Relation between Light Cone Distribution Amplitudes and Shape Function in B mesons

0.42 MB·English

by  A. Le Yaouanc

Relation between Light Cone Distribution Amplitudes
and Shape Function in B mesons
A. Le Yaouanc, L. Oliver and J.-C. Raynal
Laboratoire de Physique Th´eorique
∗
Universit´e de Paris XI, Bˆatiment 210, 91405 Orsay Cedex, France
8
0
0 Abstract
2

n The Bakamjian-Thomas relativistic quark model provides a Poincar´e rep-
a
J resentation of bound states with a fixed number of constituents and, in the

9
2 heavy quark limit, form factors of currents satisfy covariance and Isgur-Wise

] scaling. We compute the Light Cone Distribution Amplitudes of B mesons
h
p ϕB(ω)as well as the ShapeFunction S(ω), thatenters inthe decay B X γ,
s
- ± →
p that are also covariant in this class of models. The LCDA and the SF are re-
e
h lated through the quark model wave function. The former satisfy, in the
[

limit of vanishing constituent light quark mass, the integral relation given
3
v
by QCD in the valence sector of Fock space. Using a gaussian wave func-
7
2 tion, the obtained S(ω) is identical to the so-called Roman Shape Function.
0
3 From the parameters for the latter that fit the B X γ spectrum we pre-
s
. →
7
dict the behaviour of ϕB(ω). We discuss the important role played by the
0
±
7 constituent light quark mass. In particular, although ϕB(0) = 0 for van-
0 − 6
:
v ishing light quark mass, a non-vanishing mass implies the unfamiliar result
i
X ϕB(0) = 0. Moreover, we incorporate the short distance behaviour of QCD
r −
a to ϕB(ω), which has sizeable effects at large ω. We obtain the values for the
+
parameters Λ ∼= 0.35 GeV and λ−B1 ∼= 1.43 GeV−1. We compare with other
theoretical approaches and illustrate the great variety of models found in the
literature for the functions ϕB(ω); hence the necessity of imposing further
±
constraints as in the present paper. We briefly review also the different phe-
nomenathat are sensitive to the LCDA. Thevalue that we findfor λ 1 fulfills
−B
the upper bound recently measured by BaBar.
LPT Orsay 07-37
June 2007
∗
Unit´e Mixte de Recherche UMR 8627 - CNRS
1 Introduction.
The Light Cone Distribution Amplitudes (LCDA) of heavy-light mesons ΦB(ξ),
+
ΦB(ξ) [1] or, in the heavy quark limit ϕB(ω) and ϕB(ω) [2], are fundamental func-
+
− −
tions that enter in the large energy limit of amplitudes of semileptonic decays [2]
and in non-leptonic decays of B mesons [3], in the determination of the form factor
FB π(0) [4] and, more directly, in the decay B γℓν [5, 6, 7, 8].
+→ − → ℓ
On the theoretical side, these functions have been studied in the perturbative
regime at large ω [7, 9], and a number of very varied Ans¨atze have been proposed for
the dominant part of them at low ξ or ω, where the function is peaked at ξ ΛQCD
∼ mB
or ω Λ [2, 7, 8].
QCD
∼
On the other hand, one can obtain model independent information on these
functions from the measurement of the spectrum in the decay B γℓν , that is
− ℓ
→
directly related to one of the LCDA [5, 6, 7, 8].
On the theoretical side, although rigorous results are known in the perturbative
regime ω Λ , the guesses advanced for the main non-perturbative part of
QCD
≫
the LCDA amplitudes come essentially from QCD Sum Rules, imposing continuity
between the perturbative and long distance regimes [7, 9].
The motivation of the present work is to compute the LCDA in a class of rela-
tivistic quark models, namely the Bakamjian-Thomas (BT) quark models [10, 11,
12, 13, 14]. This is a class of models with a fixed number of constituents where the
states are covariant under the Poincar´e group. The model relies on an appropriate
Lorentz boost of eigenfunctions of a Hamiltonian describing the spectrum at rest.
