Paper
0
Exploring New Physics in the $B\to \phi K$ System
Authors
Robert Fleischer,Thomas Mannel
Katrine Søborg Spang,Ritsuko Kakuma,George Patton,Капиталина Константиновна Ильяшенко,Hermógenes Oliveira,Amina Suleymanova,Arnd Junghans,Thomas Schulz,Tur-Fu Huang,Per Sodersten,Leif Oxburgh,Waldemar Turski,Dean Karlen,Sylvester Gates,Michael Hauschild,Graham Wilson,Sijbrand de Jong,Kirill Kolosov,Olga Lozovskaya,Nina Kulchenko,Irina Snegireva,Odile Sergent,Steven Zinkle,Alan Aitken,Osvaldo de Melo,Evandro Salvador Alves de Oliveira,Ron Folman,ANDRES ARNAU PINO,Irena Zubel,Takeshi Hata,Osamu Yasuda,S.A. Bedina,Silvia PONDRELLI,Julienne Grant,Jaël Muls,Ana Pintar,Tahir Bachar Issa,Michael Milona,Elisabeth Rathmayr,Cristin Chall,Wendy Flores-Fuentes,Roman Drnovsek,Konrad Reszka,Yiqi Pan,Akimichi Takemura,Yang Dai,Christophe Dutang,Galina Kutekhova,Mengqi Li,Timo Baumann,Roderick Hooker,Natalia Volodina,Silvio Alejandro Alcivar Molina,Shaik Sayed Md Rashidul Hossain,Philip Cole,Riccardo Ferretti,Jiafeng Pang,John Taylor,Mikhail Zhilin,Алена Яхненко,Kudzai Kusena,Janet Scull,Xueping Liu,Ricardo Silva,Inesa Sheremet,Laurence Kok,Kara Wunderlich,Jouni Tuisku,Xing-Li Zhao,Héctor Sánchez,valentina gonzalez jaramillo,Grigory Vasiliev,Ida Pesonen,Kelsey Alland,Alexander Bukreyev,Rajmund Porada,Jehad Azar,Takao Kasuga,Uta Sailer,Jinshan Wen,Jonas Rieger,Fredrik Eika,Hamed Heydari,Jesús Brezmes,Xavier Vilanova,Izabella Grzegory,Sylwester Porowski,Vladimir Korzh,Maya Kashuba,Alla Omelchenko,Diane Smith,Taane Clark,Anna Lepekhina,Roland LE BORGNE,Yixian Zheng,Сирануш Нигарян,Svitlana Stavytska,Tim Muris,Inga Mühlenpfordt,Olga Lytvynova,Moamen Gouda,Zhi-wen Chen,Zhihe Jin,Susan Ford,Shigeyuki KON,Viet Khoi Nguyen,Goyitom Gebremedhn,Supriya Sharma,D.B. Avdeev,Tian Long,George Kiroff,腾 李,Yasuaki Akaji,Kayley McMahan,Barbara Salas Salas,Gabriela Vlase,Wei Wang,Kirsten Thonicke,Ignasi Mundet Riera,Adam Cawte,Sileshi Degu,Chris Beaumont,Carolyn Chapman,Wenlong Ma,GEORGIOS KRANIOTIS,José Germán Serrano-Gamboa,Naoto Tominaga,Amire Seyedfaraji,Yinghui Peng,Ricardo Cepedello,Augusto Tamayo,Janet Hemingway,Kathrin Glüer-Pagin,Maximino Manzanera,Igor García Irastorza,Sergey Volkov,Rochelle Basden,Jingxiu Xie,WENDY MILAGROS TERCERO MARTINEZ,José Augusto Afonso,Charlie Siu,Григорий Колесников,Werner Grass,Jeanne Kagle,Cecilia Bonolo de Campos,红 张,Grzegorz Moroz,Itzhak Barkana,Hemalatha Bhagavan,Gwyndalyn Phillips,Néstor Antonio Luque Vergara,David Little,Ahmet Nusret Bulgurcuoğlu,Reade Roberts,Mohammad Ali Mahjoub,Татьяна Димитриади,Miguel Villaplana Perez,Majid Abbasi,Ravindra Belur,zhengcai lou,Marie ,shekhar gogna,Richard Howard,Simon Lidberg,William Matern,Keiji Nakatsugawa,Matthias Held,Dennis Barrett,Stuart Rowlands,Johann Peltier,Winship Herr,David Cook,Esko Lehtonen,Witold Nazarewicz,Peter Kort,A.F. ,Bingsong Zou,Stefan Schmitt,Ana Patricia Fernández,Ann Roselle,Jean-Louis Frendo,Marek Malik,Ahmad Nasri,Mariwan Al-bajalan,Gianmarco Tesone,Lloyd Trefethen,meryem etoz,Shan Chen,Nayoung Kim,Leslie Banks-Sills,Mark Portelli,yasar durmaz,Michael Levine,Junchao Chen,Xiangshang Li,Dmitry Izergin,Jennifer Im,Pawan Kumar Chamling,Darío Ramirez,Colleen Hunt,Yonglong Qu,Sinan Ertemel,Zeki Gökçil,Yang Li,Zhao Yang,Theodore Stanoev,OZNUR ULUDAG,Elina KC,Tunca Karasu,Wentao Cao,Сердар Кулиев,Christina Dieli-Conwright,Hemamalini Mariyappa Jagadeesh,Shigeaki Ogibayashi,Juliana Cardoso,Alice Lin,Paula Larrañaga,Nuno Anjos,Viktoras Grigaliūnas,M. 