General Analysis of New Physics in $B\to J/ψK$

14 Pages·0.2 MB·English

by  Robert Fleischer

DESY 01–006
CERN–TH/2001–012
TTP01–06
hep–ph/0101276
January 2001
1
0 General Analysis of New Physics in B → J/ψK
0
2

n
a
J

4
2
Robert Fleischer
∗

1
v Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D–22607 Hamburg, Germany
6
7
2
1 Thomas Mannel†
0
1 CERN Theory Division, CH–1211 Geneva 23, Switzerland
0
/ and
h
Institut fu¨r Theoretische Teilchenphysik, Universit¨at Karlsruhe,
p
- D–76128 Karlsruhe, Germany
p
e
h
:
v
i
X
r
a Abstract
We present a model-independent parametrization of the B± J/ψK±, B J/ψK decay
d S
→ →
amplitudes by taking into account the constraints that are implied by the isospin symmetry
of strong interactions. Employing estimates borrowed from effective field theory, we explore
the impact of physics beyond the Standard Model and introduce – in addition to the usual
mixing-induced CP asymmetry mix in B J/ψK – a set of three observables, allowing
ACP d → S
a general analysis of possible new-physics effects in the B J/ψK system. Imposing a
→
dynamical hierarchy of amplitudes, we argue that one of these observables may already be
accessible at the first-generation B-factories, whereas the remaining ones will probably be left
for second-generation B experiments. However, in the presence of large rescattering effects,
all three new-physics observables may be sizeable. We also emphasize that a small value of
mix could be due to new-physics effects arising at the B J/ψK decay-amplitude level. In
ACP →
order to establish such a scenario, the observables introduced in this paper play a key role.
∗E-mail: [email protected]
†E-mail: [email protected]

1 Introduction
At present, we are at the beginning of the B-factory era in particle physics, which will provide
valuable insights into CP violation and various tests of the Kobayashi–Maskawa picture of
this phenomenon [1]. Among the most interesting B-decay channels is the “gold-plated” mode
B J/ψK [2], which allows the determination of the angle β of the unitarity triangle of the
d S
→
Cabibbo–Kobayashi–Maskawa (CKM) matrix [3]. In the summer of 2000 – after important
steps at LEP [4] and by the CDF collaboration [5] – the first results on CP-violating effects
in B J/ψK were reported by the BaBar [6] and Belle [7] collaborations, which already
d S
→
led to some excitement in the B-physics community [8].
In this paper, we consider the neutral mode B J/ψK together with its charged coun-
d S
→
terpart B± J/ψK±. Making use of the isospin symmetry of strong interactions, we derive
→
a model-independent parametrization of the corresponding decay amplitudes. After a careful
analysis of the Standard-Model contributions, we include possible new-physics amplitudes and
introduce – in addition to the usual “mixing-induced” CP asymmetry in B J/ψK – a
d S
→
set of three observables, allowing a general analysis of new-physics effects in the B J/ψK
→
system; the generic size of these effects is estimated with the help of arguments borrowed
from effective field theory. The four observables provided by B J/ψK decays are af-
→
fected by physics beyond the Standard Model in two different ways: through B0–B0 mixing,
d d
and new-physics contributions to the B J/ψK decay amplitudes. Usually, B0–B0 mixing
→ d d
is considered as the preferred mechanism for new physics to manifest itself in the mixing-
induced B J/ψK CP asymmetry [9]. Here we focus on new-physics effects arising at the
d S
→
B J/ψK decay-amplitude level [10]; we emphasize that the extraction of the CKM angle
→
β from mixing-induced CP violation in B J/ψK may be significantly disturbed by such
d S
→
effects. As is well known, the value of “β” itself, i.e. the CP-violating weak B0–B0 mixing
d d
phase, may deviate strongly from the Standard-Model expectation because of new-physics
contributions to B0–B0 mixing.
d d
The threeobservables introduced inthis paper allowusto search for“smoking-gun” signals
of new-physics contributions to the B J/ψK decay amplitudes, which would also play an
→
important role for mixing-induced CP violation in B J/ψK . Employing an isospin
d S
→
decomposition and imposing a hierarchy of amplitudes, we argue that one of these observables
may be sizeable and could already be accessible at the first-generation B-factories. On the
otherhand,theremainingtwoobservablesareexpectedtobedynamicallysuppressed, butmay
be within reach of the second-generation B experiments, BTeV and LHCb. In the presence
of large rescattering effects, all three observables could be of the same order of magnitude.