On the other hand, one has demonstrated that the matrix elements of currents be-
tween hadrons are covariant in the heavy quark limit and exhibit Isgur-Wise scaling
[15] in this limit [14]. Given a Hamiltonian describing the spectrum, the model pro-
vides an unambiguous result for the elastic Isgur-Wise function ξ(w) [14, 16]. On
the other hand, the sum rules (SR) in the heavy quark limit of QCD, like Bjorken
or Uraltsev SR are satisfied in the model [17, 18], as well as SR involving higher
derivatives of ξ(w) at zero recoil [19].
The interest of computing the LCDA functions ϕB(ω) in this framework is to
±
directly relate them, in an unambiguous way, to the Shape Function S(ω) [20, 21,
22, 23, 24, 25, 26] in B X γ, that can also be computed in the BT class of models.
s
→
2
One could calculate the LCDA within the BT scheme using a Hamiltonian de-
scribing the spectrum, like the Godfrey-Isgur (GI) model [27], that has been used
elsewhere to compute the elastic Isgur-Wise function ξ(w) and the inelastic ones
τ (w), τ (w) [16]. However, we have tested the GI model to compute the Shape
1/2 3/2
Function S(ω) and have realized that this model does not fit the B X γ spec-
s
→
trum. This is the reason why we have decided another phenomenological approach,
namely to relate the Light Cone Distribution Amplitudes ϕB(ω) to the Shape Func-
±
tion S(ω), a relation that is provided by the BT scheme. Using this relation, and a
SF S(ω) that fits the B X γ spectrum, one can predict the LCDA. The problems
s
→
for the SF and also a discussion of heavy-to-light form factors within the BT scheme
for the GI spectroscopic model will be given in detail elsewhere.
The paper is organized as follows. In Section 2 we give the master formulae
defining the theoretical framework of BT quark models. In Section 3 we review
the definitions of the LCDA at finite mass and in the heavy quark limit of QCD
and in Section 4 we set a main natural hypothesis to compute the LCDA within
quark models. In Section 5, to introduce the technicalities of the BT model, we
review the calculation of the heavy meson decay constant. In Section 6 we obtain
our main results, namely the expressions for the LCDA in the BT quark models. In
Section 7 we compute the SF S(ω) in BT models and show that, in the case of the
harmonic oscillator, it is identical with the so-called Roman Shape function, used
to fit the B X γ spectrum [20, 21, 22, 26]. In Section 8 we use the parameters of
s
→
the latter to predict the LCDA functions ϕB(ω) and their moments. In Section 9,
±
following Braun, Ivanov and Korchemsky [7] and Lee and Neubert [9], we introduce
the radiative tail of ϕB(ω). In Section 10 we compare our results with proposals
+
for the LCDA in other theoretical schemes. In Section 11 we review the different
phenomena that are sensitive to the LCDA, and in Section 12 we conclude.
2 The Bakamjian-Thomas relativistic quark model.
Asexplainedin[14], theconstructionoftheBTwave functioninmotioninvolves
a unitary transformation that relates the wave function Ψ(P) (p , ,p ) in terms
s1, ,sn 1 ··· n
···
of one-particle variables, the spin S and momenta p to the so-called internal wave
i i
3
function Ψint (P,k , ,k ) given in terms of another set of variables, the total
s1,···,sn 2 ··· n
momentum P and the internal momenta k ,k , ,k ( k = 0). This property
1 2 n i
···
i
ensuresthat,startingfromanorthonormalsetofinternalwPavefunctions, onegetsan
orthonormalset ofwave functions inanyframe. Thebase Ψ(P) (p , ,p )isuse-
s1, ,sn 1 ··· n
···
ful to compute one-particle matrix elements like current one-quark matrix elements,
while the second Ψint (P,k , ,k ) allows to exhibit Poincar´e covariance. In
s1, ,sn 2 ··· n
···
order to satisfy the Poincar´e commutators, the unique requirement is that the mass
operator M, i.e. the Hamiltonian describing the spectrum at rest, should depend
only on the internal variables and be rotational invariant, i.e. M must commute
with P, ∂ and S. The internal wave function at rest (2π)3δ(P)ϕ (k , ,k )
∂P s1,···,sn 2 ··· n
is an eigenstate of M, P (with P = 0), S2 and S , while the wave function in mo-
z
tion of momentum P is obtained by applying the boost B , where P0 = √P2 +M2
P
involves the dynamical operator M.
