × 3 = 1 + 3 + 5, refering to I = 0, 1, 2.
However, the I = 1 projection does not contribute because of Bose symmetry.
26

• From the experimental decay widths, one can then extract the value of Re (ɛ ′ /ɛ)
from the ratios
leading to
η 00 = A(K L → π 0 π 0 )
A(K S → π 0 π 0 ) , η +− = A(K L → π + π − )
A(K S → π + π − )
( )
∣ ɛ
′
Re ≈ 1 −
η 00 ∣∣∣
2
ɛ ∣η +−
(43)
ɛ from H eff
The value of ɛ can be related to the matrix element of the weak effective Hamiltonian
for kaon mixing,
2m K M ∗ 12 = 〈 ¯K 0 |H eff (∆S = 2)|K 0 〉 , (44)
where the factor 2m K comes from the normalization of the hadronic states.
• In the case of K 0 - ¯K 0 mixing, we have to take into account the specific CKM
hierarchy for s → d transitions:
– The top contribution yields a CKM factor (V td V ∗
ts) 2 ∼ λ 10 .
– Although the charm contribution is suppressed by factors of m 2 c/m 2 t , this is
compensated by a much larger CKM factor, (V cd V ∗
cs) 2 ∼ λ 2 .
As a consequence, also the box diagrams with 2 charm-quark propagators or with
one charm-quark and one top-quar propagator will be important.
This effect also contributes to the relative phase between M 12 and Γ 12 .
• The hadronic matrix element M 12 can then be expressed in terms of function
S 0 arising from the calculation of the box diagrams with different masses x c,t =
m 2 c,t/MW 2 Ṁ 12 = G2 F
12π 2 F 2 K ˆB K m K M 2 W
× [ (λ ∗ c) 2 η 1 S 0 (x c ) + (λ ∗ t ) 2 η 2 S 0 (x t ) + 2λ ∗ cλ ∗ t η 3 S 0 (x c , x t ) ] (45)
27

Figure 1: Determination of ¯ρ and ¯η from ɛ K (green band), ∆m d,s (yellow/orange) and
sin 2β (blue).
Here η i are short-distance QCD corrections, and F 2 K ˆB K parametrizes the relevant
hadronic matrix element
〈 ¯K 0 |(¯sd) (V −A) (¯sd) (V −A) |K 0 〉 ≡ 8 3 m K F 2 K ˆB K (1 + QCD corr.) (46)
where F K is the kaon decay constant, and the normalization factors are chosen in
such a way that in the naive factorization approximation, 〈 ¯K 0 |(¯sd) (V −A) (¯sd) (V −A) |K 0 〉 ≈
〈 ¯K 0 |J|0〉〈0|J|K 0 〉, the quantity ˆB K is unity,
ˆB K = 1 + non-factorizable corrections (47)
• The actual numerical estimate for ˆB K ≈ 80% is obtained from non-perturbative
methods. The theoretical formula for M 12 can then be turned into a theoretical
prediction for ɛ. Comparison with the experimental results provides a constraint on
the CP-phase δ entering the CKM element V td (in the standard parametrization).
The constraint takes the form of a hyperbola in the ¯ρ-¯η plane for the standard
CKM triangle with an error band dominated by the uncertainties on ˆB K , see the
green band in Fig. 1.
28

• Finally, the experimental measurement for direct CP violation in kaon decays yields
( ) ɛ
′
Re ∼ 2 · 10 −3 ,
ɛ
i.e. the effect is further suppressed by 3 orders of magnitude compared to the
indirect CP violation. A theoretical prediction requires the quantitive knowledge
of the hadronic matrix elements responsible for the I = 0 and I = 2 amplitudes,
and takes the generic form
ɛ ′
ɛ ∼ Im λ t [〈· · · 〉 I=0 − 〈· · · 〉 I=2 ]
Due to the cancellation effects occuring in the brackets, the theoretical predictions
are thus rather uncertain.
Summary of the different types of CP-violation observed in the neutral kaon system:
• Re ɛ = Re ¯ɛ: CP violation in mixing
• Im ɛ: CP-violation in the interference between mixing ang decay
• Re ɛ ′ : direct CP violation in decay, where
Re ɛ ′ = − 1 √
2
∣ ∣∣∣ A 2
A 0
∣ ∣∣∣
sin(φ 2 − φ 0 ) sin(δ 2 − δ 0 ) (48)
i.e. direct CP violation requires a non-vanishing difference φ 2 − φ 0 ≠ 0 of weak
phases and a non-vanishing difference δ 2 − δ 0 ≠ 0 of strong phases in the I = 0
and I = 2 amplitudes.
Questions/Exercises:
• Verify the eigenvalues and eigenvectors for the matrix Ĥ in (32).
• What is the difference between direct and indirect CP violation in kaon
decays
• Why do we have to take into account the box diagrams with charm quarks
for kaon mixing
Further Reading:
• [2, chapter 3.1-3.4]; [7]
29