However, we do not consider this as a very likely scenario and note that also the “QCD
factorization” approach is not in favour of such large rescattering processes [11].
The outline of this paper is as follows: in Section 2, we investigate the isospin structure
of the B± J/ψK±, B J/ψK decay amplitudes, and parametrize them in a model-
d S
→ →
independent manner; a particular emphasis is given to the Standard-Model case. In Section 3,
we then have a closer look at the impact of new physics. To this end, we make use of
dimensionalestimatesfollowingfromeffectivefieldtheory, andintroduceaplausibledynamical
hierarchy of amplitudes. The set of observables to search for “smoking-gun” signals of new-
physics contributions to the B J/ψK decay amplitudes is defined in Section 4, and is
→
discussed in further detail in Section 5. In Section 6, our main points are summarized.
1
2 Phenomenology of B → J/ψK Decays
The most general discussion of the B± J/ψK±, B J/ψK system can be performed in
d S
→ →
terms of an isospin decomposition. Here the corresponding initial and final states are grouped
in the following isodoublets:
1/2;+1/2 B+ B0 J/ψK+ J/ψK0
| i : | i , | di , | i , | i , (1)
1/2; 1/2 B0 B− J/ψK0 J/ψK−
| − i! | di ! −| i! | i ! −| i!
CP CP
| {z } | {z }
which are related by CP conjugation. As usual, the K state in B J/ψK is the superpo-
S d S
→
sition of the two neutral kaon states K0 and K0 corresponding to CP eigenvalue +1. The
| i | i
decays B+ J/ψK+ and B0 J/ψK0 are described by an effective low-energy Hamiltonian
→ d →
of the following structure:
G
= F V V∗ c pen pen +V V∗ u pen pen , (2)
Heff √2 cs cb QCC −QQCD −QEW us ub QCC −QQCD −QEW
h (cid:16) (cid:17) (cid:16) (cid:17)i
where the are linear combinations of perturbatively calculable Wilson coefficient functions
Q
and four-quark operators, consisting of current–current (CC), QCD penguin and electroweak
(EW) penguin operators. For an explicit list of operators and a detailed discussion of the
derivation of (2), the reader is referred to [12]. For the following considerations, the flavour
structure of these operators plays a key role:
c (cc)(bs), u (uu)(bs), (3)
QCC ∼ QCC ∼
pen (cc)+ (uu)+(dd) +(ss) (bs), (4)
QQCD ∼ { }
h i
1
pen 2(cc)+ 2(uu) (dd) (ss) (bs), (5)
QEW ∼ 3 { − }−
h i
where the factors of +2/3 and 1/3 in (5) are due to electrical quark charges. In (3)–(5), we
−
have suppressed all colour and spin indices. Since
1 1 1 3
(uu) = uu+dd + uu dd , 2(uu) (dd) = uu+dd + uu dd , (6)
2 2 − − 2 2 −
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
I=0 I=1 I=0 I=1
| {z } | {z } | {z } | {z }
we conclude that the Hamiltonian (2) is a combination of isospin I = 0 and I = 1 pieces:
= I=0 + I=1, (7)
Heff Heff Heff
where I=0 receives contributions from all of the operators listed in (3)–(5), whereas I=1
Heff Heff
is only due to u and pen. Taking into account the isospin flavour symmetry of strong
QCC QEW
interactions, we obtain
J/ψK+ I=0 B+ = + J/ψK0 I=0 B0 (8)
h |Heff | i h |Heff | di
J/ψK+ I=1 B+ = J/ψK0 I=1 B0 , (9)
h |Heff | i −h |Heff | di
2
and finally arrive at
G
A(B+ J/ψK+) = F V V∗ A(0) A(1) +V V∗ A(0) A(1) (10)
→ √2 cs cb c − c us ub u − u
h n o n oi
G
A(B0 J/ψK0) = F V V∗ A(0) +A(1) +V V∗ A(0) +A(1) , (11)
d → √2 cs cb c c us ub u u
h n o n oi
where
A(0) = Ac Apen A(0) , A(1) = A(1) (12)
