ar X iv :0 70 7. 12 86 v1 [ he p- ph ] 9 J ul 2 00 7 Light 1−+ exotics: molecular resonances Ignacio J. General†1,Ping Wang†, Stephen R. Cotanch†, Felipe J. Llanes-Estrada‡ † Department of Physics, North Carolina State University, Raleigh, North Carolina 27695-8202 ‡ Departamento de Fisica Teorica I, Universidad Complutense 28040 Madrid, Spain PACS: 12.39.Mk; 12.39.Pn; 12.39.Ki; 12.40.Yx Keywords: Exotic mesons, tetraquarks, QCD Coulomb gauge, effective Hamiltonian Abstract Highlights in the search for nonconventional (non qq̄) meson states are the π1(1400) and π1(1600) exotic candidates. Should they exist, mounting theoretical arguments suggest that they are tetraquark molecular resonances excitable by meson rescat- tering. We report a new tetraquark calculation within a model field theory ap- proximation to Quantum Chromodynamics in the Coulomb gauge supporting this conjecture. We also strengthen this claim by consistently contrasting results with exotic state predictions for hybrid (qq̄g) mesons within the same theoretical frame- work. Our findings confirm that molecular-like configurations involving two color singlets (a resonance, not a bound state) are clearly favored over hybrid or color- exotic tetraquark meson (qq̄qq̄ atoms) formation. Finally, to assist needed further experimental searches we document a useful off-plane correlator for establishing the structure of these exotic systems along with similar, but anticipated much narrower, states that should exist in the charmonium and bottomonium spectra. The existence and understanding of exotic (non qq̄ and qqq) hadrons is one of the few remaining closures to the standard model. Such states are expected according to Quantum Chromodynamics (QCD) and are of intense experimen- tal interest. In the light quark sector, there are two solid isovector candidates at 1.3-1.4 and 1.6 GeV, π1(1400) and π1(1600), each having JPC = 1−+ [1,2]. Although doubts exist about the 1.4 GeV candidate [3], a new analysis [4] supports it. The signature is observed as a p-wave resonance in the ηπ0 sys- tem and it therefore has odd parity P but even charge conjugation C yielding 1 current address: Bayer School of Natural and Environmental Sciences, Duquesne University, Pittsburgh, Pennsylvania 15282 Preprint submitted to Elsevier 11 February 2013 http://arxiv.org/abs/0707.1286v1 the quantum numbers JPC = 1−+. Since a qq̄ state with orbital L and spin S coupled to J = 1 must have P = (−1)L+1 and C = (−1)L+S, this state is clearly exotic. Assuming these states exist, the theoretical situation is even more controver- sial. The debate concerning their structure is among four possible scenarios: 1) a hybrid (qq̄g) meson; 2) a tetraquark atom (qq̄qq̄ involving intermediate color states that are not singlets); 3) a tetraquark molecular bound state of two conventional mesons; 4) a tetraquark molecular resonance (qq̄qq̄ involving two intermediate color states that are singlets but not observed mesons). All four scenarios can produce JPC = 1−+ states but the first two are more exotic since the tetraquark molecule is color equivalent to a conventional meson- meson two-body state (see Fig. 1). Lattice results, now performed with more realistic lower quark masses [5], have focused upon the hybrid scenario but find that hybrid correlators can only produce a state as low as 1.9 to 2.1 GeV. A more recent lattice calculation [6] claims to find two hybrid meson masses below 2 GeV, however they use an ad-hoc extrapolation that is quadratic in the pion mass which increases uncertainties. Similarly, the lightest Flux Tube model 1−+ predictions [7,8,9] are also near 2 GeV spanning the region of 1.8 to 2.1 GeV. This is consistent with agreement among other model approaches, based on either the concept of constituent gluons [10] or field-theory calculations (see below) generating mass-gaps [11], that the lightest hybrid mesons with just one constituent gluon should be somewhat heavier than the 1.4, 1.6 GeV experimental candidates. Finally, both well-established spin [12] and flavor [13] selection rules indicate that the above mentioned ηπ signature cannot be due to a hybrid meson decay. Therefore it would appear that the two π1 states can not be theoretically explained as hybrid mesons. Fig. 1. Four independent tetraquark color schemes. One is a singlet-singlet molecule while the other three are more exotic atoms (octet and two diquark schemes). 