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arXiv:hep-ph/0103121v1 12 Mar 2001
DESY 01–030
CERN–TH/2001–072
TTP01–07
hep–ph/0103121
March 2001
Exploring New Physics in the B → φK System
Robert Fleischer∗
Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D–22607 Hamburg, Germany
Thomas Mannel†
CERN Theory Division, CH–1211 Geneva 23, Switzerland
and
Institut f¨ur Theoretische Teilchenphysik, Universit¨at Karlsruhe,
D–76128 Karlsruhe, Germany
Abstract
Employing the SU(2) isospin symmetry of strong interactions and estimates borrowed from
effective field theory, we explore the impact of new physics on the decays B± → φK± and
Bd → φKS in a model-independent manner. To this end, we introduce – in addition to
the usual mixing-induced CP asymmetry in Bd → φKS – a set of three observables, which
may not only provide “smoking-gun” signals for new-physics contributions to different isospin
channels, but also valuable insights into hadron dynamics. Imposing dynamical hierarchies
of amplitudes, we discuss various patterns of these observables, including also scenarios with
small and large rescattering processes. Whereas the B → φK system provides, in general, a
powerful tool to search for indications of new physics, there is also an unfortunate case, where
such effects cannot be distinguished from those of the Standard Model.
∗E-mail: Robert.Fleischer@desy.de
†E-mail: Thomas.Mannel@cern.ch
DESY 01–030
CERN–TH/2001–072
TTP01–07
hep–ph/0103121
March 2001
Exploring New Physics in the B → φK System
Robert Fleischer∗
Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D–22607 Hamburg, Germany
Thomas Mannel†
CERN Theory Division, CH–1211 Geneva 23, Switzerland
and
Institut f¨ur Theoretische Teilchenphysik, Universit¨at Karlsruhe,
D–76128 Karlsruhe, Germany
Abstract
Employing the SU(2) isospin symmetry of strong interactions and estimates borrowed from
effective field theory, we explore the impact of new physics on the decays B± → φK± and
Bd → φKS in a model-independent manner. To this end, we introduce – in addition to
the usual mixing-induced CP asymmetry in Bd → φKS – a set of three observables, which
may not only provide “smoking-gun” signals for new-physics contributions to different isospin
channels, but also valuable insights into hadron dynamics. Imposing dynamical hierarchies
of amplitudes, we discuss various patterns of these observables, including also scenarios with
small and large rescattering processes. Whereas the B → φK system provides, in general, a
powerful tool to search for indications of new physics, there is also an unfortunate case, where
such effects cannot be distinguished from those of the Standard Model.