c CC − QCD − EW c − EW
A(0) = Au(0) Apen A(0) , A(1) = Au(1) A(1) (13)
u CC − QCD − EW u CC − EW
denote hadronic matrix elements J/ψK B , i.e. are CP-conserving strong amplitudes. The
h |Q| i
CKM factors in (10) and (11) are given by
λ2
V V∗ = 1 λ2A, V V∗ = λ4AR eiγ, (14)
cs cb − 2 ! us ub b
with
λ V = 0.22, A V /λ2 = 0.81 0.06, R V /(λV ) = 0.41 0.07, (15)
us cb b ub cb
≡ | | ≡ | | ± ≡ | | ±
and γ is the usual angle of the unitarity triangle of the CKM matrix [3]. Consequently, we
may write
G λ2 λ2R A(0) A(1)
A(B+ J/ψK+) = F 1 λ2A A(0) A(1) 1+ b u − u eiγ (16)
→ √2 − 2 ! c − c " 1 λ2/2 (A(0) A(1)) #
n o − c − c
G λ2 λ2R A(0) +A(1)
A(B0 J/ψK0) = F 1 λ2A A(0) +A(1) 1+ b u u eiγ . (17)
d → √2 − 2 ! c c " 1 λ2/2 (A(0) +A(1)) #
n o − c c
Let us note that (17) takes the same form as the parametrization derived in [13], making,
however, its isospin decomposition explicit. Animportant observation is that the CP-violating
phase factor eiγ enters in (16) and (17) in a doubly Cabibbo-suppressed way. Moreover, the
A(0,1) amplitudes are governed by penguin-like topologies and annihilation diagrams, and are
u
hence expected to be suppressed with respect to A(0), which originates from tree-diagram-like
c
topologies [14]. In order to keep track of this feature, we introduce a “generic” expansion
parameter λ = (0.2) [15], which is of the same order as the Wolfenstein parameter λ = 0.22:
O
A(0,1)/A(0) = (λ). (18)
u c O
(cid:12) (cid:12)
(cid:12) (cid:12)
Consequently, the eiγ terms in (16) a(cid:12)nd (17) ar(cid:12)e actually suppressed by (λ3). Since A(1) is
O c
due to dynamically suppressed matrix elements of EW penguin operators (see (12)), we expect
A(1)/A(0) = (λ3). (19)
c c O
(cid:12) (cid:12)
(cid:12) (cid:12)
Therefore, we obtain – up to negligib(cid:12)ly small c(cid:12)orrections of (λ3) – the following expression:
O
A(B+ J/ψK+) = A(B0 J/ψK0) = A(0), (20)
→ d → SM
3
with
G λ2
A(0) F 1 λ2AA(0). (21)
SM ≡ √2 − 2 ! c
Let us note that the plausible hierarchy of strong amplitudes given in (18) and (19) may be
spoiledbyvery largerescattering processes [16]. Intheworst case, (20)may receive corrections
2 3
of (λ ), and not at the λ level. However, we do not consider this a very likely scenario and
O
note that also the “QCD factorization” approach developed in [11] is not in favour of such
large rescattering effects.
The purpose of this paper is to explore the impact of physics beyond the Standard Model
on B J/ψK decays. In the next section, we investigate the generic size of such effects using
→
dimensional arguments following from the framework of effective field theory.
3 Effects of Physics Beyond the Standard Model
The generic way of introducing physics beyond the Standard Model is to use the language of
effective field theory and to write down all possible dim-6 operators. Of course, this has been
knownalreadyforalongtimeandlistsofthedim-6operatorsinvolving alltheStandard-Model
particles have been published in the literature [17]. After having introduced these additional
operators, we construct the generalization of the Standard-Model effective Hamiltonian (2) at
the scale of the b quark. The relevant operators are again dim-6 operators, the coefficients of
which now contain a Standard-Model contribution, and a possible piece of “new physics”.
