The Direct Coupling of Light Quarks to Heavy Di-quarks

In the limit $m_Q>m_Q v_{\rm rel}>m_Q v_{\rm rel}^2 \gg \Lambda_{QCD}$ hadronic states with two heavy quarks $Q$ should be describable by a version of HQET where the heavy quark is replaced by a di-quark degree of freedom. In this limit the di-quark is a small (compared with $1/\Lambda_{QCD}$) color anti-triplet, bound primarily by a color Coulomb potential. The excited Coulombic states and color six states are much heavier than the color anti-triplet ground state. The low lying spectrum of hadrons containing two heavy quarks is then determined by the coupling of the light quarks and gluons with a momentum of order $\Lambda_{QCD}$ to this ground state di-quark. In this short paper we calculate the coefficient of leading local operator $\left( S_v^{\dagger} S_v\right) \left({\bar q} {\gamma^{\mu} v_{\mu} } q\right) $ that couples this color-triplet di-quark field $S_v$ (with four-velocity $v$) directly to the light quarks $q$ in the low energy effective theory. It is ${\cal O}(1/( \alpha_s(m_Q v_{\rm rel}) m_Q^2))$. While our work is mostly of pedagogical value we make an estimate of the contribution of this operator to the masses of $\Xi_{bbq}$ baryon and $T_{QQ{\bar q} {\bar q}}$ tetraquark using the non-relativistic constituent quark model.


I. INTRODUCTION
The lowest lying 1 Ξ QQq baryons containing two heavy quarks are stable with respect to the strong interactions. For very heavy quarks Q the lowest lying T QQqq tetraquark states are also stable with respect to the strong interactions [1,2]. The reason for this is quite simple. If m Q Λ QCD then, when the heavy di-quark is in a color3 configuration, because of the attractive one gluon color Coulombic potential the di-quark has a large binding energy (compared with Λ QCD ) and a small size (compared with 1/Λ QCD ). Strong decay of the lowest lying T QQqq tetraquark states to a baryon with two heavy quarks and an anti-nucleon (when q = u, d), Ξ QQq +N qqq , is kinematically forbidden since the final state has an additional qq pair, which costs an additional ∼ 600 MeV of mass. Strong decay to two heavy mesons M Qq + M Qq does not require an additional qq pair but now the final state does not have the large color Coulombic binding energy proportional to m Q that the tetraquark state does and so this channel is also kinematically forbidden.
In nature the heavy quarks that are long lived are the charm and bottom quarks and whether they are heavy enough for tetraquarks containing them to be stable with respect to the strong interactions is not certain but widely believed to be the case for the lowest lying tetraquarks with two bottom quarks. See [3][4][5] for recent support for this hypothesis.
There are indications that this is not true for the tetraquarks with charm quarks from the previous mentioned studies and [6].
Effective field theory methods (discussed mostly in the context of theQQ channel) have been developed [7][8][9] to take advantage of the fact that for very heavy quarks Q the color anti-triplet di-quark QQ has a size small compared with 1/Λ QCD . In this paper we work in the limit m Q > m Q v rel > m Q v 2 rel Λ QCD where at leading order the light quarks and gluons with momentum of order Λ QCD (which we call Λ QCD degrees of freedom) in hadrons containing this di-quark regard the di-quark as a point object.
By the sequence of inequalities m Q > m Q v rel > m Q v 2 rel Λ QCD we mean that while we do treat v rel as small compared with unity, Λ QCD /m Q is much smaller. Hence we will not treat logarithms of the relative velocity as small and resume them. In this case one can match full QCD directly onto an HQET like theory at a scale µ which we take to be µ = m Q v rel .
In this limit the color six QQ configurations and Coulombic excitations above the lowest lying color anti-triplet di-quark state (described by principal quantum numbers n > 1) are much heavier than the ground state and can be integrated out of the theory. Hence (in the single di-quark sector) one arrives at a theory like HQET [10,11] with Lagrange density, Including the heavy quark fields 2 the leading terms would not only have the familar heavy quark spin-flavor symmetry [12] but also an enlarged heavy quark and di-quark spin-flavor symmetry [4,[13][14][15][16]. Λ QCD gluons coupling directly to the heavy di-quark already occur at leading order through the covariant derivative D = ∂ − igT B A B where the bar denotes that the SU (3) generators are in the3 representation. Of course the Λ QCD quarks q interact with the gluons, so even at leading order they interact with the di-quark. The purpose of this paper 2 After including the heavy quark fields Q the effective field theory is used in the one heavy quark and one heavy di-quark sectors.
is to calculate the direct coupling of the Λ QCD quarks q to the di-quark field S which occurs in the ellipses of eq. (1.1). We find that this operator is of the form, S † v S v (qγ µ v µ q), and that its coefficient occurs at order O(1/(α s (m Q v rel )m 2 Q )) which is between O(1/m Q ) and O(1/m 2 Q ). The 1/(α s (m Q v rel )) arises because this term is suppressed by the di-quark size 3 . We find it interesting to compute the coefficient of this term because it gives rise to dependence on the heavy quark mass and hence a breaking of heavy quark di-quark flavor symmetry that arises from the size of the di-quark system. The pattern and heavy quark mass dependence of its contribution to the breaking of heavy quark di-quark flavor symmetry is different from that of the heavy quark and di-quark kinetic terms that arise at . For example, the term we are focussing on contributes to the Ξ QQq baryon mass but not to the M Qq meson mass, while ∆L kin contributes to both.
In nature the heavy quarks are the top, bottom and charm quarks. While the top is very heavy compared with the QCD scale, it is short lived and does not form hadronic bound states. That leaves the bottom and charm quarks. Dimensional analysis suggests that for neither of these quarks will approximations based on m b,c v 2 rel Λ QCD be valid and even predictions based on the condition m b,c v rel Λ QCD are suspect although it is likely that in the bottom quark case that they have some utility. Hence we view our work as mostly of pedagogical value. Despite these cautionary remarks we will make an estimate of the importance of the operator S † v S v (qγ µ v µ q) to the masses of the lowest lying Ξ bbq baryon and T QQqq tetraquark using the non-relativistic constituent quark model.

