Improved Estimates of The $B_{(s)}\to V V$ Decays in Perturbative QCD Approach

0.33 MB·English

by  Zhi-Tian Zou

DESY 15-001
Improved Estimates of The B V V Decays
(s)
→
in Perturbative QCD Approach
Zhi-Tian Zoua∗, Ahmed Alib†, Cai-Dian Lu¨ c‡, Xin Liud, Ying Lia§
a. Department of Physics, Yantai University, Yantai 264005,China
b. Theory Group, Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, FRG
5 c. Institute of High Energy Physics and Theoretical Physics
1
0 Center for Science Facilities, CAS, Beijing 100049, China
2

n d. School of Physics and Electronic Engineering,
a
J Jiangsu Normal University, Xuzhou 221116, China

6

Abstract
]
h
p We reexamine the branching ratios, CP-asymmetries, and other observables in a large number
-
p
of B VV(q = u,d,s) decays in the perturbative QCD (PQCD) approach, where V denotes a
e q
→
h
[ light vector meson (ρ,K∗,ω,φ). The essential difference between this work and the earlier similar

2
works is of parametric origin and in the estimates of the power corrections related to the ratio
v
4
r2 = m2 /m2 (i= 2,3)(m andm denotethemassesofthevectorandB meson,respectively). In
8 i Vi B V B
7
0 particular, we use up-to-date distribution amplitudes for the final state mesons and keep the terms
0
. proportional to the ratio r2 in our calculations. Our updated calculations are in agreement with
1 i
0
the experimental data, except for a limited numberof decays which we discuss. We emphasize that
5
1
: thepenguinannihilationandthehard-scatteringemissioncontributionsareessentialtounderstand
v
Xi the polarization anomaly, such as in the B φK∗ and Bs φφ decay modes. We also compare
→ →
r
a our results with those obtained in the QCD factorization (QCDF) approach and comment on the
similarities and differences, which can be used to discriminate between these approaches in future
experiments.
PACS numbers: 13.25.Hw, 12.38.Bx
∗ [email protected]
† [email protected]
‡ [email protected]
§ [email protected]
1
I. INTRODUCTION
Exclusive B (q = u,d,s)mesondecays, especially B VV modes, whereV standsfora
q q
→
light vector meson (ρ,K∗,ω,φ), have aroused a great deal of interest for both theorists [1–9]
and in experiments [10]. In contrast to the scalar and pseudoscalar mesons, vector mesons
canbeproducedinseveralpolarizationstates. Thus, thefractionofagivenpolarizationstate
is an interesting observable, apart from the decay widths. Phenomenology of the B VV
q
→
decays offers rich opportunities for our understanding of the mechanism for hadronic weak
decays and CP asymmetry and searching for the effect of new physics beyond the standard
model. In general, the underlying dynamics for such decays is extremely complicated, but
in the heavy quark limit (m ), it is greatly simplified due to the factorization of the
b
→ ∞
hadronic matrix elements in terms of the decay constants and form factors. Based on this, a
number of two-body hadronic B decays had been calculated in this so-called naive factoriza-
tion approach [11]. However these calculations encounter three major difficulties: (i) for the
so-called penguin-dominated, and also for the color-suppressed tree-dominated decays, the
predicted branching ratiosaresystematically below themeasurements, (ii)thisapproachcan
not account for the direct CP asymmetries measured in experiments, and (iii) the predic-
tions of transverse polarization fraction in penguin-dominated charmless B VV decays
q
→
are too small to explain the data, in which large such fractions are measured. All these in-
dicate that this factorization approach needs improvements, for example by including some
more perturbative QCD contributions [12]. In the current market, there are essentially three
approaches to implement perturbative improvements: QCD factorization (QCDF) [13, 14],
perturbativeQCDapproach(PQCD)[15], andthesoft-collineareffective theory(SCET) [16].
All these frameworks have been employed in the literature to quantitatively study the dy-
namics of the B PP,VP,VV decays, having light pseudoscalar (P)and/or Vector (V)
q
→
mesons in the final states.