The problem with this point of view is that the list of possible dim-6 operators contains
close to one hundred entries, not yet taking into account the flavour structure, making this
general approach almost useless. However, we are dealing with non-leptonic decays in which
we are, because of our ignorance of the hadronic matrix elements, sensitive neither to the
helicity structure of the operators nor to their colour structure. The only information that is
relevant in this case is the flavour structure, and hence we introduce the notation
(Qq )(q q ) = [Wilson coefficient] [dim-6 operator mediating Q q q q ]. (22)
1 2 3 × → 1 2 3
h i X
Clearly, this sum is renormalization-group-invariant, and involves, at the scale of the b quark,
Standard-Model as well as possible non-Standard-Model contributions. In particular, it allows
us to estimate the relative size of a possible new-physics contribution.
3.1 B0–B0 Mixing: ∆B = ±2 Operators
d d
Using this language, we now consider the ∆B = 2 operators relevant to B0–B0 mixing. We
± d d
have [18]
(∆B = +2) = G (bd)(bd) (23)
eff
H
h i
as the relevant flavour structure. Within the Standard Model, (23) originates from the well-
known box diagrams, which are strongly suppressed by the CKM factor (V V∗)2, as well as
td tb
by a loop factor
g2 G M2
2 = F W 1 10−3, (24)
64π2 √128π2 ≈ ×
4
making the Standard-Model contribution very small, of the order of1
G G M2
G = F F W (V V∗)2. (25)
SM √2 √128π2! td tb
A new-physics contribution therefore could in principle be as large as the Standard-Model
piece. If Λ is the scale of physics beyond the Standard Model, we have
G G M2 M2
G = F F W We−i2ψ, (26)
NP
√2 √128π2! Λ2
where ψ is a possible weak phase, which is induced by the new-physics contribution. Finally,
we arrive at
G G M2 M2
G = F F W (V V∗)2 + We−i2ψ R e−iφM. (27)
√2 √128π2!" td tb Λ2 # ≡ | |
The relevant quantity is the (weak) phase φ of this expression, which enters the “mixing-
M
induced” CP-violating asymmetries [3]. Using the standard parametrization
V V∗ = λ3AR e−iβ, (28)
td tb t
where R V /(λV ) = (1), we obtain
t td cb
≡ | | O
sin(2β)+̺2sin(2ψ)
tanφ = , (29)
M
cos(2β)+̺2cos(2ψ)
with
1 M
W
̺ = . (30)
λ3AR Λ
(cid:18) t(cid:19)(cid:18) (cid:19)
Since the factor ̺ can be of order one even for large Λ, there can be a large phase shift in the
mixing phase. If we assume, for example, AR = 1, this term equals 1 for a new-physics scale
t
of Λ 8TeV. Such contributions affect of course not only the CP-violating phase φ , but
M
∼
also the “strength” R of the B0–B0 mixing, which would manifest itself as an inconsistency
| | d d
in the usual “standard analysis” of the unitarity triangle [19].
3.2 Decay Amplitudes: ∆B = ±1 Operators
We can discuss the operators mediating ∆B = 1 processes on the same footing as B0–B0
± d d
mixing. In the presence of new physics, the corresponding low-energy effective Hamiltonian
can also be composed in I = 0 and I = 1 pieces, as in (7). In Section 2, we had a closer look
at the Standard-Model contributions, arising from current–current, QCD and EW penguin
operators. New physics may affect the corresponding Wilson coefficients, and may introduce
new dim-6 operators, modifying (20) as follows:
A(B+ J/ψK+) = A(0) 1+ r(k)eiδ0(k)eiϕ(0k) r(j)eiδ1(j)eiϕ(1j) (31)
→ SM 0 − 1 
k j
X X
 
A(B0 J/ψK0) = A(0) 1+ r(k)eiδ0(k)eiϕ(0k) + r(j)eiδ1(j)eiϕ(1j) . (32)
d → SM 0 1 
k j
X X
1Note that we do not write the Inami–Limfunction comingfrom the box diagram;this function is of order
one and is ignored in our estimates. A similar comment applies to perturbative QCD corrections.
5
(k) (j) (k) (j)
Here r and r correspond to the I = 0 and I = 1 pieces, respectively, δ and δ are
0 1 0 1
(k) (j)
CP-conserving strong phases, and ϕ and ϕ the corresponding CP-violating weak phases.
0 1
The amplitudes for the CP-conjugate processes can be obtained straightforwardly from (31)
and (32) by reversing the signs of the weak phases. The labels k and j distinguish between
different new-physics contributions to the I = 0 and I = 1 sectors.