4.2 B 0 - ¯B 0 Mixing
The description of mixing between neutral B d or B s mesons is similar to the formalism
for kaon mixing. However, there are certain differences related to the different size of
the relevant CKM element and the different hadronic decay channels. Especially, in the
B 0 - ¯B 0 system, one has
∆Γ ≪ ∆M
and the eigenstates will be classified by masses (heavy and light) rather than lifetimes,
B H = pB 0 + q ¯B 0 ,
B L = pB 0 − q ¯B 0 , (49)
where we have introduced a new convenient parametrisation in terms of two complex
parameters p, q which can be related to the ¯ɛ notation via
p, q =
1 ± ¯ɛ B
√
2(1 + |¯ɛB | 2 ) .
As a consequence of Γ 12 ≪ M 12 in the B q -system (q = d, s), we can approximate
and
∆M q = M Bq
H
∆Γ q = Γ Bq
H
− M Bq
L
− ΓBq L
(q)
≃ 2|M 12 | ,
(q)
Re (M 12 Γ (q)∗
12 )
≃ 2
|M (q)
12 |
(50)
q
p ≃ M (
12
∗ 1 − 1 |M 12 | 2 Im Γ )
12
≈ −arg M 12 . (51)
M 12
As a consequence,
• the semi-leptonic asymmetry is tiny, a SL (B) ≈ 10 −4 , and therefore it is difficult
measure this observable precisely and to quantify CP violation in B-meson mixing.
• The ratio q/p is to a very good approximation a pure phase, i.e. |q/p| ≃ 1.
Since, in the B-system, the top-quark always dominates the box diagrams, the
weak phases are obtained from
30

(M ∗ 12) (d) ∝ (V td V ∗
tb) 2 ,
(M ∗ 12) (s) ∝ (V ts V ∗
tb) 2 (52)
such that
( q
p)
d,s
≃ e 2iφd,s M
with φ d M = −β , φ s M = −β s ≈ 0 (53)
where β is one of the angles in the standard unitarity triangle, and β s the according
tiny angle in the (squashed) triangle for b → s transitions.
• ∆M d and ∆M s are directly proportional to the relevant CKM elements |V td | 2 and
|V ts | 2 , respectively. The main uncertainties again stem from the hadronic matrix
element of the operator in the ∆B = 2 weak effective Hamiltonian, which are
parametrized analagously as for the kaon case by f 2 B q
ˆBq .
– ∆M d alone restricts the length of the right side of the UT, |V tb V ts /V cb V cs .
Comparison with experiment gives the yellow band in Fig. 1, with a large
uncertainty from f 2 B ˆB d .
– A sizeable part of the uncertainties drops out in the ratio ∆M d /∆M s which
again basically constrains the same side of the UT, since |V td /V ts | ≈ |V td /V cb |.
Comparison with experiment yields the (narrower) orange band in Fig. 1.
From Fig. 1, we see that ∆M d /∆M s and ɛ K alone are not sufficient to precisely determine
¯ρ and ¯η.
Time-dependent CP asymmetries at B-factories
The new idea is to exploit the fact that the B d mesons have a relatively large life-time. In
so-called asymmetric “B-factories”, one produces B- ¯B mesons with a large Lorentz
boost which allows to identify the spatial displacement between the production and decay
vertex of the B-mesons. This can be turned into information of the time-evolution of (a
statistical sample of) B-mesons. The idea of asymmetric B-factories is illustrated below:
31

• e + e − collisions at the Υ(4S) resonance, M 2 e + e − ≃ M 2 Υ(4S) .
• Υ(4S) almost exclusively decays coherently into B- ¯B mesons.
• The decay of one of the mesons into a flavour eigenstate can be used to “tag” the
charge of the b-quark at a time τ 1 defined by the corresponding displacement of
the vertex.
• Exploiting the EPR paradoxon, this fixes the charge of the other b-quark at that
time to be opposite.
• Measuring the decay vertex of the other B-meson, we can thus test the quantum
mechanical time evolution,
|B meas. (τ 2 )〉 = U(τ 2 , τ 1 ) |B meas (τ 1 )〉
In particular, we consider decays into CP-eigenstates |f〉, and define the time-dependent
CP asymmetries for t = (τ 2 − τ 1 ) as
a CP (t, f) = Γ(B0 (t) → f) − Γ( ¯B 0 (t) → f)
Γ(B 0 (t) → f) + Γ( ¯B 0 (t) → f)
(54)
Now, the time-evolution of the flavour eigenstates follows from the evolution of the mass
eigentstates,
32

|B 0 (t)〉 = f + (t) |B 0 〉 + q p f −(t) | ¯B 0 〉 ,
| ¯B 0 (t)〉 = p q f −(t) |B 0 〉 + f + (t) | ¯B 0 〉 , (55)
with
f ± (t) = 1 2
(
e
−iM H t ± e −iM Lt ) . (56)
With this, the time-dependent CP asymmetry takes a relatively simple form
a CP (t, f) = A decay
CP
(B → f) cos(∆Mt) + Ainterf. CP (B → f) sin(∆Mt) (57)
If we define
ξ f ≡ q p
A( ¯B 0 → f)
A(B 0 → f) ≃ A( ¯B 0 → f)
e2iφ M
A(B 0 → f)
(58)
one obtains
A decay
CP
= 1 − |ξ f| 2
1 + |ξ f | , 2 Ainterf CP = 2Im ξ f
(59)
1 + |ξ f | 2
The Golden Decay Mode B → J/ψK s
The expression are particularly simple if we consider decay modes which are dominated
by decay amplitudes with a single CKM factor. The well-known example is the decay
B 0 / ¯B 0 → J/ψK S which is dominated by the operator inducing b → c¯cs transitions,
involving V cb Vcs ∗ (being real in the standard parametrisation). As a consequence, for
such decays we have
A( ¯B 0 → f)
A(B 0 → f) = −η f e −2iφ decay
where η f = ±1 refers to the CP-parity of the final state. From this, we have |ξ f | = 1,
and therefore A decay
CP
= 0, while the interference term simply yields
33