The final output of the formalism that gives the total wave function in motion
Ψ(P) (p , ,p ) interms oftheinternalwave functionat restϕ (k , ,k )
s1,···,sn 1 ··· n s1,···,sn 2 ··· n
is the formula
p0
i k0
Ψ(P) (p , ,p ) = (2π)3δ p P i i
s1,···,sn 1 ··· n Xi i − !vuutPM0 Yi qp0i 
 q 
s′1X,···,s′n[Di(Ri)]si,s′iϕs′1,···,s′n(k2,···,kn) (1)
where p0 = p2 +m2 and M is the free mass operator is given by M = ( p )2.
i i i 0 0 i
i
q r
The internal momenta of the hadron at rest are given in terms of the moPmenta
of the hadron in motion by the free boost k = B 1 p where the operator B is
i − pi i p
i
the boost (√p2,0) p, the Wigner rotations R inPthe preceding expression R =
i i
→
B 1B 1 B and the states are normalized by < P,S P,S > = (2π)3δ(P
−pi − pi ki ′ ′z| z ′ −
i
P)δSz,PSz′.
The current one-quark matrix element acting on quark 1 between two hadrons
is then given by the expression
dp dp n dp
< Ψ(P,S ) J(1) Ψ(P,S ) > = ′1 1 i
′ ′ z′ | | z (2π)3 (2π)3 (2π)3!
Z i=2
Y
ΨP′ (p , ,p ) < p ,s J(1) p ,s > ΨP (p , ,p ) (2)
s′1,···,sn ′1 ··· n ∗ ′1 ′1| | 1 1 s1,···,sn 1 ··· n
4
where ΨP (p , ,p ) is given in terms of the internal wave function by (1) and
s1,···,sn 1 ··· n
< p ,s J(1) p ,s > is the one-quark current matrix element.
′1 ′1| | 1 1
As demonstrated in [14, 28], in this formalism, in the heavy quark limit, current
matrix elements are covariant and exhibit Isgur-Wise scaling, and one can compute
Isgur-Wise functions like ξ(w), τ (w), τ (w) [16].
1/2 3/2
In the present paper, as far as the LCDA are concerned, we are dealing with
current matrix elements between one meson and the vacuum, i.e. < 0 J Ψ >. In
| |
[29] such matrix elements were considered and it was demonstrated that the decay
constants of heavy-light mesons are covariant – independent of the frame – in the
heavy quark limit, and exhibit heavy quark scaling, i.e. f √m is a constant in this
B B
limit. This quantity was calculated for various Hamiltonians describing the meson
spectrum.
We want now to go beyond and compute the LCDA ϕB(ω) starting from the
±
meson-to-vacuummatrixelements < 0 J Ψ >. Wecanhereadvanceourmainresult,
| |
thatisparalleltotheoneobtainedformeson-to-mesoncurrent matrixelements. The
LCDAarecovariantintheheavyquarklimit,andcanthereforebecomputedwithout
any arbitrary parameter once the Hamiltonian giving the internal wave function is
given. The same statement holds for the Shape function S(ω) in B X γ, that can
s
→
beexpressed asa meson-to-meson matrixelement < Ψ O Ψ >. Forthelatter we will
| |
use the harmonic oscillator that gives the Roman Shape Function [20, 21, 22, 26].