For the following discussion, we have to make assumptions about the size of a possible
new-physics piece. We shall assume that the new-physics contributions to the I = 0 sector are
smaller compared to the leading Standard-Model amplitude (21) by a factor of order λ, i.e.
(k)
r = (λ). (33)
0 O
In the case where the new-physics effects are even smaller, it is difficult to disentangle them
from the Standard-Model contribution; this will be addressed, together with several other
scenarios, in Section 5. Parametrizing the new-physics amplitudes again by a scale Λ, we have
G M2 G
F W λ F λ2A , (34)
√2 Λ2 ∼ "√2 #
corresponding to Λ 1TeV. Consequently, as in the example given after (30), also here we
∼
have a generic new-physics scale in the TeV regime.
As far as possible new-physics contributions to the I = 1 sector are concerned, we assume
a similar “generic strength” of the corresponding operators. However, in comparison with the
I = 0 pieces, the matrix elements of the I = 1 operators, having the general flavour structure
(uu dd)(bs), (35)
Q ∼ −
are expected to suffer from a dynamical suppression. As in (18) and (19), we shall assume
that this brings another factor of λ into the game, yielding
(j) 2
r = (λ ). (36)
1 O
Employing this kind of counting, the new-physics contributions to the I = 1 sector would be
enhanced by a factor of (λ) with respect to the I = 1 Standard-Model pieces. This may
O
actually be the case if new physics shows up, for example, in EW penguin processes.
Consequently, we obtain
(0) 2 3
A(B J/ψK) = A 1+ (λ)+ (λ )+ (λ ) . (37)
→ SM O O O
(cid:20) (cid:21)
NPI=0 NPI=1 SM
In the presence of large rescattering effects, the| a{zss}um|ed{zdy}nam| i{czal}suppression through a
factor of (λ) would no longer be effective, thereby modifying (37) as follows:
O
(0) 2
A(B J/ψK) = A 1+ (λ)+ (λ)+ (λ ) . (38)
→ |res. SM O O O
(cid:20) (cid:21)
NPI=0 NPI=1 SM
However, as we have noted above, we do not consid|e{rzt}his|as{za}ver|y{lzike}ly scenario, and shall
3
use (37) in the following discussion, neglecting the Standard-Model pieces of (λ ), which are
O
not under theoretical control.
Concerning the analysis of CP violation, it is obvious that possible weak phases appearing
inthenew-physics contributions playthekeyrole. Aswasthecaseforthe∆B = 2operators,
±
also the ∆B = 1 operators could carry such new weak phases, which would then affect the
±
CP-violating B J/ψK observables.
→
6
4 Observables of B → J/ψK Decays
The decays B+ J/ψK+, B0 J/ψK0 and their charge conjugates provide a set of four
→ d →
decay amplitudes A . Measuring the corresponding rates, we may determine the A 2. Since
i i
| |
we are not interested in the overall normalization of the decay amplitudes, we may construct
three independent observables with the help of the A 2:
i
| |
A(B+ J/ψK+) 2 A(B− J/ψK−) 2
(+)
| → | −| → | (39)
ACP ≡ A(B+ J/ψK+) 2 + A(B− J/ψK−) 2
| → | | → |
A(B0 J/ψK0) 2 A(B0 J/ψK0) 2
dir | d → | −| d → | (40)
ACP ≡ A(B0 J/ψK0) 2 + A(B0 J/ψK0) 2
| d → | | d → |
A(B J/ψK) 2 A(B± J/ψK±) 2
d
B h| → | i−h| → | i, (41)
≡ A(B J/ψK) 2 + A(B± J/ψK±) 2
d
h| → | i h| → | i
where the “CP-averaged” amplitudes are defined as follows:
1
A(B J/ψK) 2 A(B0 J/ψK0) 2 + A(B0 J/ψK0) 2 (42)
h| d → | i ≡ 2 | d → | | d → |
h i
1
A(B± J/ψK±) 2 A(B+ J/ψK+) 2 + A(B− J/ψK−) 2 . (43)
h| → | i ≡ 2 | → | | → |
h i
If we consider the neutral decays B J/ψK , where the final state is a CP eigenstate with
d S
→
eigenvalue 1, thefollowing time-dependent CPasymmetry provides anadditionalobservable,
−
which is due to interference between B0–B0 mixing and decay processes [3]:
d d
Γ(B0(t) J/ψK ) Γ(B0(t) J/ψK )
d → S − d → S = dir cos(∆M t)+ mix sin(∆M t). (44)
Γ(B0(t) J/ψK )+Γ(B0(t) J/ψK ) ACP d ACP d
d → S d → S
Here the rates correspond to decays of initially, i.e. at time t = 0, present B0- or B0-mesons,
d d
and ∆M denotes the mass difference between the B mass eigenstates. The “direct” CP-
d d
violating contribution dir was already introduced in (40), and the “mixing-induced” CP
ACP
asymmetry is given by
2Imξ
mix = , (45)
ACP 1+ ξ 2
| |
with
ξ = e−iφ 1+ kr0(k)eiδ0(k)e−iϕ(0k) + jr1(j)eiδ1(j)e−iϕ(1j) . (46)
1+Pkr0(k)eiδ0(k)e+iϕ(0k) +Pjr1(j)eiδ1(j)e+iϕ(1j)
 
In (46), we have used the parametrPization (32) to exprePss the corresponding decay amplitudes.
