Heavy Quark Hadronic Lagrangian for S-wave Quarkonium

We use Heavy Quark Effective Theory (HQET) techniques to parametrize certain non-perturbative effects related to quantum fluctuations that put both heavy quark and antiquark in quarkonium almost on shell. The large off-shell momentum contributions are 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. The decay constants and the matrix elements of bilinear currents 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 $O(\alpha^2/\Lambda_{QCD} a_{n})$, $a_{n}$ being the Bohr radius and $\alpha$ the strong coupling constant, times the non-perturbative contribution coming from the multipole expansion (gluon condensate). We discuss the physical applications to $\Upsilon$, $J/\Psi$ and $B_{c}$ systems.

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 QCD which are described by the HQET [3][4][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 suitable approximation. The reason being that a system of two heavy quarks is mainly governed 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 QCD . Still, if one is interested in subleading non-perturbative contributions related to the "on-shellness" of the heavy quarks, the HQET may provide some useful information. 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 QCD a n /α 2 ), a n being the Bohr radius and α the strong coupling constant, 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 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 formalism below can be readily applicable. Let us denote by ψ Q , η Q , Q * Q ′ and Q Q ′ the vector Q Q, pseudoscalarQQ, vectorQQ ′ and pseudoscalarQQ ′ 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 Q in a model independent way from m ψ Q , and hence fix the parameterΛ relating m Q with the mass of theQq systems [6].
(ii) The new non-perturbative effects from the on-shell momentum region in the decay constants f ψ Q , f η Q , f Q * Q ′ and f Q Q ′ 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 H . In particular, this implies that the semileptonic 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 c , B * c , J/Ψ and η c physics. Section 8 is devoted to the conclusions. In 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.

Short distance contributions in the on-shell momentum regime
As mentioned in the introduction, what makes aQQ system qualitatively different from aQq system are the short distance contributions. In aQq system these are well understood. They amount to Wilson coefficients in the currents and in the operators of the HQET lagrangian, with anomalous dimensions which are computable in the loop expansion of QCD. For aQQ 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 Q , the two quarks in aQQ system appear 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 for the range of momentum 2) k 1 and k 2 being small.
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 We insert the identity between the current and the fields and we approximate it by the vacuum 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.
The Coulomb state in the center of mass frame (CM) reads where E ab,n , Ψ ab,n ( x) andΨ ab,n ( k) are the energy, the coordinate space wave function and the momentum space wave function of a Coulomb-type state with principal quantum number n. v is the bound state 4-vector velocity. a † α (p 1 ) and b † β (p 2 ) are creation operators of particles and anti-particles respectively. u α (p 1 ) and v β (p 2 ) are spinors normalized in such a way that in the large m limit the following holds 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 n = m ab,n v|r, P m = m ab,m v ′ = 2m ab,n v 0 (2π) 3 δ (3) (m ab,n ( v − v ′ ))δ nm δ rs . (2.8) We have to consider the following kind of matrix elements s, m ab,n v|Q a α 2 (x 2 )Q b α 1 (x 1 )|0 = e im ab,n v.X s, m ab,n v|Q a 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 ab,n v have the explicit expresion (2.5) only in the CM frame [16,17]. Factors of the kind m ab /m ab,n appearing in several expressions above have been approximated to 1 in the rest of the paper. Finally, performing the x 1 , x 2 integral and taking into account that In the last expression we approximatedΨ ab,n (e i .k) ≃Ψ ab,n (0) (we neglect O(( n|e i .k| mα ) 2 )). 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 Q (n) state. Then we take 12) so that in the limit k ′ i → 0, (i = 1, 2) we obtain Notice from (2.2) and (2.12) that we must subtract from the momentum of the quark (m a − m a m ab E ab,n )v in order to get an expression suitable to be reproduced in the HQET. This may be interpreted as if 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 This restricts the vality of our approximation to the case E ab,n ∼ µ ab α 2 /n 2 ≫ Λ QCD ( µ ab is the reduced mass), otherwise momentum fluctuations of the order of Λ QCD would take us from one pole to another. Notice also that for arbitrary large but fix µ ab there is 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. (2.14) For the momenta .., 4) working in the same way we obtain (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.