∗E-mail: Robert.Fleischer@desy.de
†E-mail: Thomas.Mannel@cern.ch
1 Introduction
The experimental data collected at the B factories will allow stringent tests of the Kobayashi–
Maskawa picture of CP violation [1]. Among the various B decays that can be used to achieve
this goal [2], B → φK decays play an important role. In these modes, which are governed
by QCD penguin processes [3], also electroweak (EW) penguins are sizeable [4], and physics
beyond the Standard Model may have an important impact [5]. In the summer of 2000, the
observation of the B± → φK± channel was announced by the Belle and CLEO collaborations.
The present results for the CP-averaged branching ratio are given as follows:
BR(B± → φK±) =
(
1.39+0.37+0.14
−0.33−0.24
)
× 10−5 (Belle [6])
(
5.5+2.1
−1.8 ± 0.6)
× 10−6 (CLEO [7]). (1)
The Belle and CLEO results are only marginally compatible with each other. Evidence
for the neutral mode B0
d → φK0 at the 2.9σ level, corresponding to a branching ratio of(
5.4+3.7
−2.7 ± 0.7)
× 10−6, was also reported by CLEO, whereas a significant signal for this decay
has not yet been observed by the Belle collaboration.
In our discussion of the B → φK system, we follow closely our recent B → J/ψK analysis
[8], and make use of the SU(2) isospin symmetry of strong interactions to derive a model-
independent parametrization of the B+ → φK+, B0
d → φK0 decay amplitudes. After reca-
pitulating the structure of the Standard-Model amplitudes, we include new-physics effects in
a general manner, and estimate their generic size with the help of arguments borrowed from
the picture of effective field theory. In order to deal with hadronic matrix elements, we im-
pose certain dynamical hierarchies of decay amplitudes, where we distinguish between small
and large rescattering effects. Although we do not consider the latter case, which is also not
favoured by the “QCD factorization” approach [9] and the present experimental upper bounds
on B → KK branching ratios [10], as a very likely scenario,1 it deserves careful attention to
separate possible new-physics effects from those of the Standard Model. Moreover, following
the strategies proposed below, we may not only obtain insights into new physics, but also into
hadron dynamics. To this end, we introduce – in addition to the usual mixing-induced CP
asymmetry in Bd → φKS – a set of three observables, providing “smoking-gun” signals for
new-physics contributions to different isospin channels. Two of these new-physics observables
may be significantly enhanced by large rescattering processes. In general, the B → φK system
offers powerful tools to search for new physics. However, there is also an unfortunate case,
where such effects cannot be disentangled from those of the Standard Model.
The outline of this paper is as follows: in Section 2, we employ a low-energy effective
Hamiltonian and the isospin symmetry of strong interactions to parametrize the B± → φK±,
Bd → φKS decay amplitudes arising within the Standard Model. The impact of new physics
on these amplitudes is explored in a model-independent way in Section 3, where we make
use of dimensional estimates following from effective field theory, and introduce plausible dy-
namical hierarchies of amplitudes. The set of observables to search for “smoking-gun” signals
of new-physics contributions to different isospin channels of the B → φK decay amplitudes
is introduced in Section 4, and is discussed in further detail in Section 5. In Section 6, our
conclusions are summarized.
1Arguments against this possibility, i.e. large rescattering effects, were also given in [11].
1
The experimental data collected at the B factories will allow stringent tests of the Kobayashi–
Maskawa picture of CP violation [1]. Among the various B decays that can be used to achieve
this goal [2], B → φK decays play an important role. In these modes, which are governed
by QCD penguin processes [3], also electroweak (EW) penguins are sizeable [4], and physics
beyond the Standard Model may have an important impact [5]. In the summer of 2000, the
observation of the B± → φK± channel was announced by the Belle and CLEO collaborations.