II. THE GROUND STATE COLOR3 DI-QUARK
Di-quarks can be in either a color triplet or color six representation. The 3×3 →3 channel is attractive and after tracing over the color the short range color Coulombic potential is, This is half as strong as the attractive potential in the 3 ×3 → 1 channel, which is relevant for quarkonium. While it is a bit odd to consider the qualitative reason for a factor of two difference between the potentials in these two channels, one can be found in the large number of colors N c limit [17] . Assuming (just for simplicity) that all the quark flavors are different in the large N c limit the interpolating field for a Ξ Q 1 Q 2 q 3 ,...q Nc−2 baryon is (suppressing constants, and all indices except for flavor and color) β 1 β 2 α 1 ...α Nc−2 Q 1β 1 Q 2β 2 q 1α 1 . . . q Nc−2α Nc−2 and the short range effective potential between the two heavy quarks in this state is, This is suppressed by a factor of 1/N c in the large number of colors limit where N c α s is held fixed as N c → ∞. At large N c the appropriate interpolating field for tetraquarks with two ) and eq. (2.2) still applies for the color Coulombic potential between the heavy quarks. On the other hand, for the case of QQ quarkonium with N c colors the appropriate interpolating field isQ α Q α and the color Coulombic potential is which does does not vanish in the large N c limit.
Returning to the real world where N c = 3 the color3 di-quark states are twice the size of the color singlet quarkonium states and so the multipole expansion should be somewhat less reliable in the di-quark case. The ground state di-quark has a spatial wave-function where the Bohr radius is a 0 = 3 2α s µ Q , (2.5) and the reduced heavy quark mass is In the case where the heavy quarks are the same flavor they must be in a spin-one state. When they are different there are degenerate spin-zero and spin-one cases. Spin is usually inert for our purposes in this paper and we will usually not keep track of those labels in our equations.
Later we will need the square of the charge radius, Since were are not treating v rel as very small the argument of the strong coupling can be taken to be either m Q or m Q v rel , in the equations given in this section. However the latter is physically more appropriate so we will use it for any quantitative estimates we make using these formulae going forward.

