Heavy Quark Hadronic Lagrangian for S-wave Quarkonium

38 Pages·0.25 MB·Norwegian (Bokmål)

by  A. Pineda

UB-ECM-PF-94/19
Heavy Quark Hadronic Lagrangian
for S-Wave Quarkonium
A. Pineda
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9
and
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J. Soto

n
a
J
Departament d’Estructura i Constituents de la Mat`eria

3
and
2
Institut de F´ısica d’Altes Energies

2
Universitat de Barcelona
v
6 Diagonal 647, 08028 Barcelona, Catalonia, Spain
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2
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0
Abstract
4
9
/
h
p
-
p We use Heavy Quark Effective Theory (HQET) techniques to parametrize certain
e
h
: non-perturbative effects related to quantum fluctuations that put both heavy quark and
v
i
X antiquark in quarkonium almost on shell. The large off-shell momentum contributions are
r
a
calculated using Coulomb type states. The almost on-shell momentum contributions are
evaluated using an effective ’chiral’ lagrangian which incorporates the relevant symmetries
of the HQET for quarks and antiquarks. The cut-off dependence of both contributions
matches perfectly. Thedecay constants andthematrixelements ofbilinearcurrents at zero
recoil are calculated. The new non-perturbative contributions from the on-shell region are
parametrized by a single constant. They turn out to be O(α2/Λ a ), a being the Bohr
QCD n n
radius and α the strong coupling constant, times the non-perturbative contribution coming
from the multipole expansion (gluon condensate). We discuss the physical applications to
Υ, J/Ψ and B systems.
c
1. Introduction
The so-calledHeavyQuark EffectiveTheory (HQET)[1-5]has becomea standardtool
to study the properties of hadrons containing a single heavy quark (see [6] for reviews).
The hadron momentum is essentially the momentum of the heavy quark which may then
be considered almost on-shell. The dynamics becomes independent of the spin and the
mass of the heavy quark giving rise to the so-called Isgur-Wise symmetries [1,2]. The
relevant modes are momentum fluctuations of the order of Λ which are described by
QCD
the HQET [3-5]. One cannot actually carry out reliable perturbative calculations at that
scale, but one can certainly use the Isgur-Wise symmetries to obtain relations between
physical observables.
For hadrons containing two heavy quarks or more the HQET is not believed to be a
suitableapproximation. Thereason being that a systemoftwo heavyquarksismainlygov-
erned by the perturbative Coulomb-type interaction. The relevant modes are momentum
fluctuations of the order of the invers Bohr radius, which is flavor dependent, and not of
the order of Λ . Still, if one is interested in subleading non-perturbative contributions
QCD
relatedtothe”on-shellness” oftheheavyquarks, theHQETmayprovidesomeuseful infor-
mation. Irrespectively of the above, the HQET has already been used in phenomenological
approaches to two heavy quark systems [7].
We shall argue that the leading non-perturbative contributions in the on-shell region
to the quarkonium decay constants and to the matrix elements of heavy-heavy currents
between quarkonia states can be described by a suitably modified HQET. The well-known
non-perturbative contributions in the off-shell region arising from the multipole expansion
[8,9] are O(Λ a /α2), a being the Bohr radius and α the strong coupling constant,
QCD n n
times the contributions we find. The key observation is that when the heavy quarks
are almost on-shell the non-perturbative effects must be important. In that regime the
multipole expansion breaks down, but it is precisely there where HQET techniques become
1
applicable.
In ref. [10] it was pointed out that when fields describing both heavy quarks and
heavy antiquarks with the same velocity are included in the HQET lagrangian, the latter
has extra symmetries beyond the well known flavor and spin symmetries [1,2]. In ref.
[11] the extra symmetries were thoroughly analysed (see [12] for related elaborations). It
was shown that they are spontaneously broken down to the spin and flavor symmetries,
even if the gluons are switched off. The Goldstone modes turn out to be two particle
states with the quantum numbers of s-wave quarkonia. Translating these findings into
phenomenologically useful statements was the original motivation of this work.