HQET for quarks and antiquarks
The lagrangian of the HQET for quarks and antiquarks moving at the same velocity v µ (v µ v µ = 1) reads [4] L , with e µ j , j = 1, 2, 3 being an orthonormal set of space like vectors orthogonal to v µ , and ǫ i ± and θ ± are arbitrary real numbers corresponding to the parameters of the transformations.
The lagrangian (3.1) is also invariant under the following set of transformations The whole set of transformations (3.2)-(3.7) corresponds to a U (4) symmetry for a single flavour. For N hf heavy flavours they correspond to a U (4N hf ) group. In the latter case h v must be considered a vector in flavour space and the parameters of the transformations (3.2)-(3.7) as hermitian matrices in that space.
When the gluons are switched off it is easy to prove that the U (4N hf ) symmetry breaks [11]). The following currents correspond to the broken generators 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
where µ is an ultraviolet symmetric cut-off in three-momentum (see [11] for more details).
Consider also (3.10) The flavor indices (a , b , c) are not summed up unless otherwise indicated. Colour indeces are not explicitly displayed in the colour singlet currents. Otherwise they will be denoted by i 1 , i 2 , ...= 1....N c , N c being the number of colours. We shall drop the subscript v from h v and change the superscripts ± into subscripts in the following.
The strong cut-off dependence of (3.9)-(3.10) is puzzling. We shall see later on that it cancels against suitable short distance contributions.
As claimed before, it is easy to see that (2.13) is reproduced by the following HQET Green function at tree level with C Γ being a Wilson coeficient.
Analogously, (2.16) is reproduced in the HQET by * * One may be tempted to include (3.13) as a perturbation in the HQET lagrangian. This is not quite correct. The Green function gives a non-zero contribution in the HQET which does not correspond to (2.14)-(2.16). It is (3.13) which gives the leading contribution to (2.14) in the HQET and hence the last term in (3.13) must not be included in the lagrangian. This means that unlike in the case of heavy-light systems, the short distance effects here cannot always be accounted for by only modifications of the currents and the lagrangian, as we may have naïvely expected.
We have to content ourselves by identifying for a given Green function, the Green function in the HQET which gives the same result. (3.13)

Effective hadronic lagrangian for the on-shell contributions of s-wave quarkonia
We have seen that for the on-shell kinematical regime certain correlators can be reproduced in the HQET. We shall see in the sect. 5 and 6 that the contributions from this region to the decay constants and matrix elements reduce to the evaluation of heavy quarkantiquark currents in the HQET. For the range of momentum we are interested in these Green functions cannot reliable be evaluated in perturbation theory. We shall use in this section effective lagrangian techniques, very similar to those used in Chiral perturbation theory, to parametrize the nonperturbative contribution.
There are well-known rules [18] (see also [19]) to construct phenomenological lagrangians for Goldstone bosons associated to the symmetry breaking of a group G down to a subgroup H for relativistic theories. These rules need two slight modifications to become applicable to our case: (i) The HQET is formally relativistic only after assigning transformation properties to the fix velocity v µ . We must take into account that the velocity v µ as well as the e i µ can also be used to build up relativistic invariant terms. With the above modifications (i) and (ii) we shall apply the rules [18] to the case (4.1) We build up the following object We assign non-linear transformations under the full group U (4N hf ) in the standard manner [18] g(θ)e H =: e H ′ h(H, θ) , where H ′ is the transformed field. Then whereḡ = γ 0 g † γ 0 . The following property holds which implies that Because of the local gauge symmetry we can only build the following connection and covariant tensor Notice that any derivative with respect to x i := e i µ x µ acting on functions of x i which are not scalars will not be covariant under the local transformations.
The u(4N hf ) algebra and the HQET lagrangian are invariant under the following discrete symmetry which is reminiscent of charge conjugation. They are also invariant under the SO(3) All these symmetries should also be implemented in the effective lagrangian.
We can start at this point the construction of the effective lagrangian, order by order in derivatives, using the objects defined above. At first order it turns out that there is no invariant term. Still there is a term which is invariant up to a total derivative. It reads T r means trace over flavour and Dirac indices whereas tr means trace over flavour indices only. We keep tr for trace over Dirac indices only. It is not difficult to prove that T r(/ vhv.∂h −1 ) is indeed a total derivative. This is analogous to the case of the Heisenberg ferromagnet where the leading order term in the effective lagrangian for the Goldstone mode is also invariant up to a total derivative [20]. Then at leading order the long distance properties of heavy quarkonia are governed by a single constant. At next to leading order we have the term Terms containing x i derivatives start appearing at sixth order. Notice that there is no vertex involving an odd number of fields. This holds at any order in derivatives and it is a consequence of the separate conservation of the number of heavy quarks and antiquarks.
For convenience we normalize the effective lagrangian as follows  L is now locally invariant under U (4N hf ) if we assign to a the transformation property a −→ gag −1 + giv.∂g −1 . Then we may identifyh where T ab is the zero matrix in flavor space except for a 1 in row a column b. It is interesting to observe that the U (4N hf ) symmetry is so large that any bilinear current of the kind Let us next calculate for further convenience the correlators (3.9) and (3.10) in the hadronic effective lagrangian. For (3.9) we have For (3.10) we have Notice at this point that we may obtain (3.9) and (3.10) from (4.18) and (4.19) by taking Hence f 2 H at the hadronic level plays the role of the cut-off µ at quark level. Observe also that the dependence on the Γ-matrices in (4.18)-(4.19) is explicit. All decay constants and matrix elements of bilinear currents are given in terms of the only non-perturbative parameter f H . This is a direct consequence of the U (4N hf ) symmetry being spontaneously broken down to U (2N hf ) ⊗ U (2N hf ).

