Rare B -> K^* l^+ l^- Decay in Light Cone QCD

19 Pages·0.21 MB·

by  T. M. Aliev

RARE B K∗l+l− DECAY IN LIGHT CONE QCD
→
T. M. ALI˙EV ∗, A. O¨ZPI˙NECI˙ † and M. SAVCI
Physics Department, Middle East Technical University
7
9 06531 Ankara, Turkey
9
1
February 1, 2008

n
a
J

2

2 Abstract
v
0
We investigate the transition formfactors for the B K∗l+l−(l = µ, τ) decay in
8
→
4
thelightconeQCD.Itisfoundthatthelightconeand3-pointQCDsumrulesanalyses
2
1
for some of the formfactors for the decay B K∗l+l− lead to absolutely different
6
→
9
q2 dependence. The invariant dilepton mass distributions for the B K∗µ+µ− and
/
h →
p B K∗τ+τ− decays and final lepton longitudinal polarization asymmetry, which
- →
p includes both short and long-distance contributions, are also calculated.
e
h
:
v
i
X
PACS numbers: 13.20.He, 11.55.Hx, 12.38.Cy
r
a
∗e-mail:[email protected]
†e-mail:[email protected]
1 Introduction
Experimental observation [1] of the inclusive and exclusive radiative decays B X γ and
s
→
B K∗γ stimulated the study of rare B decays on a new footing. These flavor changing
→
neutral current (FCNC) b s transitions in the SM do not occur at tree level and appear
→
only at loop level. Therefore the study of these rare B-meson decays can provide a means of
testing the detailed structure of the SM at the loop level. These decays are also very useful
for extracting the values of the Cabibbo-Kobayashi-Maskava (CKM) matrix elements [2],
as well as for establishing new physics beyond the Standard Model [3].
Currently, the main interest on rare B-meson decays is focused on decays for which the
SM predicts large branching ratios and can be potentially measurable in the near future.
The rare B Kl+l− and B K∗l+l− decays are such decays. The experimental situation
→ →
for these decays is very promising [4], with e+e− and hadron colliders focusing only on
the observation of exclusive modes with l = e, µ and τ final states, respectively. At
quark level the process b sl+l− takes place via electromagnetic and Z penguins and W
→
box diagrams and are described by three independent Wilson coefficients C , C and C .
7 9 10
Investigations allow us to study different structures, described by the above mentioned
Wilson coefficients. In the SM, the measurement of the forward-backward asymmetry
and invariant dilepton mass distribution in b ql+l− (q = s, d) provide information on
→
the short distance contributions dominated by the top quark loops and are essential in
separating the short distance FCNC process from the contributing long distance effects [5]
and also are very sensitive to the contributions from new physics [6]. Recently it has been
emphasized by Hewett [7] that the longitudinal lepton polarization, which is another parity
violating observable, is also an important asymmetry and that the lepton polarization in
b sl+l− will be measurable with the high statistics available at the B-factories currently
→
under construction. However, in calculating the Branching ratios and other observables in
hadron level, i.e. for B K∗l+l− decay, we have the problem of computing the matrix
→
element of the effective Hamiltonian, , between the states B and K∗. But this problem
eff
H
is related to the non-perturbative sector of QCD.
These matrix elements, in the framework of different approaches such as chiral theory
[8], three point QCD sum rules method [9], relativistic quark model by the light-front
formalism [10, 11], have been investigated. The aim of this work is the calculation of these
matrix elements in light cone QCD sum rules method and to study the lepton polarization
asymmetry for the exclusive B K∗l+l− decays.