a interf
CP (t, f) = η f sin 2β sin(∆Mt) (60)
We thus have a simple theoretical prediction for a time-dependent oscillation, with a
frequency set by the mass difference, and an amplitude yielding a direct measure for one
of the CKM angles. Fitting the theoretical prediction to the experimental measurments
(like the following one from the BaBar experiment),
one obtains a precise value
sin 2β = 0.681 ± 0.025
which translates into the blue stripe in Fig. 1. Together with ɛ K and ∆M d /∆M s this
already gives a satisfactory determination of ¯ρ and ¯η.
Comment on sub-leading decay topologies in B → JψK
• The main decay mechanism is through the current-current operator for b → c¯cs, yielding
a “tree” amplitude
T c V cb V ∗
cs ∼ T c λ 2 .
• The final state can also be reached via penguin operators for b → q¯qs (including q = c)
P t V tb V ∗
ts ∼ P t λ 2 .
As the penguin operators have smaller Wilson coefficients, we have P t ≪ T c .
• Finally, we also encounter current-current operators for b → uūs transitions,
T u V ub V ∗ us ∼ T u λ 4 .
As the uū quarks have to be turned into c¯c quarks by strong interactions, T u ≪ T c
34

In total, we thus have
A tot = T c V cb V ∗
cs + P t V tb V ∗
ts + T u V ub V ∗
us = (T c − P t ) V cb V ∗
cs + (T u − P t )V ub V ∗
us
≃ (T c − P t ) V cb V ∗
cs # (61)
where the neglected term is doubly suppressed by a relative factor λ 2 and by (T u −P t )/T c .
Questions/Exercises:
• Derive the expressions for A decay,interf.
CP
in (59).
Further Reading:
• [2, chapter 3.4-3.7,4.1,4.4]
35

5 Semi-Leptonic Decays
Semi-leptonic decay amplitudes q → q ′ lν are proportional to V qq ′. From semi-leptonic
B-meson decays, we can extract |V cb | and |V ub |.
(a) exclusive B → π(ρ)lν yields |V ub |; exclusive B → D(D ∗ )lν yields |V cb |
(b) inclusive B → X u lν yields |V ub |, inclusive B → X c lν yields |V cb |
(c) exclusive B + → τ + ν τ yields |V ub |; (exclusive B c → τ + ν τ yields |V cb |)
5.1 |V cb | from exclusive decays and HQET
In order to determine |V cb | we need, for instance,
〈D ∗ lν|Heff
b→clν |B〉 = G F
√ 〈l¯ν|¯lγ µ (1 − γ 5 )ν|0〉〈D ∗ |¯cγ µ (1 − γ 5 )b|B〉 (62)
2
where the leptonic matrix element is straight-forward, while the hadronic matrix element
involves a B → D ∗ transition form factor which parametrizes the non-perturbative QCD
dynamics. The idea is to exploit the hierarchy Λ QCD ≪ m b,c . In the limit m b,c → ∞,
the heavy quarks in the transitions act as “static sources of a colour field”, and one
expects analogous simplifications as for the transitions between hydrogen-like atoms in
non-relativistic quantum mechanics.
Heavy-Quark Effective Theory (HQET)
Within relativistic quantum field theory, we can formalize the heavy quark limit by
constructing an appropriate effective theory. In contrast to the case of the effective weak
Hamiltonian, the heavy quark field is not completely removed from the theory. To this
end, one first performs a field redefinition for (Q = b, c)
Q(x) = e −im Q v·x (h v (x) + H v (x)) (63)
with
h v = e im Q v·x 1 + /v
2
H v = e im Q v·x 1 − /v
2
Q(x) ,
Q(x) (64)
Here, one has factored out the plane-wave behaviour corresponding to the on-shell motion
of a heavy quark (i.e. the static limit in the heavy quark’s rest frame). Furthermore, v µ
36

efers to the 4-velocity of the heavy quark (v 2 = 1), and (1±/v)/2 are projectors in Dirac
space.
Inserting the new parametrization into the QCD Dirac-Lagrangian, and using the projection
properties, one obtains
L = (¯hv + ¯H v
)
e
im Q v·x (i /D − m Q ) e −im Qv·x (h v + H v )
= ¯h v (iv · D)h v − ¯H v (iv · D + 2m Q )H v + ¯h v i /D ⊥ H v + ¯H v i /D ⊥ h v . (65)
where /D ⊥ = /D − /v (v · D). Now, only the kinetic term for the projection H v contains
the heavy-quark mass m Q , and we can thus eliminate this field by making use of the
classical equations of motion (which corresponds to tree-level matching of the effective
theory),
∂L
∂ ¯H v
= −(iv · D + 2m Q )H v + i /D ⊥ h v = 0
⇒H v =
1
iv · D + 2m Q
i /D ⊥ h v ∼ O
(
ΛQCD
)
h v (66)
m Q
where the power counting is obtained under the assumption that the residual modulation
of the heavy-quark field with respect to the plane-wave factor is generated by interactions
with soft gluons, having momenta/field strength of order Λ QCD . Therefore
H v : “small component” of the heavy-quark Dirac spinor,
h v : “large component” of the heavy-quark Dirac spinor.
If we now expand the non-local term 1/(iv·D+2m Q ) in powers of Λ QCD /m Q , we obtain an
effective theory where the field H v does not propagate anymore. As in the example of the
weak effective Hamiltonian, this will change the UV properties of Feynman integrals, and
the resulting Lagrangian corresponds to an effective theory for heavy quarks interacting
with soft gluons and light quarks.
“Heavy Quark Effective Theory” (HQET)
The quantum corrections from hard momenta (of order m b ), again, will be absorbed into
Wilson coefficients of HQET operators,
Let us look at the leading-order terms (at tree-level) by inserting H v ≃ i /D ⊥
2m Q
h v ,
37