The main hypothesis in what follows is that whenever we have a heavy quark field we
may split it in two momentum regimes. The momentum regime where the heavy quark
is almost on shell (small relative three momentum), and the momentum regime where the
heavy quark is off shell (large relative three momentum). The main observation is that the
HQET should always be a good approximation for a heavy quark in the almost on-shell
momentum regime of QCD [10,12], no matter whether the heavy quark is accompained
by another heavy quark in the hadron or not. What makes a hadron containing a single
heavy quark qualitatively different from a hadron containing, say, two heavy quarks are
the large off-shell momentum effects. In the former the large off-shell momentum effects
are small and can be evaluated order by order in QCD perturbation theory [1,5,13,14]. In
the latter the large off-shell momentum effects are dominant giving rise to Coulomb-type
bound states. However, once this is taken into account there is no a priori reason not to
use HQET in the almost on-shell momentum regime for systems with two heavy quarks.
Then the extra symmetries found in [10,11], which naturally involve quarkonium systems,
should be relevant.
Suppose we have two quarks Q and Q′ which are sufficiently heavy so that the for-
malism below can be readily applicable. Let us denote by ψQ, ηQ, Q∗Q′ and QQ′ the vector
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Q¯Q, pseudoscalar Q¯Q, vector Q¯Q′ and pseudoscalar Q¯Q′ states. Our main results follow.
(i)The masses do not receive new non-perturbative contribution from the on-shell
momentum region. Consequently, the leading non-perturbative correction comes from the
multipole expansion [8,9]. This allows to extract m in a model independent way from
Q
m , and hence fix the parameter Λ¯ relating m with the mass of the Q¯q systems [6].
ψQ Q
(ii) The new non-perturbative effects from the on-shell momentum region in the de-
cay constants fψQ, fηQ, fQ∗Q′ and fQQ′ are given in terms of a single non-perturbative
parameter f .
H
(iii) The new non-perturbative effects from the on-shell momentum region in the
matrix elements of bilinear heavy quark currents at zero recoil are given in terms of the
same non-perturbative parameter f . In particular, this implies that the semileptonic
H
decays (mQ > mQ′)
ψQ, ηQ −→ Q∗Q′ , QQ′
Q∗Q′ , QQ′ −→ ψQ′ , ηQ′
at zero recoil are known in terms of fψQ, fηQ, fQ∗Q′ and fQQ′.
We distribute the paper as follows. In sect. 2 we perform some short distances
calculations in the kinematical region we are interested in. In sect. 3 we summarize the
main results of ref. [11] and match the results from sect. 2 with the HQET. In sect. 4
we construct a hadronic effective lagrangian for on-shell modes in quarkonium. In sect.
5 we calculate the decay constant. In sect. 6 we calculate the matrix elements of any
bilinear heavy quark current between quarkonia states. This is relevant for the study of
semileptonic decays at zero recoil. In sect. 7 we briefly discuss the possible use of our
formalism for Υ, B , B∗, J/Ψ and η physics. Section 8 is devoted to the conclusions. In
c c c
Appendix A we show how to include 1/m corrections in the hadronic effective lagrangian
for the on-shell modes. A few technical details are relegated to Appendix B.
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2. Short distance contributions in the on-shell momentum regime
As mentioned in the introduction, what makes a Q¯Q system qualitatively different
from a Q¯q system are the short distance contributions. In a Q¯q system these are well
understood. They amount to Wilson coefficients in the currents and in the operators of the
HQETlagrangian, withanomalousdimensionswhich arecomputableintheloopexpansion
of QCD. For a Q¯Q system the short distance contributions cannot be accounted for by
just anomalous dimensions in Wilson coefficients. Indeed, the anomalous dimension of a
current containing a heavy quark field and a heavy antiquark field with the same velocity
¯
becomes imaginary and infinite [15]. For large m , the two quarks in a QQ system appear
Q
to be very close. Due to assymptotic freedom the system can be understood in a first
approximation as a Coulomb-type bound state. In perturbation theory this is equivalent
to sum up an infinite set of diagrams (ladder approximation) whose kernel is the tree level
one gluon exchange (see [16] for a review).