We will then use the phenomenological parameters of this function, that fit the
B X γ spectrum, to predict the LCDA.
s
→
3 B meson Light Cone Distribution Amplitudes.
Let us define the LCDA ΦB(ξ), ΦB(ξ) [1]
+
−
< 0 q(z)S (z,0)Γb(0) B (P) >
| n− | d z+=z⊥
(cid:12)
(cid:12)
f 1 n/(cid:12) ΦB(ξ) ΦB(ξ)
= − 4B Z0 dξ e−iξP+z−Tr(Γ(P/ +mB)γ5"ΦB+(ξ)− v ·−n− + −2 − #) (3)
whereS (z,0)istheWilsonlinefollowingthelight-likefourvectorn = (1,0,0, 1)
n− − −
(n2 = 0), v is the B four velocity v = P , Γ is an arbitrary Dirac matrix, and the
− mB
5
center-of-mass motion is along the Oz axis. P , z are light-cone variables defined
+
−
for any four vector p by p = p0+pz, p = p0 pz. Sometimes one takes v n = 1,
+ √2 − √−2 · −
i.e. the B rest frame, but to exhibit covariance we have adopted a general value for
v n . The LCDA Φ (ξ), Φ (ξ) satisfy the normalization conditions
· − B1 B2
1 1
dξ ΦB(ξ) = dξ ΦB(ξ) = 1 (4)
+
Z0 Z0 −
In the heavy quark limit m it is useful to use a new variable
b
→ ∞
ω = m ξ (5)
b
keeping ω fixed, that yields the definition of the LCDA [2, 4, 30],
< 0 q(z)S (z,0)Γh (0) B (v) > (6)
| n− v | d z+=z⊥
(cid:12)
(cid:12)
f m ϕ(cid:12) B(ω) ϕB(ω) n/
= − B4 B Z0∞dω e−iωv+z−Tr(Γ(1+v/)γ5"ϕB+(ω)− + −2 − v ·−n−#)
The relation between ΦB(ξ) and ϕB(ω) is
± ±
1 ω
ϕB(ω) = ΦB (7)
m m
± B ±(cid:18) B(cid:19)
and ϕB(ω) satisfy the normalization conditions
±
∞dω ϕB(ω) = 1 (8)
Z0 ±
4 LCDA in quark models.
In what follows, we will use the preceding relations to obtain the expression of
ϕB(ω) in BT quark models. In the class of BT quark models, gluon exchange is
±
included in the potential. Therefore, in a way, the gluon field is integrated out, but
one looses the explicit gauge invariance that is ensured by the Wilson line of the
preceding expressions.
In a quark model, what we can consider is the matrix element involving con-
stituent quarks, in particular constituent light quarks with a dynamical mass. Our
ansatz will be to identify the QCD matrix element with the Wilson line with a ma-
trix element involving the constituent quark field, or in more rigorous terms, one
would say that one works in the light-cone gauge,
A = 0 S (z,0) = 1 (9)
+ n−
6
and set
< 0 q(z)S (z,0)Γb(0) B (v) >
| n− | d z+=z⊥=0
(cid:12)
= < 0 q (z)Γb(0) B (v) >(cid:12) (10)
| constituent | d (cid:12) z+=z⊥=0
(cid:12)
(cid:12)
This is our main hypothesis and the starting point of th(cid:12)e quark model calculation,
that then follows in a straightforward way. From now on the constituent light quark
field q will be denoted by q. Of course, the condition (9) can hold only in
constituent
field theory, and our BT scheme is just a model. Defining Φ (ξ) by the quark model
±
expression :
< 0 q(z)Γb(0) B (P) >
d
| | z+=z⊥
(cid:12)
f 1 (cid:12) n/ ΦB(ξ) ΦB(ξ)
= − 4B Z0 dξ e−iξP+z−Tr(Γ(P/ +mB)γ5"ΦB+(ξ)−(cid:12) v ·−n− + −2 − #) (11)
where n = (1,0,0, 1), one obtains
− −
1 P n/
ΦB+(ξ) = fBmB 2π+ Z dz− eiξP+z− < 0|q(z)v ·−n−γ5b(0)|Bd(P) > (cid:12)z+=z⊥=0 (12)
(cid:12)
1 P n/ (cid:12)
ΦB(ξ) = + dz eiξP+z− < 0 q(z) 2v/ − γ5b(0) Bd(P) >
− fBmB 2π Z − | − v ·n−! | (cid:12)z+=z⊥=0
(cid:12)
(cid:12)
Calling p the four- momentum of the light quark, using translational invariance
2
and integrating over z , these expressions write
−
1 p n/
ΦB(ξ) = < 0 q(0)δ ξ 2+ γ b(0) B (P) >
+ fBmB | − P+! v −n 5 | d
· −
1 p n/
ΦB(ξ) = < 0 q(0)δ ξ 2+ ) 2v/ γ b(0) B (P) > (13)
− 5 d
− fBmB | − P+! − v n ! |
· −
These will be the starting formulas to compute these functions in the BT quark
model, from which we will deduce their heavy quark limit ϕB(ω). But let us first
±
compute the heavy meson decay constant f in the BT quark models, that will
B
provide the desired normalization for the LCDA.