In the B VV decays, as the B meson is heavy, the vector mesons are energetic
q q
→
with E m /2. As the spectator quark (u,d or s) in the B meson is soft, a hard gluon
V B q
≃
exchange is needed to kick it into an energetic one to form a fast moving light vector meson.
The theoretical picture here is that a hard gluon from the spectator quark connects with the
other quarks of the four-quark operators of the weak interaction [15]. The underlying theory
is thus a six-quark effective theory, and can be perturbatively calculated [17]. In contrast
2
to the other two approaches (QCDF and SCET), the PQCD approach is based on the k
T
factorization formalism [18–20]. The basic idea here is to take into account the transverse
momentum k of the valence quarks in the hadrons, as a result of which the end-point
T
singularity in the collinear factorization (employed in the QCDF approach) can be avoided.
On the other hand, the transverse momentum dependence introduces an additional energy
scale leading to double logarithms in QCD corrections. These terms could be resummed
through the renormalization group approach, which results in the appearance of the Su-
dakov form factor. This form factor effectively suppresses the end-point contribution of the
distribution amplitude of the mesons in the small transverse momentum region, making the
calculation in the PQCD approach reliable. It is worth mentioning that in this framework,
the so-called annihilation diagrams are also perturbatively calculable without introducing
additional parameters [21, 22]. The PQCD approach has been successfully used to study a
number of pure annihilation type decays, and these predictions were confirmed subsequently
in experiments [8, 22–25]. Thus, in our view, this method is reliable in dealing with the pure
annihilation-type and annihilation-dominated decays as well.
Several years ago, H. Y. Cheng andC. K. Chua updated[4, 5] the previous predictions [1–
3] for B VV decays in the QCDF factorization approach by taking the transverse
q
→
polarization contributions into account, and using the updated values of the parameters in
the input wave functions and the form factors. In the PQCD framework, although many
studiesofthetwo-bodyB -decaysareavailable[7–9], areappraisalisneededforthefollowing
q
reasons: (i) In the previous studies, the terms proportional to “r2 = m2 /m2 (i = 2,3)” have
i Vi B
been omitted in the amplitudes, especially in the denominator of the propagators of virtual
quarks and gluons. As we point out later, these terms do bring the earlier PQCD predictions
in better accord in terms of the measured observables in some problematic cases, such as
the B φK∗ and B φφ decays, (ii) recent progress in the study of the distribution
s
→ →
amplitudes of the vector meson, especially for the φ meson, undertaken in the context of
the QCD sum rules, may significantly impact on some of the calculations done earlier, and
(iii) Experimental data for some of the B VV decays, such as the branching ratio
q
→
and the polarization fractions of B φφ, are now available. In addition, we work out
s
→
a number of observables, such as φ , φ , A0 , A⊥ , ∆φ and ∆φ for the first time in
k ⊥ CP CP k ⊥
PQCD. Among others, we revisit the B ρ (ω)φ decay modes, the direct CP asymmetry
→
of which could help us distinguish the PQCD and competing approaches. A related issue
3
is the large fraction of the transverse polarization observed in some of these decays. In
the PQCD framework, penguin-annihilation contribution is the key to understanding this
phenomenon. Especially, the chirally enhanced (S-P)(S+P) penguin-annihilation gives rise
to large transverse polarizations. Together with the hard spectator-scattering contributions,
this could help solve the transverse polarization puzzle in the penguin-dominated B VV
q
→
decays.
This work is organized as follows. In Sec. II, we outline the framework of the PQCD
approach and specify the various input parameters, such as the wave functions and decay
constants. Details of the perturbative calculations for the B VV decays are presented
q
→
in in Sec. III, and the various input functions are given in the Appendix. Numerical re-
sults of our calculations are presented in Sec. IV and compared in detail with the available
experiments and earlier theoretical works. Finally, a short summary is given in Sec. V.