L HQET = ¯h v (iv · D) h v + ¯h v i /D ⊥
1
2m Q
i /D ⊥ h v + . . . (67)
• The leading term in this expansion
– is independent of the heavy quark mass
⇒ (approximate) Heavy Quark Flavour Symmetry
– is independent of heavy quark spin orientation (no magnetic interaction with
gluon field)
⇒ (approximate) Heavy Quark Spin Symmetry
• The O(1/m Q ) term can be rewritten, using
leading to
/D ⊥ /D ⊥ = (D ⊥ ) 2 + 1 4 [γ µ, γ ν ] [D µ ⊥ , Dν ⊥] = (D ⊥ ) 2 + g 2 σ µνG µν
L HQET
∣
∣1/mQ = ¯h v
{ (iD⊥ ) 2
2m Q
− g
4m Q
σ µν G µν }
h v (68)
– The first term represents the (generalization of the) non-relatistic kinetic energy.
It breaks the HQET flavour symmetry. Its Wilson coefficient does not
receive radiative corrections because it is protected by Lorentz symmetry.
– The second term represents the (generalization of the) non-relatistic Pauli
term. It breaks the HQET flavour and spin symmetries. Its Wilson coefficient
receives radiative corrections, and in the leading-log-approximation, it is given
by
C mag (µ) ∣ ( ) 9/(33−2nf )
αs (m b )
LLA
=
α s (µ)
38

Consequences for the the Hadron Spectrum
Consider pseudoscalar or vector mesons containing one heavy quark
H ≡ (Q¯q) spin−0 ,
H ∗ ≡ (Q¯q) spin−1
In the heavy-mass limit, they would correspond to a fundamental doublet of the heavyquark
spin symmetry. The masses of the meson can thus be written as an expansion in
1/m Q :
m H = m Q + ¯Λ − λ 1
2m Q
− 3λ 2
2m Q
+ . . .
m H ∗ = m Q + ¯Λ − λ 1
2m Q
+ λ 2
2m Q
+ . . . (69)
• m Q comes from the heavy-quark mass itself,
• ¯Λ parametrizes the expectation value of the leading HQET operator ¯h v (ivD)h v ,
• λ 1 parametrizes the 1/m Q correction from the kinetic energy operator,
• λ 2 parametrizes the 1/m Q correction from the magnetic term, where the different
coefficients for pseudoscalar and vector mesons are given by Clebsch-Gordan
coefficients of SU(2).
As a consequence, one obtains simple approximate relations, like
or
m 2 H ∗ − m2 H = const.
m 2 B ∗ − m2 B ≃ 0.49 GeV 2 ,
m 2 D ∗ − m2 D ≃ 0.55 GeV 2 , (70)
m Hs − m Hq
= ¯Λ s − ¯Λ q = const.
Feynman rules in HQET
m Bs − m B ≃ m Ds − m D ≃ 100 MeV . (71)
From the leading HQET Lagrangian, ¯h v (ivD)h v , one reads off the Feynman rules for
• the heavy-quark propagator:
i
v·k+iɛ
• heavy-quark gluon vertex: ig s v µ t A 39

Alternatively, we can obtain the HQET Feynman rules from the corresponding QCD
Feynman rules. We decompose the heavy quark momentum as
p µ = m Q v µ + k µ
where k µ is the “residual momentum” of the heavy quark. Then the heavy-quark
propagator can be approximated as
/p + m Q
p 2 − m 2 Q + iɛ = m Q(1 + /v) + /k
2m Q v · k + k 2 + iɛ ≃ 1 1 + /v
v · k + iɛ 2
(72)
which reproduces the HQET propagator times the heavy-quark projector (which is implicit
in HQET).
The heavy-quark gluon vertex is reproduced by realizing that the vertex always appears
between heavy quark propagator or on-shell heavy quark spinors, which both provide a
projector 1+/v
. One then has
2
1 + /v
2
ig s γ µ t a 1 + /v
2
= ig s v µ t a 1 + /v
2
. (73)
• The HQET Feynman rules thus reproduce the soft region of Feynman integrals,
where the residual momentum of the heavy quark k µ is small compared to its mass.
• The hard momentum region are accounted for by Wilson coefficients for the subleading
operators in the HQET Lagrangian and external decay currents (see below).
Heavy-Quark Currents
In terms of full QCD fields, the semi-leptonic b → clν transitions are induced by the
currents
J QCD = ¯c(x) γ µ (1 − γ 5 ) b(x) ≡ ¯cΓb
At leading-order in the HQET expansion, this can be replaced by
J QCD ≃ e −imbv·x imcv′·x
e ¯h(c)
v
(x)Γh (b)
′ v (x) ≡ e −imbv·x e imcv′·x J HQET (74)
Similarly, we may replace the hadronic states by the corresponding states in the heavymass
limit,
|B(p)〉 → √ m B |B(v)〉 etc.
where we have factored out the mass-dependent normalization for convenience.
40