→
The effective Hamiltonian for the b sl+l− decay, including QCD corrections [12-14]
→
can be written as
4G 10
= FV V∗ C (µ)O (µ) (1)
Heff √2 tb ts i i
i=1
X
which is evolved from the electroweak scale down to µ m by the renormalization group
b
∼
equations. Here V represent the relevant CKM matrix elements, and O are a complete
ij i
set of renormalized dimension 5 and 6 operators involving light fields which govern the
b s transitions and C (µ) are the Wilson coefficients for the corresponding operators.
i
→
1
The explicit forms of C (µ) and O (µ) can be found in [12-14]. For b sl+l− decay, this
i i
→
effective Hamiltonian leads to the matrix element
G α C
= F V V∗ Ceffs¯ γ b ¯lγµl+C s¯ γ b ¯lγµγ l 2 7s¯iσ qν(m R+m L)b ¯lγµl (2)
M √2π tb ts" 9 L µ L 10 L µ L 5 − q2 µν b s #
where q2 is the invariant dilepton mass, and L(R) = [1 (+)γ ]/2 are the projection
5
operators. The coefficient Ceff(µ, q2) C (µ)+Y(µ, q2),−where the function Y contains
9 ≡ 9
the contributions from the one loop matrix element of the four-quark operators and can be
found in [12-14]. Note that the function Y(µ, q2) contains both real and imaginary parts
(the imaginary part arises when the c-quark in the loop is on the mass shell).
The B K∗l+l− decay also receives large long distance contributions from the cascade
→
process B K∗ψ(′) K∗l+l−. These contributions are taken into account by introducing
→ →
a Breit-Wigner form of the resonance propogator and this procedure leads to an additional
contribution to Ceff of the form [15]
9
2π m Γ(V l+l−)
V
→
− α2 (q2 m2 ) im Γ
V=Xψ, ψ′ − V − V V
As we noted earlier, for the calculation of the branching ratios for the exclusive B
→
K∗l+l− decays, first of all, we must calculate the matrix elements K∗ s¯γ (1 γ )q B
µ 5
h | − | i
and K∗ s¯iσ pν(1+γ )q B . These matrix elements can be parametrized in terms of the
µν 5
h | | i
formfactors as follows (see also [9]):
2V(q2)
K∗(p,ǫ) s¯γ (1 γ )q B(p+q) = ǫ ǫ∗νpρqσ
µ 5 µνρσ
h | − | i − mB +mK∗ −
− iǫ∗µ(mB +mK∗)A1(q2)+
A (q2)
+ i(ǫ∗q)P 2 +
µ
mB +mK∗
+ i (ǫ∗q) 2mK∗ A (q2) A (q2) q , (3)
q2 3 − 0 µ
h i
K∗(p,ǫ) s¯iσ qν(1+γ )q B(p+q) = 4 ǫ ǫ∗νpρqσT (q2)+
µν 5 µνρσ 1
h | | i
+ 2 i ǫ∗(Pq) (ǫ∗q)P T (q2)+
µ − µ 2
h q2 i
+ 2 i(ǫ∗q) q P T (q2) ,
µ µ 3
" − Pq #
(4)
where ǫ∗ is the polarization vector of K∗, p + q and p are the momentum of B and K∗
µ
and P = (2p+q) . The formfactor A (q2) can be written as a linear combination of the
µ µ 3
2
formfactors A and A (see [9]):
1 2
A (q2) = mB +mK∗ A (q2) mB −mK∗ A (q2) , (5)
3 1 2
2mK∗ − 2mK∗
with the condition A (0) = A (0). In calculating these formfactors we employ the light
3 0
cone QCD sum rules.
2 QCD Sum Rules for Formfactors
According to the QCD sum rules ideology, in order to calculate the formfactors we start by
considering the representation of a suitable correlator function in terms of hadron language
and quark-gluon language. Equating these representations we get the sum rules. For this
purpose we choose the following correlators.
Π(1)(p,q) = i d4xeiqx K∗(p) s¯(x)γ (1 γ )b(x)¯b(0)iγ q(0) 0 , (6)
µ h | µ − 5 5 | i
Z
Π(2)(p,q) = i d4xeiqx K∗(p) s¯(x)iσ qν(1+γ )b(x)¯b(0)iγ q(0) 0 . (7)
µ h | µν 5 5 | i
Z
HerethefirstcorrelatorisrelevantforthecalculationoftheformfactorsV(q2), A (q2), A (q2)
1 2
and A (q2) and the second one for T , T and T .