The present results for the CP-averaged branching ratio are given as follows:
BR(B± → φK±) =
(
1.39+0.37+0.14
−0.33−0.24
)
× 10−5 (Belle [6])
(
5.5+2.1
−1.8 ± 0.6)
× 10−6 (CLEO [7]). (1)
The Belle and CLEO results are only marginally compatible with each other. Evidence
for the neutral mode B0
d → φK0 at the 2.9σ level, corresponding to a branching ratio of(
5.4+3.7
−2.7 ± 0.7)
× 10−6, was also reported by CLEO, whereas a significant signal for this decay
has not yet been observed by the Belle collaboration.
In our discussion of the B → φK system, we follow closely our recent B → J/ψK analysis
[8], and make use of the SU(2) isospin symmetry of strong interactions to derive a model-
independent parametrization of the B+ → φK+, B0
d → φK0 decay amplitudes. After reca-
pitulating the structure of the Standard-Model amplitudes, we include new-physics effects in
a general manner, and estimate their generic size with the help of arguments borrowed from
the picture of effective field theory. In order to deal with hadronic matrix elements, we im-
pose certain dynamical hierarchies of decay amplitudes, where we distinguish between small
and large rescattering effects. Although we do not consider the latter case, which is also not
favoured by the “QCD factorization” approach [9] and the present experimental upper bounds
on B → KK branching ratios [10], as a very likely scenario,1 it deserves careful attention to
separate possible new-physics effects from those of the Standard Model. Moreover, following
the strategies proposed below, we may not only obtain insights into new physics, but also into
hadron dynamics. To this end, we introduce – in addition to the usual mixing-induced CP
asymmetry in Bd → φKS – a set of three observables, providing “smoking-gun” signals for
new-physics contributions to different isospin channels. Two of these new-physics observables
may be significantly enhanced by large rescattering processes. In general, the B → φK system
offers powerful tools to search for new physics. However, there is also an unfortunate case,
where such effects cannot be disentangled from those of the Standard Model.
The outline of this paper is as follows: in Section 2, we employ a low-energy effective
Hamiltonian and the isospin symmetry of strong interactions to parametrize the B± → φK±,
Bd → φKS decay amplitudes arising within the Standard Model. The impact of new physics
on these amplitudes is explored in a model-independent way in Section 3, where we make
use of dimensional estimates following from effective field theory, and introduce plausible dy-
namical hierarchies of amplitudes. The set of observables to search for “smoking-gun” signals
of new-physics contributions to different isospin channels of the B → φK decay amplitudes
is introduced in Section 4, and is discussed in further detail in Section 5. In Section 6, our
conclusions are summarized.
1Arguments against this possibility, i.e. large rescattering effects, were also given in [11].
1
2 Phenomenology of B → φK Decays
The B → φK system is described by the following low-energy effective Hamiltonian:
Heff = GF
√2
[
VcsV ∗
cb
(
Qc
CC − Qpen
QCD − Qpen
EW
)
+ VusV ∗
ub
(
Qu
CC − Qpen
QCD − Qpen
EW
)]
, (2)
where the Q are linear combinations of perturbative Wilson coefficient functions and four-
quark operators, consisting of current–current (CC), QCD penguin and EW penguin operators.