Separating and evaluating off-shell and on-shell contributions
Consider the current-current correlator We separateQ Where (Q a ΓQ b ) on and (Q a ΓQ b ) of f means that both heavy quark fields in the current have momenta almost on-shell and off-shell respectively. Our goal is to obtain a representation in terms of the HQET of any Green function containing an (Q a ΓQ b ) on . In order to enforce "on-shellness" it is convenient to make the substitution 3) Analogously, using (2.14), (2.16) and (3.13) we have The contribution involving only off shell quarks has the familiar form The expressions (5.5) and (5.6) correspond to corrections O(Λ 3 QCD a 3 ab,n ) and O(Λ 6 QCD a 6 ab,n ) respectively to the leading result (5.7), a ab,n ∼ n/(αµ ab ) is the Bohr radius. Since we are only interested in the leading non-perturbative corrections we shall neglect (5.6) in the following. Let us only remark that the hadronization of the four quark operator in (5.6) introduces new parameters. This is because it is not a generator of the U (4N hf ) symmetry as the currents of the kind (4.17) are.
The r.h.s. of (5.5) can be hadronized and calculated using the effective lagrangian discussed in section 4. From (4.18) we obtain Notice that the result is spin independent and the flavor dependence resides only in the wave function, which is known. We finally obtain |f ψ Q (n) | 2 = 4m ab,n N c |Ψ ab,n (0)| 2 + 1 2 Ψ * ab,n (0)Ψ ab,n (0) + Ψ * ab,n (0)Ψ ab,n (0) f 2 H , (5.9) Notice that the non-perturbative correction we find to the decay constant is O(Λ 3 QCD a 3 ab,n ) and hence presumably more important that the correction arising from the multipole expansion which is O((Λ QCD a ab,n ) 4 /α 2 ) [8,9] (we count the quark condensate as O(Λ 4 QCD )).

Cut-off independence
Let us next discuss the important issue of the cut-off independence. Even though we have not written it down explicitely, the introduction of a cut-off to separete almost on-shell momenta from off-shell momenta is necessary. Of course, the final results must not depend on the particular value of the cut-off. At the short distance end of the calculation, the cut-off must exclude momenta which are almost on-shell. This is easily achieved by cutting off small momenta from the wave function ab,n (0) , (5.10) where µ is a symmetric IR cut-off in three momentum. The wave functions in (5.9) must be understood as the cut-off wave functions (5.10). On the HQET side the cut-off must be ultraviolet. It has already been displayed in the leading order perturbative calculation at quark level in section 3. In particular, from (3.9) we obtain This strong cut-off dependence, however, is totally compensated by (5.10). Indeed, once (5.10) is used we have ab,n (0) + Ψ (µ) ab,n * (0)Ψ ab,n (µ) = − µ 2 2π 2 Ψ * ab,n (0)Ψ ab,n (0) + Ψ * ab,n (0)Ψ ab,n (0) ab,n (0)( µ 3 6π 2 ) =Ψ * ab,n (0)Ψ ab,n (0)

(5.12)
Notice that the way in which the cut-off dependence cancels is remarkable. The strong cut-off dependence of (5.11) was first found in [11]. It was not clear at all which short distance contribution it should cancel against. (5.10) gives the solution to that puzzle. It is apparent from (5.8) and (5.11) that f H in the hadronic theory plays the role of the UV cut-off in the HQET at quark level. From (5.12) it is clear that the cut-off µ must be much smaller than the invers Bohr radius. Therefore our formalism becomes exact in the following situation Furthermore, we have to assume that µ can be taken large enough so that we may enter the asymptotic freedom regime from the HQET side. Otherwise the matching we have carried out at tree level would not make much sense.
From the discussion above it should also be clear that (5.9) can be written in a cut-off independent way at O((µa ab,n ) 3 ) by just replacing wheref 2 H need not be positive.