II. FORMALISM AND WAVE FUNCTION
Our goal is to calculate the transition matrix elements:
VV B , (1)
eff q
M ∝ h |H | i
with the weak effective Hamiltonian written as [26]
eff
H
10
G
= F V∗V [C (µ)Oq(µ)+C (µ)Oq(µ)] V∗V C (µ)O (µ) . (2)
Heff √2 ( ub uX 1 1 2 2 − tb tX " i i #)
i=3
X
Here, V and V (X = d,s) are the CKM matrix elements, C (µ) are the effective
ub(X) tb(X) i
Wilson coefficient calculated at the scale µ, and the local four-quark operators O (j =
j
1,...,10) are defined and classified as follows:
Current-current (tree) operators,
•
Ou = (¯b u ) (u¯ X ) , Ou = (¯b u ) (u¯ X ) , (3)
1 α β V−A β α V−A 2 α α V−A β β V−A
QCD penguin operators,
•
O = (¯b X ) (q¯′q′) , O = (¯b X ) (q¯′q′ ) , (4)
3 α α V−A β β V−A 4 α β V−A β α V−A
q′ q′
X X
O = (¯b X ) (q¯′q′) , O = (¯b X ) (q¯′q′ ) , (5)
5 α α V−A β β V+A 6 α β V−A β α V+A
q′ q′
X X
4
Electroweak penguin operators,
•
3 3
O7 = 2(¯bαXα)V−A eq′(q¯β′qβ′)V+A, O8 = 2(¯bαXβ)V−A eq′(q¯β′qα′ )V+A, (6)
q′ q′
X X
3 3
O9 = 2(¯bαXα)V−A eq′(q¯β′qβ′)V−A, O10 = 2(¯bαXβ)V−A eq′(q¯β′qα′ )V−A, (7)
q′ q′
X X
with the SU(3) color indices α and β and the active quarks q′ = (u,d,s,c). The left-handed
(right-handed) current V A are defined as γ (1 γ ). Following [27], we introduce the
µ 5
± ±
following combinations a of the Wilson coefficients:
i
a = C +C /3, a = C +C /3,
1 2 1 2 1 2
a = C +C /3, i = 3,5,7,9/4,6,8,10. (8)
i i i±1
IntheperturbativeapproachtohadronicB decays, several typical scales areencountered
q
with large logarithms involving the ratios of these scales. They are resummed using the
renormalization group (RG) techniques. Standard model specifies the Wilson coefficients at
the electroweak scale m , the W boson mass, and the RG equations enable us to evaluate
W
the dynamical effects in scaling the Wilson coefficients in Eq. (2) from m to m , the
W b
b-quark mass. The physics between the scale m and the factorization scale Λ , taken
b h
typically as Λ m Λ , can be calculated perturbatively and included in the so-called
h b QCD
≃
hard kernel in thepPQCD approach. The soft dynamics below the factorization scale Λ
h
is nonperturbative and is described by the hadronic wave functions of the mesons involved
in the decays B VV. Finally, based on the factorization ansatz, the decay amplitudes
q
→
are described by the convolution of the Wilson coefficients C(t), the hard scattering kernel
H(x ,b ,t) and the light-cone wave functions Φ (x ,b ) of the mesons [28]:
i i Mi,B j j
dx dx dx b db b db b db
1 2 3 1 1 2 2 3 3
A ∼
Z
Tr C(t)Φ (x ,b )Φ (x ,b )Φ (x ,b )H(x ,b ,t)S (x )e−S(t) , (9)
× B 1 1 M2 2 2 M3 3 3 i i t i
(cid:2) (cid:3)
where Tr denotes the trace over Dirac and color indices, b are the conjugate variables of
i
the quark transverse momenta k , x are the longitudinal momentum fractions carried by
iT i
the quarks, and t is the largest scale in the hard kernel H(x ,b ,t). The jet function S (x )
i i t i
coming fromthethreshold resummation of thedouble logarithms ln2x smears the end-point
i
singularities in x [29]. The Sudakov form factor e−S(t) from the resummation of the double
i
5
logarithms suppresses the soft dynamics effectively i.e. the long distance contributions in
the large-b region [30, 31].