5 B decay constant in BT models.
The calculation of the matrix elements to obtain the LCDA ϕB(ω) is just rem-
±
iniscent of the one made to obtain the corresponding decay constant [29]. In this
7
latter case one needs the matrix element
< 0 q(0)Γb(0) B (v) >
d
| |
dp p0 k0k0 m m 1
= N 2 i i 1 2 1 2 ϕ(k )
q cZ (2π)3sPM0 vuup01p02s p01p02 √2 2
1+γ0 t 1+γ0
Tr Γγ5B B B 1B B 1B 1 (14)
" u k2 2 −u u 2 −k1 −u #
where ϕ(k ) is the internal wave function at rest, with the normalization
2
dk
2 ϕ(k ) 2 = 1 (15)
(2π)3 | 2 |
Z
p and p (m and m ) are the quark four-momenta (masses ) of respectively the
1 2 1 2
heavy and light quarks, B is the 4 4 boost matrix associated with the four-vector
p
×
p, and the four vector u, M and the relation between k and p are given by
0 i i
p +p
u = 1 2 M = (p +p )2 B k = p (i = 1,2) (16)
0 1 2 u i i
M
0 q
where k and k are the four-momenta of the quarks in the rest frame of the B
1 2
meson and B is the boost associated with the four-vector u. The products of 4 4
u
×
matrices under the trace read
1+γ0 m +p/ 1+u/
B B B 1 = 2 2
u k2 2 −u 2m (k0 +m ) 2
2 2 2
1+γ0 q1+u/ m +p/
B B B 1 = 1 1 (17)
u 2 k1 −u 2 2m (k0 +m )
1 1 1
q
This yields the expression
< 0 q(0)Γb(0) B (P) >
d
| |
√N dp √u0 k0k0
= c 2 √2 1 2
− 8 (2π)3 p0p0v(k0 +m )(k0 +m )
Z 1 2u 1 1 2 2
u
t
Tr γ5Γ(m +p/ )(1+u/)(m +p/ ) ϕ(k ) (18)
2 2 1 1 2
h i
Using the current Γ = γ γ we obtain, after some algebra and the definition of the
µ 5
four vector u (16),
√N dp √u0 k0k0
< 0 q(0)γ γ b(0) B (P) >= c 2 1 2
| µ 5 | d √2 (2π)3p0p0v(k0 +m )(k0 +m )
Z 1 2u 1 1 2 2
u
m +m t
1 2
1+ (p m +p m )ϕ(k ) (19)
1µ 2 2µ 1 2
M
(cid:18) 0 (cid:19)
8
Since the BT states are normalized acccording to
< B (P ) B (P) > = (2π)3δ(P P) (20)
d ′ d BT ′
| −
while the covariant normalization is < B (P ) B (P) >= (2π)32P0δ(P P), we
d ′ d ′
| −
have to identify the former matrix element with the definition of the decay constant
f P
B µ
< 0 q(0)γ γ b(0) B (P) >= (21)
µ 5 d
| | √2P0
one obtains
dp √u0 k0k0 m +m
f √m = N √v0 2 1 2 1+ 1 2
B B c (2π)3p0p0v(k0 +m )(k0 +m ) M
q Z 1 2uu 1 1 2 2 (cid:18) 0 (cid:19)
[m (p v)+m (p v)]ϕ(k ) t (22)
1 2 2 1 2
· ·
This expression is not covariant, but becomes covariant in the heavy quark limit.