0 1 2 3
The main task in QCD is the calculation of the correlation functions (6) and (7). This
problem can be solved in the deep Euclidean region, where both virtualities q2 and (p+q)2
are large and negative. The virtuality of the heavy quark in the correlators (6) and (7) is
large, of order m2 (p+q)2, and one can use the perturbative expansion of its propagator in
b−
the external field of slowly varying fluctuations inside the vector meson. Then, the leading
contribution is
d4xd4k ei(q−k)x
Π(1)(p,q) = i K∗ s¯(x)γ (1 γ )(k +m )γ q(0) 0 , (8)
µ (2π)4 (m2 k2)h | µ − 5 6 b 5 | i
Z b −
d4xd4k ei(q−k)x
Π(2)(p,q) = qν K∗ s¯(x)σ (1+γ )(k +m )γ q(0) 0 . (9)
µ − (2π)4 (m2 k2) h | µν 5 6 b 5 | i
Z b −
It is obvious from the above expressions that the problem is reduced to the calculation of
the matrix elements of the gauge-invariant non-local operators, sandwiched in between the
vacuum and the meson states. These matrix elements define the vector meson light cone
wave functions. Following [16, 17] we define the meson wave functions as:
1
0 q¯(0)σ q(x) K∗(p,ǫ) = i (ǫ p ǫ p )f⊥ due−iupxφ (u,µ2) , (10)
h | µν | i µ ν − ν µ K∗ ⊥
Z0
ǫx 1
0 q¯(0)γµq(x) K∗(p,ǫ) = pµ fK∗mK∗ due−iupxφk(u,µ2)+ (11)
h | | i px
Z0
ǫx 1
+ ǫµ −pµpx!fK∗mK∗ Z0 due−iupxg⊥(v)(u,µ2) ,
1 1
h0|q¯(0)γµγ5q(x)|K∗(p,ǫ)i = − 4 ǫµνρσǫνpρxσfK∗mK∗ due−iupxg⊥(a)(u,µ2) . (12)
Z0
3
The functions φ (u, µ2) and φ (u,µ2) give the leading twist distributions in the fraction
⊥ k
of total momentum carried by the quark in the transversaly and longitudinally polarized
meson, respectively. In [17] it was shown that
3
gv(u) = 1+(2u 1)2 , (13)
⊥ 4 −
ga(u) = 6uh(1 u) , i (14)
⊥ −
which we use in the numerical analysis. For the explicit form of φ (u, µ2) we shall use the
⊥
results of [17]:
1
φ (u, µ2) = 6u(1 u) 1+a (µ)(2u 1)+a (µ) (2u 1)2 +
⊥ 1 2
− − − − 5
(cid:26) (cid:20) (cid:21)
7
+ a (µ) (2u 1)3 (2u 1) +... , (15)
3
3 − − −
(cid:20) (cid:21) (cid:27)
γn
α (µ) b
s
a (µ) = a (µ ) . (16)
n n 0
"αs(µ0)#
Here b = 11N 2n , and
3 C − 3 f
n+1 1
γ = C 1+4 , (17)
n F
 j
j=2
X
 
N2−1
where C = C .