2 In search of other explanations, a potential model lifetime calculation [14] has ruled out a molecular η(1295)π or η(1440)π bound state. However, it has been shown that meson-rescattering, specifically in the ηπ channel, could produce a resonance with this signature [15,16]. Also, it has been suggested [17] that the 1.6 GeV resonance could be interfering with a background to produce the 1.4 GeV structure. Summarizing the status of this situation, while the π1(1600) can not be firmly precluded as a hybrid meson or exotic tetraquark, the π1(1400) seems explain- able only as a molecular resonance excited by meson rescattering. The purpose of this work is to confirm the latter by a new theoretical analysis which also predicts that the π1(1600) is not a color exotic system. Our formalism, referred to as the Coulomb Gauge Model (CGM), has been successfully established in both the quark [18,19,20] and gluon [21] sectors and is based upon the exact QCD Hamiltonian in the Coulomb gauge [22] given by HQCD = Hq + Hg + Hqg + HC (1) Hq = ∫ dxΨ†(x)[−iα · ∇ + βm]Ψ(x) (2) Hg = 1 2 ∫ dx [ J −1Πa(x) ·JΠa(x) + Ba(x) · Ba(x) ] (3) Hqg = g ∫ dx Ja(x) ·Aa(x) (4) HC =− g2 2 ∫ dxdyρa(x)J −1Kab(x, y)J ρb(y) . (5) Here g is the QCD coupling, Ψ the quark field with current quark mass m, Aa the gluon fields satisfying the transverse gauge condition, ∇ · Aa = 0, a = 1, 2, ...8, Πa the conjugate fields and Ba the non-abelian magnetic fields, Ba = ∇×Aa + 1 2 gfabcAb ×Ac. The color densities, ρa(x) = Ψ†(x)T aΨ(x) + fabcAb(x) · Πc(x), and quark color currents, Ja = Ψ†(x)αT aΨ(x), entail the SUc(3) color matrices, T a = λa 2 , and structure constants, fabc. The Faddeev- Popov determinant, J = det(M), of the matrix M = ∇ · D with covariant derivative Dab = δab ∇−gfabcAc, is a measure of the gauge manifold curvature and the kernel in Eq. (5) is given by Kab(x, y) = 〈x, a|M−1∇2M−1|y, b〉. The Coulomb gauge Hamiltonian is renormalizable, permits resolution of the Gribov problem, preserves rotational invariance, avoids spurious retardation corrections, aids identification of dominant, low energy potentials and does not introduce unphysical degrees of freedom (ghosts) [23]. To make the problem tractable, the Coulomb instantaneous kernel is approximated by its vacuum expectation value, yielding an effective potential field theory, HQCD → Heff QCD HC → Heff C = − 1 2 ∫ dxdyρa(x)V̂ (|x − y|)ρa(y) , (6) with confinement described by a Cornell potential, V̂ (r) = −αs r + σr, where 3 the string tension, σ = 0.135 GeV2, and αs = 0.4 have been independently determined from conventional meson studies [18,19,20] within the same field theory approach. We also use the lowest order, unit value, for the Faddeev- Popov determinant in the Hg term and treat the Hqg interaction using per- turbation theory. Lattice data confirms the Cornell potential form between static sources [24] and further provides the scale of the gluon mass gap [25], that is needed to fit a counterterm in the gluon gap equation. The remaining parameters are the reasonably well known current quark masses at some high energy scale where the mass function runs perturbatively. The quark sector Hamiltonian then takes the form of the Cornell coupled-channel model [26]. We note that three-body forces [27,28] are omitted, however based upon suc- cessful three-body applications [21] we submit the CGM should capture the dominant features of a multi-parton spectrum. Before presenting our tetraquark results we highlight our recent hybrid me- son calculation [29]. Since the gluon carries a color octet charge, the quark and antiquark are also in a color octet wavefunction with elements T a ij (they repel each other at short distance). In the hybrid rest frame there are two independent three-momenta q+ = q+q 2 , q− = q − q and one dependent g = −q − q = −2q+. The leading hybrid Fock space wavefunction can there- fore be constructed from the respective quark, anti-quark and gluon quasipar- ticle operators B† λ1C1 (q), D† λ2C2 (q) and αa† µ (g) |ΨJPC〉 = ∫ ∫ dq+ (2π)3 dq− (2π)3 ΦJPC λ1λ2µ(q+, q−)T a C1C2 B† λ1C1 (q)D† λ2C2 (q)αa† µ (g)|Ω〉 . (7) An angular momentum expansion for the lightest 1−+ state reveals a required p-wave excitation in one of the two orbital wave functions. Consult Ref. [29] for complete details. Significantly, in agreement with earlier findings [11], the predicted hybrid masses are about 2 GeV for the ground state (parity +) quadruplet and 2.2 and 2.4 GeV, respectively, for the first 1−+ exotics. The repulsive nature of the short range qq̄ interaction potential kernel and the large gluon mass gap are responsible for these large masses. We also performed a parameter sensitivity and error analysis study and concluded there was no model possibility