In the PQCD approach, both the initial and the final state meson wave functions are
importantnon-perturbativeinputs. ForB (q = u,d,s)meson, thelight-conematrixelement
q
could be decomposed as [32, 33]
i n/ v/
d4zeik·z 0 q (z)¯b (0) B (P ) = (P/ +M )γ φ (k) − (k) , (10)
Z h | β α | q Bq i √6 ( Bq Bq 5" Bq − √2φ¯Bq #)βα
where n = (1,0,~0 ) and v = (0,1,~0 ) are the unit vectors of the light-cone coordinate
T T
system. Corresponding to the two Lorentz structures in the B meson distribution ampli-
q
¯
tudes, there are two wave functions φ (k) and φ (k), obeying the following normalization
Bq Bq
conditions:
d4k f d4k
φ (k) = Bq , φ¯ (k) = 0, (11)
(2π)4 Bq 2√6 (2π)4 Bq
Z Z
where f is the decay constant of the B meson. Due to the numerical suppression, the
Bq q
¯
contribution of φ is often neglected. Finally, for convenience, the wave function of B meson
B
can be expressed as:
i
Φ (x,b) = (P/ +M )γ φ (x,b), (12)
Bq √6 Bq Bq 5 Bq
with the light-cone distribution amplitude
M2 x2 1
φ (x,b) = N x2(1 x2)exp Bq w2b2 , (13)
Bq Bq − − 2ω − 2 q
" q #
where N is a normalization factor and ω is a shape parameter. For B0(B±) meson, we
Bq q
use ω = 0.4 0.04 GeV, which is determined by the calculation of form factor and other
q
±
well known decay modes [18, 19, 34]. Taking into account the small SU(3) breaking and
the fact that the s quark is heavier than the u or d quark, we use the shape parameter
ω = 0.5 0.05 GeV for the B meson, indicating that the s quark momentum fraction is
s s
±
larger than that of the u or d quark in the B± or B0 meson [8].
The light vector meson is treated as a light-light quark-antiquark system with the mo-
mentum P2 = M2, andits polarizationvectors ǫinclude onelongitudinal polarizationvector
V
ǫ and two transverse polarization vectors ǫ , which are defined in [7, 35]. Up to twist-3,
L T
6
the vector meson wave functions are given by [36]:
1
ΦL = M /ǫ φ (x) + ǫ/ P/φt (x)+M φs (x)
V √6 V L V L V V V
1 (cid:2) (cid:3)
Φ⊥ = M /ǫ φv (x) + ǫ/ P/φT(x) + M iǫ γ γµǫνnρvσφa(x) , (14)
V √6 V T V T V V µνρσ 5 T V
(cid:2) (cid:3)
for the longitudinal polarization and the transverse polarization, respectively. Here ǫ is
µνρσ
Levi-Civita tensor with the convention ǫ0123 = 1.
The twist-2 distribution amplitudes are given by
3f
V k 3/2 k 3/2
φ (x) = x(1 x) 1+a C (t)+a C (t) , (15)
V √6 − 1V 1 2V 2
h i
3fT
φT(x) = V x(1 x) 1+a⊥ C3/2(t)+a⊥ C3/2(t) , (16)
V √6 − 1V 1 2V 2
h i
(T)
with t = 2x 1, and f are the decay constants of the vector meson, which for V =
− V
ρ,ω,K∗,φ are shown numerically in Table I. For the Gegenbauer moments, we use the
following values[36, 37]:
k(⊥) k(⊥) k(⊥) k(⊥)
a = a = a = 0, a = 0.03 0.02(0.04 0.03) ,
1ρ 1ω 1φ 1K∗ ± ±
k(⊥) k(⊥) k(⊥)
a = a = 0.15 0.07(0.14 0.06) a = 0 (0.20 0.07) ,
2ρ 2ω ± ± 2φ ±
k(⊥)
a = 0.11 0.09 (0.10 0.08) . (17)
2K∗ ± ±
For the twist-3 distribution amplitudes, for simplicity, we adopt the asymptotic forms
3fT 3fT
φt (x) = V t2, φs (x) = V ( t),
V 2√6 V 2√6 −
3f 3f
φv (x) = V (1+t2), φa(x) = V ( t). (18)
V 8√6 V 4√6 −
TABLE I: Input values of the decay constants of the light vector mesons, taken from [37]
vector f (MeV) fT(MeV)
V V
ρ 216 3 165 9
± ±
. ω 187 5 151 9
± ±
K∗ 220 5 185 10
± ±
φ 215 5 186 9
± ±
7
V2
¯bq
V3
q
a b c d
¯b
V2
V3
q
e f g h
FIG. 1: Leading order Feynman diagrams contributing to the B VV decays in PQCD
(s)
→
III. PERTURBATIVE CALCULATION
Atleadingorder, thereareeighttypesofFeynmandiagramscontributingtotheB VV
q
→
decays, which are presented in Fig.1. The first row shows the emission-type diagrams, with
the first two contributing to the usual form factor; the last two are the so-called hard-
scattering emission diagrams. In fact, the first two diagrams are the only contributions
calculated in the naive factorization approach. The second row shows the annihilation-type
diagrams, with the first two factorizable and the last two nonfactorizable.