We shall assume that the dominant short distance contribution to heavy quarkonia
is the existence of Coulomb-type bound states. Typically we shall be interested in Green
functions of the kind
G (p ,p ) := d4x d4x eip1x1+ip2x2 0 T Q¯aΓQb(0)Q¯bi1(x )Qai2(x ) 0 , (2.1)
Γ 1 2 1 2 h | α1 1 α2 2 | i
Z
(cid:8) (cid:9)
for the range of momentum
p = m v k , p = m v k , (2.2)
1 b 1 2 a 2
− − − −
k and k being small.
1 2
Since the quarks are very massive, for the range of momentum (2.2) the leading
contribution to (2.1) is only given by the following ordering
G (p ,p ) = d4x d4x eip1x1+ip2x2θ max(x0,x0)
Γ 1 2 1 2 − 1 2 (2.3)
Z
0 Q¯aΓQb(0)T Q¯bi1(x(cid:0))Qai2(x ) 0 (cid:1).
×h | α1 1 α2 2 | i
(cid:8) (cid:9)
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We insert the identity between the current and the fields and we approximate it by the vac-
uum plus the Coulomb-type states (the states above threshold shall not give contribution
when we sit in the relevant pole). We treat then the fields as being free.
d3P~
n ~ ~
1 0 0 + s,P = m ~v s,P = m ~v (2.4)
≃ | ih | (2π)32P0| n ab,n ih n ab,n |
n,s Z n
X
The Coulomb state in the center of mass frame (CM) reads
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1 m2 d3~k 1
s,P~ = m ~v = ab v0 Ψ˜ (~k)
| n ab,n i √Nc mab,n Z (2π)3 ab,n 2p012p02 (2.5)
u¯α(p )Γ vβ(p )a†(p )b†(pp ) 0 ,
× 1 s 2 α 1 β 2 | i
α,β
X
where
~ ~
k.~v k.~v
~ ~
~p = m ~v +k + ~v , p~ = m ~v k ~v,
1 a 1+v0 2 b − − 1+v0
p0 = m v0 +~k.~v , p0 = m v0 ~k.~v,
1 a 2 b −
m := m +m ,m := m E ,Γ = iγ p ,i/eip ,
ab a b ab,n ab ab,n s 5 − −
−
1 /v
v2 = 1 , p := ± , ei.v = 0. (2.6)
±
2
E , Ψ (~x) and Ψ˜ (~k) are the energy, the coordinate space wave function and the
ab,n ab,n ab,n
momentum space wave function of a Coulomb-type state with principal quantum number
n. v is the bound state 4-vector velocity. a†(p ) and b†(p ) are creation operators of
α 1 β 2
particles and anti-particles respectively. uα(p ) and vβ(p ) are spinors normalized in such
1 2
a way that in the large m limit the following holds
uα(p )u¯α(p ) = p , vα(p )v¯α(p ) = p . (2.7)
1 1 + 1 1 −
−
α α
X X
Choosing the momenta as in (2.6) is crucial in order to take into account that the CM
of the bound state moves with a fix velocity v with respect to the laboratory frame [17].
(2.5) has the usual relativistic normalization
s,P~ = m ~v r,P~ = m ~v′ = 2m v0(2π)3δ(3)(m (~v ~v′))δ δ . (2.8)
n ab,n m ab,m ab,n ab,n nm rs
h | i −
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We have to consider the following kind of matrix elements
s,m ~v Qa (x )Q¯b (x ) 0 = eimab,nv.X s,m ~v Qa (x X)Q¯b (x X) 0
h ab,n | α2 2 α1 1 | i h ab,n | α2 2 − α1 1 − | i
= eimab,nv.Xmma32b (Γ¯s)α2α1 (2dπ3~k)3Ψ˜∗ab,n(~k)ei(~k.~vx0−~x(~k+1~k+.v~v0~v)), (2.9)
ab,n
Z
m x +m x
a 1 b 2
X = , x = x x .