As discussed in [8], this Hamiltonian is a combination of isospin I = 0 and I = 1 pieces:
Heff = HI=0
eff + HI=1
eff , (3)
where HI=0
eff receives contributions from all of the operators appearing in (2), whereas HI=1
eff
is due to only Qu
CC and Qpen
EW. If we employ the SU(2) isospin flavour symmetry of strong
interactions, we obtain
〈φK+|HI=0
eff |B+〉 = +〈φK0|HI=0
eff |B0
d 〉 (4)
〈φK+|HI=1
eff |B+〉 = −〈φK0|HI=1
eff |B0
d 〉, (5)
yielding
A(B+ → φK+) = GF
√2
[
VcsV ∗
cb
{
A(0)
c − A(1)
c
}
+ VusV ∗
ub
{
A(0)
u − A(1)
u
}]
(6)
A(B0
d → φK0) = GF
√2
[
VcsV ∗
cb
{
A(0)
c + A(1)
c
}
+ VusV ∗
ub
{
A(0)
u + A(1)
u
}]
, (7)
where the CP-conserving strong amplitudes2
A(0)
c = Ac
CC − Apen
QCD − A(0)
EW, A(1)
c = −A(1)
EW (8)
A(0)
u = Au(0)
CC − Apen
QCD − A(0)
EW, A(1)
u = Au(1)
CC − A(1)
EW (9)
can be expressed in terms of hadronic matrix elements 〈φK|Q|B〉. Taking into account that
VcsV ∗
cb =
(
1 − λ2
2
)
λ2A, VusV ∗
ub = λ4A Rb eiγ , (10)
where γ is the usual angle of the unitarity triangle of the CKM matrix [2], and
λ ≡ |Vus| = 0.22, A ≡ |Vcb|/λ2 = 0.81 ± 0.06, Rb ≡ |Vub/(λVcb)| = 0.41 ± 0.07, (11)
we finally arrive at
A(B+ → φK+) = GF
√2
(
1 − λ2
2
)
λ2A {
A(0)
c − A(1)
c
} [
1 + λ2Rb
1 − λ2/2
{ A(0)
u − A(1)
u
A(0)
c − A(1)
c
}
eiγ
]
(12)
A(B0
d → φK0) = GF
√2
(
1 − λ2
2
)
λ2A {
A(0)
c + A(1)
c
} [
1 + λ2Rb
1 − λ2/2
{ A(0)
u + A(1)
u
A(0)
c + A(1)
c
}
eiγ
]
. (13)
2The labels of A(0) and A(1) refer to the isospin channels I = 0 and I = 1, respectively.
2
The B → φK system is described by the following low-energy effective Hamiltonian:
Heff = GF
√2
[
VcsV ∗
cb
(
Qc
CC − Qpen
QCD − Qpen
EW
)
+ VusV ∗
ub
(
Qu
CC − Qpen
QCD − Qpen
EW
)]
, (2)
where the Q are linear combinations of perturbative Wilson coefficient functions and four-
quark operators, consisting of current–current (CC), QCD penguin and EW penguin operators.
As discussed in [8], this Hamiltonian is a combination of isospin I = 0 and I = 1 pieces:
Heff = HI=0
eff + HI=1
eff , (3)
where HI=0
eff receives contributions from all of the operators appearing in (2), whereas HI=1
eff
is due to only Qu
CC and Qpen
EW. If we employ the SU(2) isospin flavour symmetry of strong
interactions, we obtain
〈φK+|HI=0
eff |B+〉 = +〈φK0|HI=0
eff |B0
d 〉 (4)
〈φK+|HI=1
eff |B+〉 = −〈φK0|HI=1
eff |B0
d 〉, (5)
yielding
A(B+ → φK+) = GF
√2
[
VcsV ∗
cb
{
A(0)
c − A(1)
c
}
+ VusV ∗
ub
{
A(0)
u − A(1)
u
}]
(6)
A(B0
d → φK0) = GF
√2
[
VcsV ∗
cb
{
A(0)
c + A(1)
c
}
+ VusV ∗
ub
{
A(0)
u + A(1)
u
}]
, (7)
where the CP-conserving strong amplitudes2
A(0)
c = Ac
CC − Apen
QCD − A(0)
EW, A(1)
c = −A(1)
EW (8)
A(0)
u = Au(0)
CC − Apen
QCD − A(0)
EW, A(1)
u = Au(1)
CC − A(1)
EW (9)
can be expressed in terms of hadronic matrix elements 〈φK|Q|B〉. Taking into account that
VcsV ∗
cb =
(
1 − λ2
2
)
λ2A, VusV ∗
ub = λ4A Rb eiγ , (10)
where γ is the usual angle of the unitarity triangle of the CKM matrix [2], and
λ ≡ |Vus| = 0.22, A ≡ |Vcb|/λ2 = 0.81 ± 0.06, Rb ≡ |Vub/(λVcb)| = 0.41 ± 0.07, (11)
we finally arrive at
A(B+ → φK+) = GF
√2
(
1 − λ2
2
)
λ2A {
A(0)
c − A(1)
c
} [
1 + λ2Rb
1 − λ2/2
{ A(0)
u − A(1)
u
A(0)
c − A(1)
c
}
eiγ
]
(12)
A(B0
d → φK0) = GF
√2
(
1 − λ2
2
)
λ2A {
A(0)
c + A(1)
c
} [
1 + λ2Rb
1 − λ2/2
{ A(0)
u + A(1)
u
A(0)
c + A(1)
c
}
eiγ
]
. (13)
2The labels of A(0) and A(1) refer to the isospin channels I = 0 and I = 1, respectively.