To derive the consequences of the heavy quark symmeties, we first study the representation
of heavy-meson states under Lorentz transformations
with
M (Q)
v
(Q α¯q β ) ∼ ( )
M (Q) αβ
v
→ D(Λ)M (Q)
v D(Λ) −1
under Lorentz transformations, where D(Λ) determines the transformations of the fundamental
Dirac spinors.
• As a consequence of the HQET spin symmetry, we can combine pseudoscalar and
vector meson states into a common (momentum-space) representation matrix for
the spin doublet,
M (Q)
v = 1 + /v [ɛ/(v, λ) a V (v, λ) + iγ 5 a P (v, λ)] (75)
2
which includes the HQ-projector, the polarization vector ɛ for a vector meson, as
well as the annihilation operators for vector or pseudoscalar meson states. Notice
that M (Q)
v /v = −M (Q)
v .
• In terms of this representation matrix, the hadronic matrix element of the decay
current can be written as
〈D(v ′ )|¯h (c)
v
Γh (b)
′ v
[
|B(v)〉 ∝ tr Ξ(v, v ′ )
]
(c) ¯M
v
ΓM (b)
′ v
(76)
Here the Dirac matrix Ξ(v, v ′ ) parametrizes the (unspecified) bound-state dynamics
of the light degrees of freedom, whereas the remaining terms are dictated by the
HQET and Lorentz symmetries.
• Now, Ξ can only be a function of v and v ′ , and due to the heavy-quark projection
properties and the parity conservation of strong interactions, all allowed Dirac
structures can be reduced to the unit matrix,
Ξ(v, v ′ ) ∼ ξ(v · v ′ ) 1
With this, all transition matrix element between pseudoscalar and vector B or D-mesons,
can be expressed by a unique “Isgur-Wise“ Form Factor ξ(ω = v · v ′ ),
41

〈D(v ′ )|¯c v ′γ µ b v |B(v)〉 = ξ(ω) ( v µ + v ′ µ)
,
〈D ∗ (v ′ , ɛ)|¯c v ′γ µ γ 5 b v |B(v)〉 = −iξ(ω) ( (1 + ω)ɛ ∗ µ − (v · ɛ ∗ )v ′ µ)
,
〈D ∗ (v ′ , ɛ)|¯c v ′γ µ b v |B(v)〉 = ξ(ω) ε µναβ ɛ ∗ν v ′α v β .
We may also consider the case where Q = Q ′ = b and v = v ′ (”Isgur-Wise limit“, ω = 1).
This corresponds to the matrix element of a conserved current,
〈B(v)|¯h (b)
v γ µ h (b)
v |B(v)〉 = 2ξ(ω = 1) v µ
≃ 〈B(p)|¯bγ µ b|B(p)〉/m B = 2F (q 2 = 0)p µ /m B (77)
such that
ξ(1) = F (0) = 1
In semi-leptonic decays, the Isgur-Wise limit ω = 1 corresponds to the maximal momentum
transfer to the lepton pair,
q 2 = (p B − p D ) 2 = (m B − m D ) 2 = q 2 max .
Then, the CKM element |V cb | can be extracted from
|V cb | ≃ |F (q2 → q 2 max) V cb | exp.
|F (v · v ′ = 1)| theor.
(78)
where
|F (v · v ′ = 1)| theor. = 1 + O(1/m Q ) + O(α s ) (79)
In particular, it turns out that for B → D ∗ decays, there are no 1/m Q corrections at
ω = 1 (”Luke’s theorem“), i.e. the power corrections start at O(1/m b · 1/m c ). The
relevant form factor including the corrections is then estimated as
F D ∗(ω = 1) = 0.91 ± 0.05
42

which means a 10% deviation from unity, with a relative hadronic uncertainty of 50%,
implying a 5% theoretical uncertainty on the extraction of |V cb |.
Questions/Exercises:
• Prove that (1 ± /v)/2 are projectors in Dirac space. Verify that in the heavyquark’s
rest frame, the projections h v and H v correspond to the particle and
anti-particle components of the Dirac spinor.
• Calculate
1 ± /v
2
γ µ
1 ± /v
2
and 1 ∓ /v
2
γ µ
1 ± /v
2
• Verify the derivation of the HQET Lagrangian and derive the Feynman rules
for the HQET propagator, the leading heavy-quark gluon vertex, and the
sub-leading quark-gluon vertices from the kinetic and chromomagnetic operator.
• Verify (77).
Further Reading:
• [8, chapter 2.5, 2.6, 2.9, 3.4, 4.3]
.
5.2 |V cb | from inclusive decays
In the inclusive decay B → X c l¯ν, we do not specify the final state completely, except
for demanding open charm in the final state. The situation is somewhat similar to
deep inelastic scattering: Instead of elastic parton-electron scattering, the partonic subprocess
is given by the weak b → cl¯ν decay.
The differential decay rate can be written in terms of a ”hadronic tensor W µν “ and a
”leptonic tensor L µν “,
d 3 Γ
dq 2 dE l dE ν
= G2 F
4π 3 |V cb| 2 W µν L µν (80)
with q µ L µν = q ν L µν = 0 for m l = 0 (i.e. for electrons and muons).
• The computation of L µν is straightforward
L µν = 1 2 tr [/p νγ µ /p l γ ν (1 − γ 5 )]
• The hadronic tensor contains the hadronic matrix element and phase-space factors
43

