Semileptonic decays of heavy tetraquarks.

There exist heavy tetraquark states QQq̄q̄ in the heavy quark limit. These states are stable with respect to strong interactions and hence must decay weakly. It is shown that the semileptonic decay depends on a single Isgur– Wise form factor, which can be expressed in term of the Isgur–Wise form factors which govern the semileptonic decays of the Qqq and QQq baryons. Typeset using REVTEX 1 The possibility of the existence of exotics, i.e, hadrons with properties incompatible with a qq̄ or qqq description, is an important issue in QCD. In particular, the existence of tetraquark (or di-meson) states has been proposed by Jaffe [1] and since then investigated by many other groups [2–5]. The case of binding two heavy quarks to two light antiquarks to form a heavy tetraquark QQq̄q̄ is especially interesting. The two heavy quarks are bound by the short range Coulumbic chromoelectric attraction in the 3 ⊗ 3 → 3̄ channel. The bound state will have binding energy of order E ∼ α s(mQ)mQ. When mQ is so large such that E ≫ ΛQCD, the large binding energy forbids the dissociation QQq̄q̄ → Qq̄ + Qq̄. Moreover, the light degrees of freedom cannot resolve the closely bound QQ system, which has size of order (mQ αs(mQ)) −1 ≪ Λ QCD. This results in bound states that have “brown mucks” similar to those of Λ̄Q states, with QQ playing the role of the heavy antiquark. Hence the stability of Λ̄Q implies that QQq̄q̄ is also safe from decaying through QQq̄q̄ → QQq + q̄q̄q̄. As the result, QQq̄q̄ is stable with respect to strong interactions and must decay weakly. Since the “brown mucks” of the ground state tetraquarks are spinless, the spins of the tetraquarks are just given by the spins of the heavy quarks. As the result, the ground states are degenerate spin S = 0 or 1 tetraquarks. We will denote the tetraquarks by |T (p, S,m)〉 where m is the z-component of the spin. The normalizations of these states are given by 〈T (p′, S ′, m)|T (p, S,m)〉 = 16πδSS′δmm′δ(p− p′), (1) such that, when the masses of the tetraquarks go to infinity, the states are still well defined. Since the two heavy quarks are bound by a Coulumbic potential, we can represent the heavy degrees of freedom as |QaQb(v, S,m)〉 = ∫ dp |Qa(va, sa)〉 ⊗ |Qb(vb, sb)〉 ⊗ ψ(B;p) ( 2 , sa; 1 2 , sb|S,m). (2) The Clebsch–Gordon coefficient describes the spin structure of the tetraquark, and p = ma (v − va) = −mb (v − vb) is the relative momentum between the heavy quarks. ψ(B;p) is the ground state Coulumbic wavefunction in momemtum space 2 ψ(B;p) = 4B (p2 +B2)2 , (3) with B = μab αs(μab) the reciprocal of the Bohr radius. Together with the spinless light degrees of freedom |φ(v)〉, we get the decomposition |Tab(v, S,m)〉 = |QaQbq̄q̄(v, S,m)〉 = ∫ dp |Qa(va, sa)〉 ⊗ |Qb(vb, sb)〉 ⊗ ψ(B;p)⊗ |φ(v)〉 ( 2, sa; 12 , sb|S,m). (4) We define the Isgur–Wise form factor η̃abc(w) of the semileptonic Tab → Tac decay by 〈Tac(v, S ′, m)|Q̄cΓQb|Tab(v, S,m)〉 = η̃abc(w) δsas′a ūc(v ′, s′c)Γub(v, sb) ( 1 2 , sa; 1 2 , sb|S,m) ( 2 , s′a; 12 , sc|S , m), (5) where w = v · v. The form factor η̃abc(w) contains contributions from both perturbative QCD, which describes the attraction between the two heavy quarks, and non-perturbative QCD, which accounts for the interaction of the QQ system with the light degrees of freedom. By Eq. (4), the matrix element can also be decomposed as 〈Tac(v, S ′, m)|Q̄cΓQb|Tab(v, S,m)〉 = ∫