In the following, we shall give the general factorization amplitudes for these B VV
q
→
decays. Weusethesymbol LLtodescribetheamplitudeofthe(V A)(V A)operators,LR
− −
denotestheamplitudeofthe(V A)(V +A)operatorsandSP denotesthatof(S P)(S+P)
− −
operators resulting from the Fierz transformation of the (V A)(V + A) operators. For
−
the B VV decays, both the longitudinal polarization and the transverse polarization
q
→
contribute. The amplitudes can be decomposed as follows:
(ǫ ,ǫ ) = i L +i(ǫT∗ ǫT∗) N +(ǫ nµvνǫT∗αǫT∗β) T, (19)
A 2 3 A 2 · 3 A µναβ 2 3 A
where L is the longitudinally polarized decay amplitude, T and N are the transversely
A A A
polarized contributions, and ǫT is the transverse polarization vector of the vector meson.
The longitudinal polarizationamplitudes forthe factorizableemission diagrams inFig.(a)
8
and (b) are as follows:
1 1/Λ
LL(LR),L = 8πC M4f dx dx b db b db φ (x ,b ) [( 1+x )φ (x )
Aef − F B V2 1 3 1 1 3 3 B 1 1 { − 3 3 3
Z0 Z0
+r (2x 1)(φs(x )+φt(x )) E (t )h (x ,x (1 r2),b ,b )
3 3 − 3 3 3 3 ef a ef 1 3 − 2 1 3
+2r φs(x )E (t )h (x ,x (1 (cid:3) r2),b ,b ) , (20)
3 3 3 ef b ef 3 1 − 2 3 1
(cid:9)
where r = MVi and C = 4/3 is a color factor. The functions h , t , and E can be found
i MB F ef a,b ef
in Appendix A. There is no (S P)(S +P) type amplitude, as a vector meson can not be
−
produced through this type of operators. In the PQCD approach, the traditional emission
contribution is also the dominant one. Unknown higher order perturbative QCD corrections
will influence the emission contributions as well as those from other topologies. At present,
althoughthenext-to-leading order (NLO) contributions have not beencompleted, the vertex
correction has been done and is used to improve the predictions for the decays B πρ(ω)
→
and the B π form factors [38], which allows us to estimate the stability of the emission
→
diagram in NLO. The results quoted below are based on the leading order calculations, but
we also estimate the uncertainties from the partial NLO contributions based on the available
results, as explained in Sec. IV.