1 2
m −
ab
where it is essential to extract the CM dependence in the fields before using the explicit
expression (2.5) for the calculation of (2.9). As mention above the states s,m ~v have
ab,n
| i
the explicit expresion (2.5) only in the CM frame [16,17]. Factors of the kind m /m
ab ab,n
appearing in several expressions above have been approximated to 1 in the rest of the
paper. Finally, performing the x , x integral and taking into account that
1 2
¯
(Γ ) (Γ ) = 2(p ) (p ) (2.10)
s α2α4 s α1α3 − + α2α3 − α1α4
s
X
we obtain
G (p ,p ) = Ψ˜∗ (0)Ψ (0)(p Γp ) δ
Γ 1 2 ab,n ab,n − + α2α1 i1i2
n
X (2.11)
1 1
,
× v.k + ma E +iǫ v.k + mb E +iǫ
2 mab ab,n 1 mab ab,n
In the last expression we approximated Ψ˜ (ei.k) Ψ˜ (0) (we neglect O((n|ei.k|)2)).
ab,n ≃ ab,n mα
In (2.11) there is a sum over an infinite number of poles. Each term in the sum corresponds
to a Coulomb-type bound state. At the hadronic level we want to describe only one of
those states. This is achieved by tunning the external momenta to sit on the relevant pole.
Suppose we are interested in ψ (n) state. Then we take
Q
m m
k = k′ b E v , k = k′ a E v, (2.12)
1 1 − m ab,n 2 2 − m ab,n
ab ab
so that in the limit k′ 0, (i = 1,2) we obtain
i →
G (p ,p ) =Ψ˜∗ (0)Ψ (0)(p Γp ) δ
Γ 1 2 ab,n ab,n − + α2α1 i1i2
1 1 (2.13)
.
× v.k′ +iǫ v.k′ +iǫ
2 1
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Notice from (2.2) and (2.12) that we must subtract from the momentum of the quark
(m ma E )v inordertogetanexpressionsuitabletobereproducedintheHQET.This
a−mab ab,n
may be interpreted asif integrating out off-shell short distance degrees of freedom produces
an effective mass for the almost on-shell modes of a heavy quark inside quarkonium. This
effective mass depends on the precise bound state the quark is in. We are almost on-shell
when v.k′,ej.k′ Λ (i=1,2).
i i ∼ QCD
This restricts the vality of our approximation to the case E µ α2/n2 Λ
ab,n ab QCD
∼ ≫
( µ is the reduced mass), otherwise momentum fluctuations of the order of Λ would
ab QCD
take us from one pole to another. Notice also that for arbitrary large but fix µ there is
ab
always an n where this approximation fails. Therefore we shall always be dealing with a
finite number of low laying energy levels.
Consider the four-point function.
G(p ,p ,p ,p ) := d4x d4x d4x d4x eip1x1+ip2x2+ip3x3+ip4x4
1 2 3 4 1 2 3 4
(2.14)
Z
0 T Qbi1(x )Qai2(x )Q¯ai3(x )Q¯bi4(x ) 0 .
×h | α1 1 α2 2 α3 3 α4 4 | i
(cid:8) (cid:9)
For the momenta
m
p = (m b E )v k′ ,
1 − b − m ab,n − 1
ab
m
p = (m a E )v+k′ ,
2 a − m ab,n 2
ab (2.15)
m
p = (m a E )v k′ ,
3 − a − m ab,n − 3
ab
m
p = (m b E )v+k′ ,
4 b − m ab,n 4
ab
(k′ 0 , i = 1,...,4) working in the same way we obtain
i →
i
G(p ,p ,p ,p ) = (2π)4δ(4)( k′ +k′ k′ +k′) (Γ ) (Γ¯ )
1 2 3 4 − 1 2 − 3 4 2N n α2α4 n α1α3
c
Γn=iγX5p−,i/eip−
1 1 1 1
δ δ Ψ˜∗ (0)Ψ˜ (0) + .
× i1i3 i2i4 ab,n ab,n v.k′ +iǫ v.k′ +iǫ v.k′ +iǫ v.k′ +iǫ
3 1 (cid:18) 2 4 (cid:19)
(2.16)
We shall see in the next section that (2.13) and (2.16) can be reproduced (with suitable
changes) by a HQET for quarks and antiquarks.