2
At first sight, expressions (12) and (13) are completely analogous to the ones for the
B+ → J/ψK+ and B0
d → J/ψK0 amplitudes given in [8]. However, the dynamics, which
is encoded in the strong amplitudes A, is very different. In particular, the current–current
operators Qc
CC cannot contribute to B → φK decays, i.e. to Ac
CC, through tree-diagram-
like topologies; they may only do so through penguin topologies with internal charm-quark
exchanges, which include also
B+ → {D+
s D0, ...} → φK+, B0
d → {D+
s D−, ...} → φK0 (14)
rescattering processes [12], and may actually play an important role [13]. On the other hand,
the Au(0,1)
CC amplitudes receive contributions from penguin processes with internal up- and
down-quark exchanges, as well as from annihilation topologies.3 Such penguins may also
be important, in particular in the presence of large rescattering processes [12, 14]; a similar
comment applies to annihilation topologies. In the B → φK system, the relevant rescattering
processes are
B+ → {K+π0, ...} → φK+, B0
d → {K+π−, ...} → φK0, (15)
containing – in addition to long-distance penguins – also annihilation processes (see Figs. 1
and 2). In contrast to (14), large rescattering effects of the kind described by (15) may affect
the search for new physics with B → φK decays, since these processes are associated with
the weak phase factor eiγ . Moreover, they involve “light” intermediate states, and are hence
expected to be enhanced more easily, dynamically, through long-distance effects than (14),
which involve “heavy” intermediate states.
As is well known, the φ-meson is an almost pure ss state; the mixing angle with its isoscalar
partner ω ∼ (uu + dd)/√2 is small, i.e. at the few per cent level. Whereas ω–φ mixing does
not at all affect the isospin relations (4) and (5), which rely on the fact that the φ is an isospin
singlet, it has an impact on the size of the amplitudes Au(0,1)
CC , since an ω component of the
φ state permits current–current operator contributions through tree-diagram-like topologies.
However, the arguments given below are not modified by the small ω–φ mixing.
Let us now have a closer look at the structure of the B → φK decay amplitudes, focusing
first on the case corresponding to small rescattering effects. Looking at (8) and (9), we expect
∣
∣
∣A(0,1)
u /A(0)
c
∣
∣
∣ = O(1). (16)
In the case of the amplitude A(1)
c , the situation is different. Here we have to deal with an
amplitude that is essentially due to EW penguins. Moreover, the B → φK matrix elements
of I = 1 operators, having the general flavour structure
QI=1 ∼ (uu − dd)(bs), (17)
are expected to suffer from a dynamical suppression. In order to keep track of these features,
we introduce, as in [8], a “generic” expansion parameter λ = O(0.2) [15], which is of the same
order as the Wolfenstein parameter λ = 0.22, and suggests
∣
∣
∣A(1)
c /A(0)
c
∣
∣
∣ = O(λ)
︸ ︷︷ ︸
EW penguins
× O(λ)
︸ ︷︷ ︸
Dynamics
= O(λ2). (18)
3Note that the isospin projection operators Q ∼ (uu ± dd)(bs) involve also dd quark currents.