The last two diagrams in the first row in Fig.1 are the hard-scattering emission diagrams,
whose contributions are given below:
2 1 1/Λ
LL,L = 16 πC M4 d[x] b db b db φ (x ,b )φ (x )
Menf − 3 F B 1 1 2 2 B 1 1 2 2
r Z0 Z0
(x 1)φ (x )+r x (φs(x ) φt(x )) E (t )h (α,β ,b ,b )
× 2 − 3 3 3 3 3 3 − 3 3 enf c enf 1 1 2
+(cid:8)((cid:2)x +x )φ (x ) r x (φs(x )+φt(x ))(cid:3)E (t )h (α,β ,b ,b ) ,(21)
2 3 3 3 − 3 3 3 3 3 3 enf d enf 2 1 2
(cid:2) (cid:3) (cid:9)
2 1 1/Λ
LR,L = 16 πC r M4 d[x] b db b db φ (x ,b )
Menf 3 F 2 B 1 1 2 2 B 1 1
r Z0 Z0
r ((x x 1)(φs(x )φs(x ) φt(x )φt(x ))
× 3 2 − 3 − 2 2 3 3 − 2 2 3 3
+(cid:8)(x(cid:2) +x 1)(φt(x )φs(x ) φs(x )φt(x )))
2 3 − 2 2 3 3 − 2 2 3 3
+(x 1)φ (x )(φs(x )+φt(x )) E (t )h (α,β ,b ,b )
2 − 3 3 2 2 2 2 enf c enf 1 1 2
+ r ((x x )(φt(x )φs(x )+φs(cid:3)(x )φt(x ))
3 3 − 2 2 2 3 3 2 2 3 3
+((cid:2)x +x )(φs(x )φs(x )+φt(x )φt(x )))
2 3 2 2 3 3 2 2 3 3
+x φ (x )(φs(x ) φt(x )) E (t )h (α,β ,b ,b ) , (22)
2 3 3 2 2 − 2 2 enf d enf 2 1 2
(cid:3) (cid:9)
9
2 1 1/Λ
SP,L = 16 πC M4 d[x] b db b db φ (x ,b )φ (x )
Menf − 3 F B 1 1 2 2 B 1 1 2 2
r Z0 Z0
φ (x )(x x 1)+r x (φs(x )+φt(x )) E (t )h (α,β ,b ,b )
× 3 3 2 − 3 − 3 3 3 3 3 3 enf c enf 1 1 2
+(cid:8)φ(cid:2) (x )x +r x (φt(x ) φs(x )) E (t )h (cid:3) (α,β ,b ,b ) . (23)
3 3 2 3 3 3 3 − 3 3 enf d enf 2 1 2
(cid:2) (cid:3) (cid:9)
The functions t , E , h , α, β for the nonfactorizable emission diagrams are also listed
c(d) enf enf i
in Appendix A. As is well known, the hard-scattering emission diagrams with a light meson
(pseudoscalar or vector) are suppressed. This can be seen fromthe figures (c) and (d), which
are symmetrical. But, compared with the figure (d), the anti-quark propagator in figure (c)
has an additional negative sign. As a result, the two contributions cancel each other.
Figures (e) and (f) are the factorizable annihilation diagrams, whose factorizable contri-
butions are listed below:
1 1/Λ
LL(LR),L = 8C πf M4 dx dx b db b db
Aaf F B B 2 3 2 2 3 3
Z0 Z0
φ (x )φ (x )(x 1)+2r r φs(x )(x φt(x ) (x 2)φs(x ))
× 2 2 3 3 3 − 2 3 2 2 3 3 3 − 3 − 3 3
E(cid:8)(cid:2)(t )h (α ,β,b ,b ) (cid:3)
af e af 1 2 3
·
x φ (x )φ (x )+2r r φs(x )((x 1)φt(x )+(x +1)φs(x ))
− − 2 2 2 3 3 2 3 3 3 2 − 2 2 2 2 2
E(cid:2) (t )h (α ,β,b ,b ) , (cid:3) (24)
af f af 2 3 2
· }
1 1/Λ
SP,L = 16C f πM4 dx dx b db b db
Aaf − F B B 2 3 2 2 3 3
Z0 Z0
2r φ (x )φs(x )+r (x 1)φ (x )(φs(x )+φt(x ))
× 2 3 3 2 2 3 3 − 2 2 3 3 3 3
E(cid:8)(cid:2)(t )h (α ,β,b ,b ) (cid:3)
af e af 1 2 3
·
+ 2r φ (x )φs(x )+r x φ (x )(φt(x ) φs(x ))
3 2 2 3 3 2 2 3 3 2 2 − 2 2
E(cid:2) (t )h (α ,β,b ,b ) , (cid:3) (25)
af f af 2 3 2
· }
and the related scales and the hard functions listed in Appendix A
The last two diagrams in Fig.1 are the nonfactorizable annihilation diagrams. The ex-
10