The CP-violating weak phase φ is given by φ = φ +φ , where φ was introduced in (27),
M K M
and φ is the weak K0–K0 mixing phase, which is negligibly small in the Standard Model.
K
Owing to the small value of the CP-violating parameter ε of the neutral kaon system, φ
K K
can only be affected by very contrived models of new physics [20].
7
In order to search for new-physics effects in theB J/ψK system, it isuseful to introduce
→
the following combinations of the observables (39) and (40):
1 1
S dir + (+) , D dir (+) . (47)
≡ 2 ACP ACP ≡ 2 ACP −ACP
h i h i
Using the parametrizations (31) and (32), and assuming the hierarchy in (37), we obtain
(k) (k) (k) (l) (l) (l) 2
S = 2 r sinδ sinϕ 1 2 r cosδ cosϕ = (λ)+ (λ ) (48)
− " 0 0 0 #" − 0 0 0 # O O
k l
X X
(j) (j) (j) 2
D = 2 r sinδ sinϕ = (λ ) (49)
− 1 1 1 O
j
X
(j) (j) (j) 2
B = +2 r cosδ cosϕ = (λ ), (50)
1 1 1 O
j
X
3
where terms of (λ ), including also a Standard-Model contribution, which is not under
O
theoretical control, have been neglected. Note that if the dynamical suppression of the I = 1
contributions would be larger, B and D would be further suppressed relative to S.
The corresponding expression for the mixing-induced CP asymmetry (45) is rather com-
plicated and not very instructive. Let us give it for the special case where the new-physics
contributions to the I = 0 and I = 1 sectors involve either the same weak or strong phases:
mix = sinφ 2r cosδ sinϕ cosφ 2r cosδ sinϕ cosφ
ACP − − 0 0 0 − 1 1 1
+r2 (1 cos2ϕ )sinφ+cos2δ sin2ϕ cosφ = sinφ+ (λ)+ (λ2). (51)
0 − 0 0 0 − O O
h i
Expressions (48)–(50) also simplify in this case:
S = 2r sinδ sinϕ +r2sin2δ sin2ϕ , D = 2r sinδ sinϕ , B = 2r cosδ cosϕ . (52)
− 0 0 0 0 0 0 − 1 1 1 1 1 1
In the following section, we discuss the search for new physics with these observables in more
detail, and have also a closer look at the present experimental situation.
5 Discussion
Let us begin our discussion by turning first to the present experimental situation. Concerning
the direct CP asymmetries (39) and (40), we have
(13 14)% (BaBar [6])
dir = (26 19)% (BaBar [6]), (+) = ± (53)
ACP ± ACP (( 1.8 4.3 0.4)% (CLEO [21]).
− ± ±
Thepresentstatusofthemixing-inducedB J/ψK CPasymmetry(45)isgivenasfollows:2
d S
→
0.79+0.41 (CDF [5])
−0.44
mix = 0.10 0.42 (BaBar [6]) (54)
−ACP − ±
0.49+0.53 (Belle [7]).
−0.57
2The values for sin2β reported in [6, 7] are dominated by Bd → J/ψKS, but actually correspond to an
averageover various modes.
8