Physical state normalization
There is still a subtle point which makes eq. (5.9) with the replacement (5.14) not quite correct. It has to do with the normalization of physical states. It will be clear later on (see eq. (6.14) below) that the states we obtain by this procedure do not have the standard relativistic normalization as they are supposed to. When we evaluate the Green function (5.1) we insert resolutions of the identity which are approximated by Coulombtype states. This is all right. However the low momentum tale of these states is cut-off and substituted by a quantity evaluated using the effective hadronic theory. After doing so there is no guarantee that the resolution of the identity we introduced is still properly normalized. This can be fixed up by changing where |n n| (µ) symbolises the cut-off Coulomb states whose low energy tale is evaluated in the hadronic effective theory. We present a heuristic calculation of N n (µ, f H ).
We start from the Coulomb-type bound state (2.5) and separate high and low relative momentum according to |Γ n , P n = m ab,n v = |Γ n , P n = m ab,n v k>µ + |Γ n , P n = m ab,n v k<µ (5.16) The high momentum part of the physical state can be well approximated by the Coulombtype contribution so we may leave it as it stands. However, the low momentum part receives non-perturbative corrections, which we evaluate using the effective hadronic lagrangian.
We proceed as follows. Since a ab,n µ ≪ 1, we can approximate the low momentum region by Observe now that (5.17) is nothing but the integral of a local HQET current.
where k → 0 and only low momenta are allowed.
At this point, we can hadronize the current (see (4.17)) and calculate the low momentum contribution to N n (µ, f H ) k<µ Γ n , P n = m ab,n v|Γ n , P ′ n = m ab,n v ′ k<µ Then, putting together high and low momentum contributions, we have Γ n , P n = m ab,n v|Γ n , P ′ n = m ab,n v ′ = 2m ab, (5.20) Wheref 2 H is defined in (5.14) and notice that the result is cut-off independent.
Finally, the normalization factor reads N n (µ, f H ) can also be obtained from requiring as we shall see later on. Once we have taken into account the correct normalization (5.9) We shall relegate to section 7 the discussion on the applicability of the limit (5.13) and formula (5.23) to physical situations.

Matrix elements at zero recoil
We are interested in Green functions of the kind For the momentum range We separate each current in almost on-shell momenta and off-shell momenta according to (5.2). The leading contribution is given by the term and the next to leading contribution by the term The calculation of (6.5) and (6.7) is analogous to the ones carried out in section 2. We obtain G on,1 Notice that (6.5) and (6.7) can not be written in terms of local Green functions in the HQET. One propagator must be kept explicit.
The calculation of (6.6) is more subtle. We describe it in some detail in the Appendix B. We obtain 10) This term is the only one in (6.4) which remains in the matrix elements (see (6.14) below).
We calculate (6.8)-(6.10) using the hadronic effective lagrangian (see formulas (4.18) and (4.19)). We obtain The matrix element at zero recoil then reads Γ n = iγ 5 p − , i/ e i p − for the pseudoscalar and vector particle respectively. The integral in (6.14) must be understood with an infrared cut-off µ. From (6.14) it is apparent that our physical states are not properly normalized. Indeed, for b = c and Γ = γ 0 one should obtain (5.22) but one does not. The reason for this has been discussed at the end of sect.
5. The solution consist of introducing the normalization factor N n (µ, f H ) defined in (5.21).
The properly normalized result reads Γ n , P n = m ab,n v|Q b ΓQ c (0)|Γ m , P m = m ac,m v = − √ m ab,n m ac,m tr(Γ n ΓΓ m ) Notice that the non-perturbative correction depends only on a single parameterf 2 H which may be extracted from the decay constants calculated in section 5. This is a nontrivial prediction which turns out to be a direct consequence of the U (4N hf ) symmetry being spontaneously broken down to U (2N hf ) ⊗ U (2N hf ).