F 2NC
As in [17], we will use the following values for the parameters appearing in eqs.(10)-(12)
and eq.(15) :
f⊥ = 210 MeV, aK∗(µ = m ) = 0.57, aK∗(µ = m ) = 1.35
K∗ 1 b 2 b −
and aK∗(µ = m ) = 0.46 ,
3 b
φ (u, µ2) = 6u(1 u). (18)
k
−
Using eqs.(10-12), we get the following results from eq.(8) and eq.(9) for the theoretical
part of the sum rules:
1 du 1
Π(µ1) = − imbfK∗mK∗ ∆ ǫ∗µg⊥(v) +2(qǫ∗)pµ∆(Φk −G⊥(v)) −
Z0 (cid:20) (cid:21)
m 1 du 1 φ
− ǫµνρσǫ∗νpρqσ" 2bfK∗mK∗ Z0 ∆2g⊥(a) +fK⊥∗Z0 du ∆⊥#−
1 φ
if⊥ du ⊥ ǫ∗(pq+p2u) p (qǫ∗) (19)
− K∗ ∆ µ − µ
Z0 h i
4
1 du 1 du
Π(µ2) = ǫµνρσǫ∗νpρqσ(mbfK⊥∗Z0 ∆φ⊥ −mK∗fK∗"Z0 ∆ (cid:16)Φk −G⊥(v)(cid:17)−
1 du 1 du
ug(v) g(a) ∆+q2 +2pqu +
− Z0 ∆ ⊥ −Z0 2∆2 ⊥ (cid:16) (cid:17)#)
1 du
+ i ǫ∗(pq) (qǫ∗)p m f⊥ φ +
h µ − µi( b K∗Z0 ∆ ⊥
(a) (a)
1 du g g u(qp)
+ mK∗fK∗Z0 ∆"−(cid:16)Φk −G⊥(v)(cid:17)+ug⊥(v) + ⊥2 + ⊥2∆ #)+
1 du p2u
+ imK∗fK∗hǫ∗µ(q2)−(qǫ∗)qµiZ0 ∆ "g⊥(v) − 2∆g⊥(a)#+
1 du
+ 2imK∗fK∗(qǫ∗) pµ(q2)−(pq)qµ ∆2 Φk −G⊥(v) , (20)
h iZ0 (cid:16) (cid:17)
where
u
Φ (u) = φ (v)dv ,
k k
−
Z0
u
(v) (v)
G (u) = g (v)dv , (21)
⊥ − ⊥
Z0
and
∆ = m2 (q +pu)2 .
b −
Letusturnourattentiontothephysicalpartofthecorrelatorfunctions(6)and(7). Writing
a dispersion relation in the variable (p+ q)2, one can separate the B meson ground state
contribution to the correlator functions Π(1) and Π(2), by inserting a complete set of states
µ µ
between the currents in (6) and (7) focusing on the term B B :
| i h |
f m2
Π(1) = B B K∗(p) s¯γ (1 γ )q B(p+q) (22)
µ m [m2 (q+p)2] h | µ − 5 | i
b B −
f m2
Π(2) = B B K∗(p) s¯iσ qα(1+γ )q B(p+q) (23)
µ m [m2 (q+p)2] h | µα 5 | i
b B −
Using the definitions of the formfactors (see eqs.(3) and (4)) in (22) and (23) and equating
these expressions to eqs.(19) and (20), we get the sum rules for the formfactors. The
remaining part of the calculation follows from the QCD sum rules procedure: perform the
Borel transformation on the variable (p+q)2 and subtract the continuum and higher states
contributions invoking quark-hadron duality. (Details of these procedures can be found in
[17-19]). After this procedure we obtain the following sum rules for the formfactors:
5
V(q2) = mB +mK∗ mb emM2B2 1duexp m2b +p2uu¯−q2u¯
2 fBm2B Zδ − uM2 !×
g(a) f⊥ φ
mbfK∗mK∗ ⊥ + K∗ ⊥ , (24)
× 2u2M2 u 
 
 
1 m m2 1 m2 +p2uu¯ q2u¯
A1(q2) = mB +mK∗ fBmb2B eMB2 Zδ duexp − b uM2− !×
g(v) f⊥ φ (m2 q2 +p2u2)
mbfK∗mK∗ ⊥ + K∗ ⊥ b − , (25)
× u 2u2 
 
 
m m2 1 m2 +p2uu¯ q2u¯
A2(q2) = − (mB +mK∗) fBmb2B eMB2 Zδ duexp − b uM2− !×
mbfK∗mK∗ Φ G(v) 1 f⊥ φ⊥ , (26)
×( u2M2 k − ⊥ − 2 K∗ u )
(cid:16) (cid:17)
A3(q2)−A0(q2) = 2mq2K∗ fBmmb2B emM2B2 Zδ1duexp −m2b +pu2Muu¯2−q2u¯!×
mbfK∗mK∗ Φ G(v) 1 f⊥ φ⊥ . (27)
×( u2M2 k − ⊥ − 2 K∗ u )
(cid:16) (cid:17)
From eq.(26) and eq.(27) we get a new relation between formfactors A , A and A :
3 0 2
A (q2)q2
A (q2) A (q2) = 2 . (28)
3 0
− − 2mK∗(mB +mK∗)
For the formfactors T , T , and T , we get the following sum rules:
1 2 3
1 m m2 1 du m2 +p2uu¯ q2u¯
T1(q2) = 4 fBmb2B eMB2 Zδ u exp − b uM2− !(mbfK⊥∗φ⊥ −
g(a) ga(m2 +q2 p2u2)
− fK∗mK∗"Φk −G⊥(v) −ug⊥(v) − ⊥4 − ⊥ b 4uM2− #) ,
(29)
6
1 m m2 1 du m2 +p2uu¯ q2u¯
T2(q2) = 2(m2B∗ −m2K∗) fBmb2B eMB2 Zδ u exp(− b uM2− )×
p2 m f⊥ φ
×(fK∗mK∗"g⊥(v) − 2M2g⊥(a)#q2 + b 2Ku∗ ⊥(m2b −q2 −p2u2) +
(m2 q2 p2u2)
+ fK∗mK∗ b − −
" 2u ×
(m2 q2 p2u2)g(a)
Φ G(v) +ug(v) + b − − ⊥ , (30)
× − k − ⊥ ⊥ 4uM2 !#)
h i
1 m m2 1 du m2 +p2uu¯ q2u¯
T3(q2) = 4 fBmb2B eMB2 Zδ u exp − b uM2− !×
g(a) (m2 q2 p2u2)
×(mK∗fK∗" ⊥4 + b −4uM−2 g⊥(a)#−
g(v) p2g(a)
− 2mK∗fK∗" ⊥2 (2−u)− 2M⊥2 #−
Φ G(v) m2 q2 p2u2
2mK∗fK∗ k − ⊥ b − − +
− " uM2 u
uM2
+ q2 M2 + +m f⊥ φ . (31)
− 2 !# b K∗ ⊥)
Using the equation of motion we can relate T and A A by:
3 3 0
−
A (q2) A (q2)
T3(q2) = mK∗(mb −ms) 3 q−2 0 . (32)
m2−p2
Here M is the Borel mass parameter. The lower integration limit δ = b depends on the
s0−p2
effective threshold s above which the contributions from higher states to the dispersion
0
relation (22) and (23) are cancelled against the corresponding piece in the QCD represen-
tation (19) and (20) Note that the sum rules for V(q2) and A (q2) and T (q2) in the light
1 1
cone QCD are derived in [17]. Our results agree with that of [17]. The region of appli-
cability of these sum rules is restricted by the requirement that the value of q2 m2 be
− b
sufficiently less than zero. In order not to introduce an additional scale, we require that
q2 m2 (p+q)2 m2 which translates to the condition that m2 q2 is of the order of the
− b ≤ − b b−
typical Borel parameter M2 5 8 GeV2. From this condition we obtain that the region
∼ ÷
of applicability of the sum rules is q2 < 15 17 GeV2 , which is few GeV2 below the zero
÷
recoil point.
7
Finally we calculate the differential decay rate with longitudinal polarization of the final
leptons. The differential decay rate is given by:
dΓ G2α2 V V∗ 2√λv
= | tb ts| (2m2 +m2 s) 16 A 2 + C 2 m4 λ+
dq2 212π5 3mB ( l B " | | | | B
(cid:16) (cid:17)
λ+12rs m4 λ2
+ 2 B 2 + D 2 +2 B 2 + D 2 B
1 1 2 2
| | | | rs | | | | rs −
(cid:16) (cid:17) (cid:16) (cid:17)
m2 λ
4[Re(B B∗)+Re(D D∗)] B (1 r s) +
− 1 2 1 2 rs − − #
m2 λ
+ 6m2 16 C 2m4 λ+4Re(D D∗) B
l"− | | B 1 3 r −
m4 (1 r)λ m4 sλ m2 λ
4Re(D D∗) B − +2 D 2 B 4Re(D D∗) B
− 2 3 r | 3| r − 1 2 r −
m4 λ
24 D 2 +2 D 2 B (2+2r s)
1 2
− | | | | r − #−
m4 λ
4vξ 8Re(AC∗)λm6 s [Re(B∗D )+Re(B∗D )] B (1 r s)+
− " B − 1 2 2 1 r − −
m6 λ2 λ+12rs
+ Re(B∗D ) B + Re(B∗D )m2 , (33)
2 2 r 1 1 B r #)
where λ = 1+r2+s2 2r 2s 2rs, r = m2K∗, s = q2 , ξ is the longitudinal polarization
− − − m2B m2B
4m2
of the final lepton, m and v = 1 l are its mass and velocity, respectively. In eq.(33)
l − q2
r
A, B , B , B , C, D , D , and D are defined as follows:
1 2 3 1 2 3
V m
A = Ceff +4C bT ,
9 mB +mK∗ 7 q2 1
m
B1 = C9eff(mB +mK∗)A1 +4C7 q2b(m2B −m2K∗) ,
A m q2
eff 2 b
B = C +4C T + T ,
2 9 mB +mK∗ 7 q2 2 m2B −m2K∗ 3!