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3. HQET for quarks and antiquarks
The lagrangian of the HQET for quarks and antiquarks moving at the same velocity
v (v vµ = 1) reads [4]
µ µ
L = ih¯ /vv Dµh = ih¯+v Dh+ ih¯−v Dh−, (3.1)
v v µ v v · v − v · v
where h = h+ +h− and h± = 1±/vh . h+ contains annihilation operators of quarks with
v v v v 2 v v
small momentum about mv and h− contains creation operators of anti-quarks again with
µ v
small momentum about mv . D is the covariant derivative containing the gluon field.
µ µ
The quark and antiquark sector of (3.1) are independently invariant under the well-
known spin and flavour symmetry [1,2,4]
h± eiǫi±Si±h± and h¯± h¯±e−iǫi±Si± , (3.2)
v → v v → v
where S± = iǫ [/e ,/e ](1 /v)/2, with eµ ,j = 1,2,3 being an orthonormal set of space
i ijk j k ± j
like vectors orthogonal to v , and
µ
h± eiθ±h± and h¯± h¯±e−iθ± . (3.3)
v → v v → v
ǫi and θ are arbitrary real numbers corresponding to the parameters of the transforma-
± ±
tions.
The lagrangian (3.1) is also invariant under the following set of transformations
h eiγ5ǫh ; h¯ h¯ eiγ5ǫ, (3.4)
v v v v
→ →
h eγ5/vǫh ; h¯ h¯ eγ5/vǫ, (3.5)
v v v v
→ →
h eǫi/eih ; h¯ h¯ eǫi/ei , (3.6)
v v v v
→ →
h eiǫi/ei/vh ; h¯ h¯ eiǫi/ei/v. (3.7)
v v v v
→ →
The whole set of transformations (3.2)-(3.7) corresponds to a U(4) symmetry for a single
flavour. For N heavy flavours they correspond to a U(4N ) group. In the latter case
hf hf
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h must be considered a vector in flavour space and the parameters of the transformations
v
(3.2)-(3.7) as hermitian matrices in that space.
WhenthegluonsareswitchedoffitiseasytoprovethattheU(4N )symmetrybreaks
hf
spontaneously down to U(2N ) U(2N ) (see [11]). The following currents correspond
hf hf
⊗
to the broken generators
jab := h¯aiγ p hb and jabi := h¯ai/e p hb , (3.8)
5± v 5 ± v 5± v i ± v
a,b,c... = 1,...N are flavour indices. They transform according to two four dimensional
hf
irreducible representations of U(2N ) U(2N ). In what follows we are going to assume
hf hf
⊗
that the situation above is not modified when soft gluons are switched on. The currents
(3.8)havethequantumnumbers ofpseudoescalarandvectorquarkoniumrespectively. The
heavyquarkandantiquarkfieldsinteractwithsoftgluonsaccordingtothelagrangian(3.1).
For soft gluons, perturbation theory cannot be realiable applied. However, one can use
effective lagrangian techniques, which fully exploite the symmetries above, to parametrize
the non-perturbative contributions in this region. This shall be done in section 4.
For further purposes let us carry out some leading order perturbative calculations.
Consider first
GΓΓ′(k) = d4xe−ik.xh0|T h¯av−Γhbv+(0)h¯bv+Γ′hav−(x) |0i
Z (3.9)
µ3 (cid:8) 1 (cid:9)
= iN tr(p Γ′p Γ) ,
− c6π2 + − v.k +iǫ
where µ is an ultraviolet symmetric cut-off in three-momentum (see [11] for more details).
Consider also
GΓΓ′Γ′′(k1′,k2′)
= d4x d4x eik1′x1−ik2′x2 0 T h¯a−Γ′′hb+(x )h¯b+Γhc+(0)h¯c+Γ′ha−(x ) 0
1 2 h | v v 1 v v v v 2 | i (3.10)
Z
µ3 (cid:8)1 1 (cid:9)
= N tr(p Γ′′p Γp Γ′) .
c6π2 − + + v.k′ +iǫ v.k′ +iǫ
1 2
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