Applications
If the charm and bottom mass were large enough we could apply the results above to the physics of Υ, η b , B c , B * c , J/Ψ and η c . (The top is believed to be too heavy to form hadronic bound states and will be ignored). We analyse in this section whether this is so or not. In the systems where the formalism actually applies, we are mainly interested in estimating the importance of the new non-perturbative contribution rather than in obtaining accurate results. The latter is a much harder task which is definitely beyond the scope of the present work.
Let us first focus on bottom. The fact that the almost 'on-shell' momentum excitations in heavy quarkonium are Goldstone modes [11] implies that the Υ and η b spectrum does not received additional non-perturbative contributions. We may then extract the bottom mass from the Υ mass by means of the formulas given in [8,9] which take into account the leading order in the multipole expansion. Since we have established a link between quarkonium and the HQET we can next use m b to extractΛ, the non-perturbative parameter relating the mass of the B-meson to m b . Moreover, taking into account thatΛ is flavour independent, we may next extract the charm mass m c . We summarize the results in the Table I. In Table I the values we obtain for m b are about a 3% lower than those obtained in QCD sum rules [22] but compatible with a recent QCD-based evaluation [23] and with the lattice calculation [24]. The values we obtain forΛ are a bit lower but otherwise compatible with those extracted from QCD sum rules [6]. Our values for m c are again about a 6% lower than the typical values in QCD sum rules [22]. We should emphasize that our numbers in Table I are model independent.
We can next extract the non-perturbative parameterf 2 H from f Υ (this is done in Table   II). We use the following formula where the 1-loop QCD corrections and the leading correction from the multipole expansion [9] * are taken into account.
The numbers in Table II are  Observe that the conditions (5.13), in particular a −1 bb,0 ≫ µ ≫ Λ QCD , may be considered as reasonable well fulfilled if we take the cut-off µ ∼ 700 MeV (see table III below).
Let us next turn our attention to charm. The charm mass is known not to be heavy enough as for the multipole expansion to work [8]. This means that the non-perturbative * We use the formula given in ref. [9] which differs from the ones in ref. [8].
contribution overwhelms the perturbative one. Therefore any approximation whose leading order is a perturbative contribution, like our approach, will not be able to say much about charmonium. In particular, for the 'on-shell' contributions the difficulty lies on the second last condition in (5.13) being fulfilled. There is little room to accomodate the cut-off µ between the invers Bohr radius and Λ QCD as should be clear from Table III. We refrain ourselves from giving any numbers for charmonium.
Unfortunately, the situation is not much better for the B c , which has received considerable attention lately [25][26][27]. Nonetheless, once we havef 2 H , we shall give some numbers in this case in Table IV.
From Table IV we see that for Λ QCD = 100 , 150M eV the contribution of the condensate is too large for the approach to be reliable. For Λ QCD = 200M eV we are at the boundary of its validity since the 'on-shell' correction is large. We may thus give a rought estimate for f B c only for Λ QCD ∼ 200M eV , which turns out to be compatible with the estimate obtained by QCD sum rules [26], but about a 30% lower than potential model estimates [27].
From the Table V it follows that the new non-perturbative contribution is not very important in the matrix elements between Υ − B c states.
The decay constants and matrix elements above receive contributions from corrections of several types: (i) QCD perturbative corrections to the Coulomb potential O(α(1/a n )). These have been evaluated at one loop level in [28] (see also [23]).
(iii) QCD perturbative corrections to the Green functions O(α(m)). These corrections have been taken into account in (7.1). They correspond to the only QCD corrections in heavy-light systems. In our case they are important for the calculation of matrix elements at non-zero recoil.