W µν = (2π)3
2m B
∑ ∫
X c
δ (4) (p B − q − p X )
× 〈 ¯B|¯bγ µ P L c|X c 〉〈X c |¯cγ ν P L b| ¯B〉 (81)
and it can be decomposed into the most general Lorentz structures
W µν ≡ −g µν W 1 + v µ v ν W 2 + iɛ µναβ v α q β W 3
+ q µ q ν W 4 + (v µ q ν + q µ v ν )W 5 (82)
The W i are real function of q 2 and v · q. W 4,5 do not contribute when contracted
with L µν (for m l = 0).
• The aim is now to calculate the W i in the heavy-mass limit in terms of HQET parameters
and α s (m b ). The general procedure is to make use of the optical theorem
to relate the decay rate to the imaginary part of the forward-scattering amplitude,
where
W i = − 1 π Im T i
T µν =
−i ∫
2m B
d 4 x e −iq·x 〈 ¯B|T [ J † µ(x)J ν (0) ] | ¯B〉 (83)
Now, the time-ordered product of the two currents can be computed in perturbation
theory, by contracting the quark fields to propagators. Formally, this amounts to
an ”Operator Product Expandion“ (OPE), where, in momentum space, the
product of two operators at different space-time points is expanded into a series of
local operators with the corresponding Wilson coefficients.
∫
−i
d 4 x e −iqx T [ J † µ(x)J ν (0) ] ≡ ∑ i
C i (q) O i (84)
The operator with the lowest dimension, contributing to the imaginary part of T µν
has the form ¯bΓb and gets a tree-level contribution from contracting the charmquark
fields to a charm propagator. The Wilson coefficient is obtained from match-
44

ing the expression
ū(p b )γ µ P L
/p b − q/
(p b − q) 2 − m 2 c + iɛ γ νP L u(p b )
From this one immediately sees that Im T i corresponds to an on-shell charm quark,
(p b − q) 2 − m 2 c = 0, in other words at leading order one has
LO:
dΓ(B → X c lν) ∼ dΓ(b → clν)
i.e. the hadronic decay rate can be approximated by the partonic one. The matrix
elements of the local operators, 〈 ¯B|¯bΓb| ¯B〉, are simply determined by the trivial
probability to find a b-quark in the B-meson, and do not introduce a new hadronic
parameter (at this order), apart from the b-quark mass (see below).
Working out the Dirac and Lorentz algebra, the final result for the LO differential
rate then reads
d 3 Γ
dE e dE ν dq = G2 F |V [
] (
cb| 2
− q2 E ν
+ E 2 2π 3 e E ν δ E e + E ν − q2 + m 2 b − )
m2 c
2m b 2m b
(85)
• For instance, we can study the electron-energy spectrum by integrating over E ν and
q 2 . Introducing the dimensionless variables y = 2E e /m b and ρ = m 2 c/m 2 b ∼ 0.08,
one obtains
dΓ
dy = G2 F |V [
]
cb| 2 m 5 b
3y 2 − 2y 3 − 3y 2 ρ − 3y2 ρ 2
96π 3 (1 − y) + (3y2 − y 3 )ρ 3
θ(1 − ρ − y)
2 (1 − y) 3
(86)
45

– Notice the strong dependence of the decay width on the b-quark mass. Remember
that m b is a scheme- and scale-dependent quantity, which means that
the related uncertainties can only be resolved by calculating the higher orders
in perturbation theory.
• The experimental procedure to extract a value for |V cb | from the comparison of
experimental data and theoretical predictions goes as follows:
– Compute the α s corrections to the partonic rate.
– Include higher orders in the OPE, with operators of the form
¯bΓ(iD⊥ ) · · · (ivD) · · · (iD ⊥ )b
which can be related to HQET parameters ¯Λ, λ 1 , λ 2 etc.
– measure moments 〈y n 〉 with n = 0..3, of the electron-energy spectrum
– compare with the theoretical formula and simultaneously fit the HQET parameters
and |V cb |.
– Notice that the final result will depend on the renormalization scheme.
• The result of such analysis on average leads to [9]
|V cb | incl. = (41.5 ± 0.7) · 10 −3 (87)
which should be compared with the number obtained from exclusive B → D/D ∗ lν
decays
|V cb | incl. = (38.7 ± 1.1) · 10 −3 (88)
The slight discrepancy between the two numbers can be taken as a statistical fluctuation
in the data, an underestimate of perturbative and non-perturbative theoretical
uncertainties, or even an effect of new-physics contributions (for instance,
from right-handed weak currents).
Questions/Exercises:
• Calculate the (normalized) moments 〈y n 〉/〈y 0 〉 (n = 1..3) of the electronenergy
spectrum, approximating ρ = 0.
46