3
B+ → J/ψK+ and B0
d → J/ψK0 amplitudes given in [8]. However, the dynamics, which
is encoded in the strong amplitudes A, is very different. In particular, the current–current
operators Qc
CC cannot contribute to B → φK decays, i.e. to Ac
CC, through tree-diagram-
like topologies; they may only do so through penguin topologies with internal charm-quark
exchanges, which include also
B+ → {D+
s D0, ...} → φK+, B0
d → {D+
s D−, ...} → φK0 (14)
rescattering processes [12], and may actually play an important role [13]. On the other hand,
the Au(0,1)
CC amplitudes receive contributions from penguin processes with internal up- and
down-quark exchanges, as well as from annihilation topologies.3 Such penguins may also
be important, in particular in the presence of large rescattering processes [12, 14]; a similar
comment applies to annihilation topologies. In the B → φK system, the relevant rescattering
processes are
B+ → {K+π0, ...} → φK+, B0
d → {K+π−, ...} → φK0, (15)
containing – in addition to long-distance penguins – also annihilation processes (see Figs. 1
and 2). In contrast to (14), large rescattering effects of the kind described by (15) may affect
the search for new physics with B → φK decays, since these processes are associated with
the weak phase factor eiγ . Moreover, they involve “light” intermediate states, and are hence
expected to be enhanced more easily, dynamically, through long-distance effects than (14),
which involve “heavy” intermediate states.
As is well known, the φ-meson is an almost pure ss state; the mixing angle with its isoscalar
partner ω ∼ (uu + dd)/√2 is small, i.e. at the few per cent level. Whereas ω–φ mixing does
not at all affect the isospin relations (4) and (5), which rely on the fact that the φ is an isospin
singlet, it has an impact on the size of the amplitudes Au(0,1)
CC , since an ω component of the
φ state permits current–current operator contributions through tree-diagram-like topologies.
However, the arguments given below are not modified by the small ω–φ mixing.
Let us now have a closer look at the structure of the B → φK decay amplitudes, focusing
first on the case corresponding to small rescattering effects. Looking at (8) and (9), we expect
∣
∣
∣A(0,1)
u /A(0)
c
∣
∣
∣ = O(1). (16)
In the case of the amplitude A(1)
c , the situation is different. Here we have to deal with an
amplitude that is essentially due to EW penguins. Moreover, the B → φK matrix elements
of I = 1 operators, having the general flavour structure
QI=1 ∼ (uu − dd)(bs), (17)
are expected to suffer from a dynamical suppression. In order to keep track of these features,
we introduce, as in [8], a “generic” expansion parameter λ = O(0.2) [15], which is of the same
order as the Wolfenstein parameter λ = 0.22, and suggests
∣
∣
∣A(1)
c /A(0)
c
∣
∣
∣ = O(λ)
︸ ︷︷ ︸
EW penguins
× O(λ)
︸ ︷︷ ︸
Dynamics
= O(λ2). (18)
3Note that the isospin projection operators Q ∼ (uu ± dd)(bs) involve also dd quark currents.
3
u,d
b q
q
s
s
q
q
B π
Κ s
s
φ
K
u,d
Figure 1: Rescattering processes contributing to B → φK through penguin-like topologies
with internal q-quark exchanges (q ∈ {u, d}). The shaded circle represents insertions of the
corresponding current–current operators.
q
b s
s
s
B K
π
q
q
q
q
q
q
s
s
K
φ
Figure 2: Rescattering processes contributing to B → φK through annihilation topologies.
The shaded circle represents insertions of current–current operators (q ∈ {u, d}).
Consequently, we obtain
A(B+ → φK+) = A(0)
SM
[
1 + O(λ2)]
= A(B0
d → φK0), (19)
with
A(0)
SM ≡ GF
√2 λ2A A(0)
c . (20)
The terms entering (19) at the λ2 level contain also pieces that are proportional to the weak-
phase factor eiγ , thereby leading to direct CP violation in the B → φK system.
Let us now consider large rescattering effects of the kind given in (15). In the worst case,
(16) would be dynamically enhanced as
∣
∣
∣A(0,1)
u /A(0)
c
∣
∣
∣ = O(1/λ), (21)
and the dynamical suppression in (18) would no longer be effective, i.e.
∣
∣
∣A(1)
c /A(0)
c
∣
∣
∣ = O(λ). (22)
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