III. MATCHING
One can compute matching onto the effective HQET like di-quark effective field theory by computing an appropriate physical perturbative process in QCD. We will work in the leading Feynman diagrams that contribute (at tree level) to elastic light quark heavy di-quark scattering.
logarithmic approximation, which means tree level matching and one-loop renormalization group running. By physical we mean on-shell but not taking into account confinement.
Since we are interested in local operators involving the light quarks and the di-quark field S the appropriate process is light quark heavy di-quark elastic scattering, Fig. 1. In the heavy quark limit the scattering must be elastic, k 0 f = k 0 i and we denote the three-momentum transfer by k = k i − k f and k = |k|. We work in the rest frame of the di-quark.
Expanding the heavy quark spinors to zeroth order in three-momenta we find that the amplitude A for this process is Hereφ is the Fourier transform of the spatial wave-function for the di-quark state. Expanding eq. (3.1) in k the term at zero'th order corresponds to the leading order Lagrangian in eq. (1.1) where the light quark scatters off the anti-triplet charge of the di-quark without resolving its size. The term linear order in k vanishes because the ground state wave-function φ is s-wave. The term quadratic order in k is, where the subscript 2 denotes that we have expanded to quadratic order in k.
Matching onto the effective theory we find the contribution (generalizing to arbitrary di-quark four-velocity v) to its Lagrange density whereT A = −(T A ) T are the SU (3) generators in the anti-triplet representation and the coefficient C = πα s r 2 3 The charge radius arises from the non-zero size of the di-quark as so we feel it is appropriate to view the coefficient in eq. (3.4) as evaluated at the subtraction point µ = m Q v rel even though as mentioned earlier we will not be keeping track of factors of v rel in logarithms as far as the power counting is concerned.
The cancellation of the gluon propagator's 1/k 2 by the factor of k 2 from expanding the wave-functions has the same origin as in penguin diagrams for weak decays and can be thought of as arising from an application of the equations of motion [18]. Finally we remove the product of SU(3) generators using the identity (recallT A = (−T A ) T ), T A αβ T A µν = −(1/6)δ αβ δ µν + (1/2)δ αν δ βµ , and write the effective Lagrangian as and The subtraction point dependence of the operators O 1,2 is given by the renormalization group equations, This anomalous dimension matrix γ is computed from the one loop diagrams Feynman diagrams in Fig. 2. Using dimensional regularization with minimal subtraction we find that, where n q is the number of light quark flavors. The last is not one-particle irreducible but contributes to a local operator in the same way the penguin diagrams do in kaon decay.
Combining this with the results of the previous section we arrive at The two equations above are the main results of this paper. Including logarithmic corrections to scale the effective Lagrangian down from the scale m Q v rel is not just of academic interest. As an example of how it can matter consider the color magnetic moment term that arises in the matching from expanding the heavy quark spinors to leading order in the gluon momentum. It gives rise in the rest frame of the di-quark (when the two heavy quarks composing the di-quark are identical) to the term 4 .
where S is the di-quark spin vector. If we had not evaluated the strong coupling at µ but rather at the matching scale the anomalous dimension of the operator would be large and effectively bring the scale the coupling is evaluated at to µ. What we have seen in this section is that a large anomalous scaling like this does not occur for the local operator O − .