Further Reading:
• [8, chapter 6.1-6.2]; [9]
5.3 |V ub | from inclusive decays and SCET
• The determination of |V ub | from exclusive decay modes like B → π(ρ)lν heavily
relies on non-perturbative methods to estimate the form factors. The PDG quotes
with an uncertainty 10 − 15%.
|V ub | excl. = 3.38 · 10 −3
• From the theoretical point of view, the determination from hte inclusive decays
¯B → X u l¯ν seems more favourable, as the same theoretical framework (OPE) as
for B → X c lν can be used.
• However, because |V cb | 2 ≫ |V ub | 2 , the b → clν decays actually represent a large
background for the small b → ulν signal.
• The usual experimental strategy is to introduce kinematic cuts, e.g.
M 2 X < M 2 X c,min ≃ M 2 D .
• Unfortunately, for such kinematic cuts, most of the remaining events for B → X u lν
correspond to the final state X u forming hadronic jets, i.e. the energy E X is of
order m b /2 while the invariant mass M 2 X ≪ m2 b .
(The particles forming the jet move almost collinear to each other – in the context of the effective
theory to be constructed, we sometimes call these particle modes ”hard-collinear“.)
m b v + soft
¯ν l
l
B
soft
hard-collinear (jet)
X u
To get further insight, let us consider the LO diagram for the functions T i again:
47

– The momentum of the (now up-quark) propagator in the forward-scattering
amplitude is given by
∆ µ = (m b v µ + k µ ) − q µ = (m b v µ + k µ ) − (m B v m u − p µ X ) = pµ X + (kµ − ¯Λv µ )
(89)
The virtuality of the propagator follows as (for m u ≃ 0)
∆ 2 = p 2 X + 2k · p X − 2¯Λv · p X + O(Λ 2 QCD) (90)
– The OPE case discussed before in the context of B → X c lν corresponds to
hadronic final states which are dominated by momenta satisfying p 2 X ∼ m2 b
(notice, that the larger |p X |, the larger the available phase space, and therefore events with
smaller p 2 X are less important)
In this case, we can approximate ∆ 2 ≃ p 2 X , and the partonic decay rate does
not depend on the residual momentum k µ which justifies the OPE in terms
of local operators.
– In contrast, the jet configuration corresponds to events, where
p 2 X ∼ m b Λ ≪ m 2 b
”hard-collinear“
where for the considered case, we can consider Λ ∼ Λ QCD . In this case, the
terms depending on the residual heavy-quark momentum in (90) cannot be
neglected anymore.
As a consequence, the decay rate will depend on the distribution of the b-
quark momentum k µ in the B-meson, which can be described by a parton
distribution function (as in DIS), dubbed ”Shape Function“ in the context
of inclusive B decays.
Phenomenology for |V ub | incl.
• One can construct models for the shape functions, implementing theory constraints
from the OPE. The variance in ”reasonable“ model functions gives an estimate for
the related systematic theoretical uncertainties.
48

• Alternatively, one can extrac the shape-function from other inclusive decays, noteably
the radiative decay B → X s γ, which is based on the partonic rate for b → sγ,
and the shape-function region corresponds to large photon energies.
The PDG quotes a value
|V ub | incl. = 4.27 ± 0.38 · 10 −3
which is somewhat higher than the the exclusive value.
The effect on the CKM triangle fit is via the ratio |V ub |/|V cb | which enters the length of
the left side of the triangle, see the dark green band in the figure below:
As one can see, the determination of |V ub /V cb | from semi-leptonic decays (in particular
from the inclusive rates) favour slightly larger values than the ones corresponding to the
overall CKM triangle fit.
Soft-collinear effective theory (SCET)
The theoretical description of the shape-function region can be formulated in terms of
an effective theory which simultaneously reproduces the dynamics of soft and (hard-
)collinear particle fields, ”soft-collinear effective theory“ (SCET).
Starting point is the decomposition of Lorentz vectors according to ”light-cone kinematics“,
introducing light-like vectors (given in the B-meson rest frame)
n µ ± = (1, 0, 0, ±1) , n + · n − = 2 , n 2 + = n 2 − = 0 (91)
which allow to decompose any vector as
p µ = (n + p) nµ −
2 + pµ ⊥ + (n −p) nµ +
2 , p2 = (n + p)(n − p) + p 2 ⊥ (92)
49

We distinguish:
• (hard-)collinear particles (in jets), with
|n + p X | ∼ m b ≫ |p X,⊥ | ∼ √ m b Λ ≫ |n − p X | ∼ Λ ,
p 2 X ∼ m b Λ , (93)
• soft particles (from B-meson), within
|n + k| ∼ |k ⊥ | ∼ |n − k| ∼ Λ , k 2 ∼ Λ 2 . (94)
The virtuality of the up-quark propagator in the leading diagram for the T i in b → ulν
is then given by
∆ 2 ≃ p 2 X + (n − k − ¯Λ)(n + p X )
which shows that we actually need the distribution of the light-cone component (n − k)
of the residual b-quark momentum in the B-meson, f (B)
b
(n − k), defining the relevant
shape-function (or pdf).
The effective theory again should reproduce the low-energy (or, as we will see, more
precisely low-virtuality) region(s) of loop integrals in perturbation theory, whereas the
hard momentum region will be absorbed into coefficients (more precisely, coefficient
functions). As an example, consider the 1-loop correction to the b → u decay vertex:
• In full QCD, the resulting Feynman integral has the structure:
∫
d D l
(2π) [numerator] 1
D l 2 + iɛ ·
1
(m b v + k − l) 2 − m 2 b + iɛ ·
1
(∆ − l) 2 + iɛ
(95)
where l is the loop-momentum carried by the gluon, and the result will be a nonanalytic
function of ∆ 2 /m 2 b and (v · k)/m b.
50