For m one has
1
→ ∞
p k0
m m m , u v , 1 v , 1 1 , k0 p v (23)
0 → 1 → B → m → m → 2 → 2 ·
1 1
where v is the B meson four-velocity, and one gets the expression of the B decay
constant in the BT model in the heavy quark limit [29],
dp 1
f √m = √2 N 2 (p v)(p v +m ) ϕ (p v)2 m2 (24)
B B c (2π)3p0 2 · 2 · 2 2 · − 2
q Z 2q (cid:18)q (cid:19)
This expression is covariant, satisfies heavy quark scaling and gives the decay con-
stant in terms of the internal wave function. In the B rest frame one gets
1/2
√N 1 m + p2 +m2
f √m = c ∞dp p2 2 2 ϕ(p) (25)
B B √2 π2 Z0  pq2 +m22 
 q 
We have checked that these expressions for the decay constant hold exactly in
the equivalent light-front approach of Cardarelli at al. [13] in the heavy mass limit.
6 LCDA in the heavy quark limit in BT models.
According to (13), we have to compute a generic matrix element
1 p
2+
Φ (ξ) = < 0 q(0)δ ξ Γb(0) B (P) > (26)
B d
fBmB | − P+! | z+=z⊥=0
(cid:12)
(cid:12)
(cid:12)
9
where Γ is a Dirac matrix. In the BT quark model, this expression writes, taking
into account the appropriate normalizations,
1 √N dp 1 p
Φ(ξ) = √2P0 c 2 δ ξ 2+ (27)
− fBmB 8 Z (2π)3p02 − P+!
√u0 k0k0
= √2 1 2 Tr γ5Γ(m +p/ )(1+u/)(m +p/ ) ϕ(k )
p0 v(k0 +m )(k0 +m ) 2 2 1 1 2
1 u 1 1 2 2
u h i
t
Changing the measure and integrating relatively to p and p , one obtains
2+ 2
−
1 √N 1 1 dp √u0 k0k0
Φ(ξ) = √2P0 c 2⊥√2 1 2
− f m 8 ξ 2π (2π)2 p0 v(k0 +m )(k0 +m )
B B hZ 1 uu 1 1 2 2
t
Tr γ5Γ(m +p/ )(1+u/)(m +p/ ) ϕ(k ) (28)
h 2 2 1 1 i 2 ip2+=ξP+,p2−=(p2⊥2P)+2+ξm22
Making use of the definition of the four-vector u (16) and computing the trace
particularizing respectively to Γ = n/− γ and Γ = 2v/ n/− γ , one gets
vn− 5 − vn− 5
· ·
(cid:16) (cid:17)
1 √N 1 1 dp √u0 k0k0
ΦB(ξ) = √2P0 c 2⊥√2 1 2
+ f m 2 ξ 2π  (2π)2 p0 v(k0 +m )(k0 +m )
B B Z 1 uu 1 1 2 2
t
m +m m (p n )+m (p n )
1+ 1 2 1 2 · −  2 1 · − ϕ(k2)
(cid:18) M0 (cid:19)" v ·n− # #p2+=ξP+,p2−=(p2⊥2P)+2+ξm22
1 √N 1 1 dp √u0
ΦB(ξ) = √2P0 c 2⊥√2
− fBmB 2 ξ 2π "Z (2π)2 p01
k0k0 m +m
1 2 1+ 1 2 2[m (p v)+m (p v)]
v(k0 +m )(k0 +m ) M { 1 2 · 2 1 ·
u 1 1 2 2 (cid:18) 0 (cid:19)
u
t
m (p n )+m (p n )
1 2 2 1
· − · − ϕ(k2) (29)
− v ·n− ) #p2+=ξP+,p2−=(p2⊥2P)+2+ξm22
6.1 Heavy quark limit.
At finite mass the expressions (29) are not covariant. In the heavy quark limit,
using now the variable ω = m ξ,
1
u v v k0 m , k0 p v
→ 1 → 1 → 1 2 → 2 ·
M m m m ξ ω (30)
0 1 B 1
→ → →
one obtains, denoting p = p
2
⊥ ⊥
√N 1
ϕB(ω) = c √2 dp2
+ fB√mB 8π2 Z ⊥
10