eff2mK∗ mb
B = C (A A )+4C T ,
3 − 9 q2 3 − 0 7 q2 3
V
C = C ,
10
mB +mK∗
D1 = C10(mB +mK∗)A1 ,
A
2
D = C ,
2 10
mB +mK∗
8
2mK∗
D = C (A A ) .
3 − 10 q2 3 − 0
For the dileptonic decays of the B mesons, the longitudinal polarization asymmetry, P , of
L
the final lepton, l, is defined as
dΓ dΓ
(ξ = 1) (ξ = 1)
P (q2) = dq2 − − dq2 , (34)
L dΓ dΓ
(ξ = 1)+ (ξ = 1)
dq2 − dq2
where ξ = 1(+1) corresponds to the left (right) handed lepton in the final state. In the
−
Standard Model, this polarization asymmetry comes from the interference of the vector or
magnetic moment and axial vector operators. If in eq.(33) the lepton mass is equated to
zero, our results coincide with the results in [20] and if m = 0 they coincide with the results
l
6
in [11].
3 Numerical Analysis
For the input parameters which enter the sum rules for the formfactors and the expressions
of the decay width we have used the following values :
m = 4.8 GeV, m = 1.35 GeV, m = 0.106 GeV, m = 1.78 GeV,
b c µ τ
ΛQCD = 225 MeV, mB = 5.28 GeV, mK∗ = 0.892 GeV, s0 = 36 GeV2, M2 = 8 GeV2
In Fig.1 we present the q2 dependence of the formfactors V(q2), A (q2), A (q2) and
1 2
A (q2) (the formfactor A can be easily obtained from eq.(28)). All these formfactors
0 3
increase with q2. From these figures we see that A (q2) increases with q2 strongly, but
2
A (q2) and A (q2) do so smoothly. At this point let us compare our results on these
1 0
formfactors with the results which are obtained from 3-point QCD sum rules analysis in
[9]. In our case A (q2) increases with q2, but in [9] it decreases with q2. The behaviour of
1
the other formfactors are similar.
In Fig.2 we depict the dependence of the formfactors T , T , and T on q2. In this
1 2 3
case also all formfactors increase with q2. For formfactors T and T , our predictions
2 3
on their q2 dependence also differ from the predictions of [9]. In [9], T is positive and
2
smoothly decreases, the value of T is negative for all q2. Note that our predictions on the
3
q2 dependence ofallformfactorscoincide withrelativistic quark modelpredictions [11]. The
source of discrepeancy of our results with the predictions of [9] on A , T , and T should
1 2 3
be carefully analysed. This lies out of the scope of this paper. We are planning to come
back to the analysis of these points in our forthcoming works.
In Fig.3(4) we present the q2 dependence of the branching ratios for B K∗µ+µ−
→
(B K∗τ+τ−) decay, with and without the long distance effects, respectively.
→
In Fig.5 we plot the longitudinal polarization asymmetry P as a function of q2 for
L
B K∗µ+µ− and B K∗τ+τ−, with m = 176 GeV, with and without the long distance
t
→ →
9