• The hard momentum region is obtained by expanding the integrand under the assumption
|l µ | ∼ m b ∼ (n + ∆), and putting all sub-leading momentum components
(∆ ⊥ , (n − ∆), k) to zero.
hard:
∫
× 1
l 2 + iɛ ·
d D l
(2π) D [numerator]
1
(m b v − l) 2 − m 2 b + iɛ ·
1
l 2 − (n + ∆)(n − l) + iɛ
(96)
As usual, we will absorb the contribution of the hard region into a Wilson coefficient
for the current operator in the effective theory. An important difference to the cases
discussed so far is, that the Wilson coefficient now depends on the large momentum
component (n + ∆) of the u-quark jet,
C i = C i (µ, m b , n + ∆) (97)
• The hard-collinear region corresponds to the case
which leads to
|n + l| ∼ m b ∼ |n + ∆| ≫ |l ⊥ | ∼ |∆ ⊥ | ≫ |n − l| ∼ |n − ∆| ∼ |k|
hard-collinear:
× 1
l 2 + iɛ ·
∫
d D l
(2π) D [numerator]
1
−m b (n + l) + iɛ ·
1
(∆ − l) 2 + iɛ
(98)
It should be reproduced by the Feynman rules in SCET (including the unconventional
factor 1/(n + l)). The result for the hard-collinear region is independent of
the heavy-quark residual momentum, and only depends on the virtuality ∆ 2 of the
u-quark jet. It thus represents a universal perturbative correction to the propagator
of the hard-collinear u-quark. In SCET, these effects are collected in the
so-called ”Jet Function“
51

(hard-collinear) Jet-Function: J = J(µ, ∆ 2 )
• Finally, the soft region of the integral corresponds to |l µ | ∼ |k µ ∼ Λ QCD , which
gives rise to
soft:
∫
d D l
(2π) D [numerator]
× 1
l 2 + iɛ ·
1
2v · (k − l) + iɛ ·
1
−(n + ∆)(n − l) + iɛ
(99)
which contains the familiar HQET propagator for a quasi-static b-quark, and again
an unconventional 1/(n − l) factor to be reproduced by the SCET Feynman rules.
The result for the soft integral depends only on the residual momentum k (apart
from a global kinematic 1/(n + ∆) factor) and thus represents an inherent property
of the B-meson which can be attributed to a non-trivial ”Shape Function“
(soft) Shape-Function: S = S(µ, n − k)
describing the distribution of the residual b-quark momentum in the B-meson.
Putting everything together, one ends up with a ”Facotrization Theorem“, which
takes the schematic form
dΓ ∝ C · J ⊗ S (100)
which takes the form of a convolution in the residual-momentum variable (n − k) (every
allowed value of (n − k) for fixed external kinematics (q 2 , E e , E ν ) corresponds to another
value of ∆ µ ). Here the Wilson coefficient C can be calculated in full QCD, the jet
function J can be calculated in SCET, and the shape-function S can be caluclated
in HQET. SCET/HQET also describes the dependence of the various functions under
changes of the factorization scale µ, and therefore the above formula can be exploited to
calculate the decay rate in renormalization-group improved perturbation theory, where
52

• The hard function is matched at µ ∼ m b and evolved down to µ ∼ √ ∆ 2 .
• The jet function is matched at µ ∼ √ ∆ 2 .
• The soft function is modelled at µ ∼ 1−2 GeV and evolved (upwards) to µ ∼ √ ∆ 2 .
Questions/Exercises:
• Compare the analysis for the b → u vertex in the shape-function region in
SCET, with the corresponding vertex correction for the b → c current in
HQET. Verify that the required Wilson coefficient does not depend on the
residual momentum of the heavy quark.
Further Reading:
• [10] and refs. therein.
53

References
[1] G. Buchalla, A. J. Buras, M. E. Lautenbacher, Rev. Mod. Phys. 68 (1996) 1125-
1144. [hep-ph/9512380].
[2] A. J. Buras, “Flavor dynamics: CP violation and rare decays,” [hep-ph/0101336].
[3] The CKM Quark-Mixing Matrix, in C. Amsler et al., Phys. Lett. B667, 1 (2008),
Available online at:
http://pdg.lbl.gov/2010/reviews/rpp2010-rev-ckm-matrix.pdf.
[4] M. E. Peskin, D. V. Schroeder, Reading, USA: Addison-Wesley (1995) 842 p.
[5] M. Beneke, G. Buchalla, M. Neubert, C. T. Sachrajda, Nucl. Phys. B591 (2000)
313-418. [hep-ph/0006124].
[6] M. Neubert, “Lectures on the theory of nonleptonic B decays,” [hep-ph/0012204].
[7] CP violation in meson decays, in C. Amsler et al., Phys. Lett. B667, 1 (2008),
Available online at:
http://pdg.lbl.gov/2010/reviews/rpp2010-rev-cp-violation.pdf.
[8] A. V. Manohar, M. B. Wise, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 10
(2000) 1-191.
[9] Determination of V cb and V ub , in C. Amsler et al., Phys. Lett. B667, 1 (2008),
Available online at:
http://pdg.lbl.gov/2010/reviews/rpp2010-rev-vcb-vub.pdf
[10] M. Neubert, [hep-ph/0512222].
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