Conclusions
We have demostrated that, contrary to the common belief, HQET techniques are also useful for the study of systems composed of two heavy quarks. In particular, we have identified new nonperturbative contributions to the decay constants and to certain matrix elements which are described by a hadronic lagrangian based on the HQET. All these new contributions are parametrized at leading order by a single constant f H . This is non-trivial and can be traced back to the fact that a U (4N hf ) symmetry is spontaneously broken down It is remarkable that strong cut-off dependences coming from a totally different origin match perfectly. Indeed, at the off-shell end the cut-off arises from an integral over a Coulomb type wave function, whereas at the on-shell end it arises from a Feynman integral.
We should also stress that we have been able to put in the same context (i.e. the HQET) both heavy-heavy and heavy-light systems. This allows for a model independent determination of heavy quark masses from quarkonium, which may then be used to extract the parameterΛ relating the mass of the heavy-light systems to the mass of the heavy quark.
As far as practical applications is concerned, our formalism is suitable for the ground state of the Υ and η b family. Unfortunately the charm mass is too small for the formalism to become applicable in general to J/Ψ and B c systems. Nevertheless one may stretch it in some cases to obtain information on the mass and decay constant of the latter.
We have presented a technique which allows to disentangle the on-shell contributions from the rest and match them to the HQET. The matching has been carried out at tree level. We have already shown in [29] that the matching also goes through at one loop level.
Nevertheless, a word of caution is needed. It would be desirable to have a more direct and systematic derivation of these results from QCD. Progress in this direction is being made [30].
Let us finally mention that the hadronic HQET lagrangian can easily incorporate heavy-light mesons. The formalism can then be extended to the calculation of matrix elements between quarkonium and heavy-light systems. The leading non-perturbative contributions to those are also given by f H and another non-perturbative parameter which is related to heavy-light decay constants. Non-recoil contributions can also be evaluated within the formalism.
Within this model, the interactions between (η Q , η Q ′ ), (η Q , ψ Q ′ ), (ψ Q , ψ Q ′ ), when the two particles move roughtly at the same velocity, are described by a single unknown constant. This is analogous to the fact that at lowest order in 3-flavour chiral perturbation theory the elastic scattering of (π, π), (K, K) and (π, K) is also described by a single constant. When heavy-light mesons are included in the effective lagrangian the same constant describes the elastic scattering of heavy-light mesons with quarkonium. This is also analogous to the fact that the local vertex π-π-N -N at leading order in the Chiral Lagrangian is described by the same constant as the (π, π) elastic scattering. Let us mention at this point that when one actually calculates the scattering amplitudes, one obtains zero. This has to do with the universality of the leading order effective lagrangians for Goldstone modes [18,19,20]. Any theory undergoing a U (4N hf ) spontaneous symmetry breaking down to U (2N hf ) ⊗ U (2N hf ) has the same low energy effective lagrangian (4.12) provided the rest of the symmetries in the theory are also the same. It was shown in [11], that even when the gluons are switched off the spontaneous symmetry breaking occurs in the HQET. In that case there is no interaction in the fundamental theory and hence it is not surprising that the scattering amplitudes in the effective lagrangian vanish. Universality implies that there will be vanishing scattering amplitudes when the gluons are switched on as well.
Within this model one can also treat 1/m corrections in a similar way that small quarks masses are dealt with in Chiral perturbation theory. At the quark level the leading 1/m corrections to the HQET are given by a kinetic term and a spin breaking term The kinetic term (A.1) does not break the global U (2N hf ) ⊗ U (2N hf ) symmetry but it breaks its local version. In order to construct at the hadronic level terms which break the U (4N hf ) symmetry in the same fashion as (A.1) does, we introduce the u(4N hf )-valued sources φ and a i transforming as Then the term is on one hand invariant under U (4N hf ) and on the other reduces to (A.1) upon setting At the hadronic level, we must then construct invariant terms linear in φ, which may also contain a i . Up to two space derivatives we have We have not written down terms which coincide or vanish upon using (A.5).
For the spin breaking term (A.2) we may introduce a u(4N hf )-valued source R l transforming as so that (A.2) is substituted byh We recover (A.2) upon setting There are no terms at the hadronic level with the same symmetry transformation properties at lower orders in derivatives. The first possible term appears at third order.
Therefore The procedure above can easily be extended to any finite order in 1/m.

Appendix B
We present in this appendix some technical details on the evaluation of the off-shell short distance effects carried out in sect. 6.
Consider the following matrix element Any β is equaly good since we are eventually interested in the limit x 3 = x 4 = x. Notice that the ambiguity in α in (B.6) is proportional to the ambiguity in β in (B.9). Since we can choose β at will, we do it in such a way that the dependence in both α and β cancels. This is how we are able to obtain a representation of (6.6) in terms of the HQET (6.10). Table I. We use Λ QCD as an input and take the one loop running coupling constant α at the scale of the invers Bohr radius, i.e. α = α(1/a bb,0 ). For the gluon condensate we take the fix value < B 2 >= (585M eV ) 4 Table II. We display the relative weight, with its sign, of the 1-loop (α(m b )), the condensate (< B 2 >) and the 'on-shell' (f H ) contribution with respect to the Coulomb type contribution (normalized to 1). The last columns display the mass m cr from which the 'on-shell' contribution dominates over the condensate and the value of (f 2 H ) 1/3 . Table III. We give thecc,bc,bb invers Bohr radius as a function of Λ QCD . Table IV. We display the analogous to Table II for B c . We have also given our predictions for f B c in the last column. Table V. We give the relative weight, with its sign, of the 'on-shell' contribution with respect to the Coulomb-type contribution (normalized to 1) in the matrix elements (6.15) between Υ − B c states.