V. A NON-RELATIVISTIC CONSTITUENT QUARK MODEL ESTIMATE
We can get a rough idea about how large the contribution of eq. (4.4) is to the mass of the Ξ bbq baryon by making a non-relativistic quark model estimate of the matrix element 4 In this case the anomalous scaling is the same as in the heavy quark case [19,20].
of O − which presumably we should view as reasonable for a subtraction point µ around the QCD scale. In the non relativistic constituent quark model color is included through a color factor and then the light quarks are viewed as non-relativistic quasi-particles bound by some potential. By relating various physical quantities in the model an estimate can be made independent of the particular potential that binds the constituent quarks in a hadron. The estimate we make in this section is similar in spirit to using the vacuum insertion approximation [21] for the K −K matrix element of the four-quark operator The color configuration for the Ξ bbq is (1/ √ 3)S α q α . In the non-relativistic quark model, for a Ξ bbq at rest, we find the O − expectation value to be where n S (x) = δ 3 (x) is the number density of di-quarks, n q (x) = |φ q (x)| 2 is the number density of light quarks q, φ q is the wave function of q, and −8 is the color factor. Using heavy quark symmetry φ(0) is related to the B-meson decay constant, Combining these results, neglecting the renormalization group running and setting m b = m B , we have that the contribution of the Lagrange density in eq. (4.4) to the Ξ bbq mass, ∆m Ξ bbq , is estimated to be Here we used f B 190 MeV and α s (m b v) 0.35 for the numerical result. The numerical result in eq. (5.3) above is only a little smaller than a typical order Λ 2 QCD /m b contribution to the Ξ bbq mass. This should not be particularly surprising given that the Bohr radius for such a color Coulombic bound state is The color configuration for the T bbqq tetraquark is (1/ √ 6) αβγ S αqβqγ (there are spin flavor labels on the light quarks that we have suppressed). In the non-relativistic constituent quark model, for a T bbqq at rest expectation value, we find that The color factor is −1/2 what it was for the baryon case and there is an additional minus sign becauseqγ 0 q = n q − nq. Since T bbqq contains two antiquarks (while Ξ bbq contains a single quark) a contribution of about 30 MeV to the mass of the T bbqq tetraquark from this term is a reasonable estimate. Note that if the contribution of O − to the mass of the T QQqq tetraquark and Ξ QQq baryon are the same then ∆L in eq. (4.4 ) does not correct the leading order sum rule [4], m QQqq − m Qqq = m QQq − m Qq .
Of course there are additional contributions to the masses of the Ξ bbq and T bbqq hadrons from the leading terms explicitly displayed in eq. (1.1) and the familiar (from HQET) terms of order 1/m Q . However these do not arise from the size of the heavy di-quark and have a different pattern of contributions to the masses of hadrons containing one heavy quark or di-quark and a different dependence on the heavy quark mass.
This paper is about the effective field theory for the ground state anti-triplet di-quark and the direct coupling of light Λ QCD quarks to the ground state di-quark degrees of freedom in that effective HQET like theory. If we did not take 5 m Q v 2 rel Λ QCD then such an effective field theory would not be appropriate. One could still write an effective theory [15,16] for the lowest lying baryons (or tetraquarks) containing two heavy quarks interacting with low momentum photons and pions, or an effective theory containing the possible di-quark configurations (pNRQCD) but matching the latter to an effective field theory just containing the lowest lying di-quark configuration and the Λ QCD gluon and light quark degrees of freedom would not be justified.
To illustrate this let us consider the case where the two heavy quarks are different flavors.
Then expanding eq. (3.1) to linear order in k we match onto a transition operator taking the lowest lying (n=2) L = 1 color anti=triplet di-quark 6 S j to the lowest lying (n=1) L = 0 di-quark field S we have been considering. In the rest frame of the di-quarks, where the transition charge radius is 128 81 a 0 (6.2) eq. (6.1) contributes to the mass of a Ξ Q 1 Q 2 q baryon at second order in ∆L an amount of order Here the strong coupling g in eq. (6.1) is evaluated at the subtraction point (i.e., near the QCD scale) and not the matching scale since we know from HQET that there is a large anomalous dimension that makes this appropriate. Recall that the contribution from the matrix element of O − estimated in the previous section (see 5 Including numerical factors in the ground state di-quark color Coulombic binding energy we need Λ QCD when the two heavy quarks are the same. 6 We take the two heavy quarks to be in the spin-zero configuration so the total spin of the initial di-quark is one and the final di-quark is zero.

eq. (5.3)) is of order ∆m
. So the impact on the Ξ Q 1 Q 2 q mass from eq. (6.1) at second order in perturbation theory is suppressed by a factor of rel )) when compared with the contribution of O − . This contribution and the contribution of other excited di-quark states (including the color six continuum and color anti-triplet scattering states) would not be suppressed if we did not work in the limit m Q v 2 rel Λ QCD .

VII. CONCLUDING REMARKS
In this paper we have computed the leading direct coupling of the quarks that have momenta of order Λ QCD to the effective color anti-triplet di-quark degree of freedom S assuming the hierarchy of scales, m Q > m Q v rel > m Q v 2 rel Λ QCD . In the effective HQET like theory for di-quarks this comes from the operator S † v S v (qγ µ v µ q) which corresponds in the baryon Ξ QQq to a repulsive delta function potential between the heavy di-quark and the light quarks and in the T QQqq a repulsive delta function potential between the heavy di-quark and the light anti-quarks. It arises from the finite size of the di-quark and has ). Its coefficient is anomalously large because the factor of 1/α s (m Q v rel ) originates from g(m Q v rel ) 2 /α s (m Q v rel ) 2 which gives an additional 4π when written in terms of color fine structure constant 7 . We estimated, using the non-relativistic quark model, that this term would contribute around 30 MeV to the mass of tetraquarks and baryons containing two bottom quarks. It gives rise to the leading violation of heavy quark, di-quark flavor symmetry arising from the finite size of the di-quark.
If the stability (with respect to the strong interactions) of tetraquarks containing two heavy bottom quarks is firmly established then it will still be interesting to study other aspects of their physical properties. For example, will they correspond more to the small (compared with 1/Λ QCD ) di-quark picture or to a di-meson molecule. The latter is possible since the long range potential from one pion exchange is attractive in some channels and capable of giving rise to bound states [2]. Perhaps tetraquarks that contain two heavy bottom quarks and are stable with respect to the strong interactions will lie between these two extremes.

ACKNOWLEDGMENTS
We thank Aneesh Manohar, Tom Mehen, Ira Rothstein, and Mikhail Solon for some useful comments. HA is supported by the Recruitment Program for Young Professionals of 7 Hence higher order terms in the multipole expansion that arise from expanding eq. (3.1) to higher orders in k will not be even more enhanced.