to lower one of these states near the 1.6 GeV candidate and therefore rule out this, and even more clearly the 1.4 GeV, state as a hybrid. Returning to our thrust, we report results for qqq̄q̄ spectroscopy (see also a preliminary study [30]). This system was first investigated in the bag model [31] and then more extensively by Ref. [32] with subsequent potential model applications reported by Refs. [33,34]. While these studies have some similarity to our model, we submit our results are more robust since our approach is much more comprehensive, has many QCD elements with no new parameters to be determined and employs a realistic potential kernel extracted from lattice gauge theory. The leading tetraquark Fock space wavefunction is [35,36] 4 |ΨJPC〉 = ∫∫∫ dqA (2π)3 dqB (2π)3 dqI (2π)3 ΦJPC λ1λ2λ3λ4 (qA, qB, qI) × RC1C2 C3C4 B† λ1C1 (q1)D † λ2C2 (q2)B † λ3C3 (q3)D † λ4C4 (q4)|Ω〉. (8) In the cm there are three independent momenta which we take to be qA = q1−q2 2 , qB = q3−q4 2 and qI = q3+q4 2 − q1+q2 2 (q1, q3 for the quarks and q2, q4 for the anti-quarks). The color matrices RC1C2 C3C4 yielding color-singlet wave- functions follow from the SUc(3) algebra depicted in Fig. 1. The radial part of ΦJPC λ1λ2λ3λ4 (qA, qB, qI) is chosen to be a gaussian, exp(− q2 A α2 A − q2 B α2 B − q2 I α2 I ), with variational parameters αA, αB and αI for s-wave states and a gaussian multi- plied by q2 i /α 2 i (i = A, B, I) corresponding to orbital Li = 1, when treating p-wave states. Note, as for hybrid mesons, the ground state tetraquark multi- plet has positive parity and that constructing 1−+ exotics requires one of the three orbitals to be a p-wave. Using the variational principle, the tetraquark mass is then given by MJPC ≤ 〈ΨJPC|Heff QCD|Ψ JPC〉 〈ΨJPC|ΨJPC〉 = Mself + Mqq + Mq̄q̄ + Mqq̄ + Mannih . (9) Contributions to the Hamiltonian expectation value are summarized in Fig. 2 and correspond to 4 self-energy, 6 scattering, 4 annihilation and 70 exchange terms, each of which can be reduced to 12 dimensional integrals that are eval- uated in momentum space. Because of the computationally intensive nature of this analysis, the hyperfine interaction was not included. Complete expressions will be given in another publication, but as an example note the annihilation contribution, not possible in standard quark models, is + + ++ + + exchange terms + + + + + + + + =Mtetra−quark Fig. 2. Equal-time diagrams for the expectation value of the model Hamiltonian. 5 Mannih = ∫∫∫∫ dq1dq2dq3dk (2π)12 V (q1 + q2)u † λ ′ 1 (q1 + k)vλ ′ 2 (q2 − k) × v† λ2 (q2)uλ1 (q1)Φ JPC† λ1λ2λ3λ4 (q1, q2, q3)Φ JPC λ ′ 1 λ ′ 2 λ3λ4 (q1 + k, q2 − k, q3), (10) involving Dirac spinors uλ1 and vλ2 . This raises the mass of the isoscalar states relative to the states with higher isospin, which would otherwise typically be heavier due to the exclusion principle applied to equal-flavor quarks. Performing large-scale Monte Carlo calculations (typically 50 million samples), has conclusively determined that the molecular representation (i.e. singlet- singlet) produces the lightest mass for a given JPC . This is due to suppression of certain interactions in this color scheme from vanishing color factors for every parton pair which does not occur in the other representations. Also, there are additional, repulsive forces in the more exotic color schemes. Using mu = md = 5 MeV, the predicted tetraquark ground state is the non-exotic vector 1++ state in the molecular representation with mass around 1.2 GeV. Figure 3 depicts the predicted tetraquark spectra for states having conven- tional and exotic quantum numbers in both singlet and octet color represen- tations. The quark annihilation interactions (qq̄ → g → qq̄) in the Iqq̄ = 0 channel generate isospin splitting contributions, up to several hundred MeV, in the octet but not singlet scheme as slightly illustrated in the figure. Isospin splitting is a consequence of a more proper field theory treatment, not present in conventional quark models. The annihilation interaction terms are repul- sive, yielding octet states with I = 2 lower than the I = 1 which are lower than the I = 0. This is intuitively contrary to expectations that I= 2 states are higher based upon the Pauli principle that identical quarks repel. The molecular states are all isospin degenerate and the lightest exotic molecule is an intriguing 0−− with mass 1.35 GeV that could be detected in a sophisti- cated p-wave analysis of an ωπ spectrum. Because of the isospin degeneracy, there will be several molecular tetraquark states with the same JPC in the 1 to 2 GeV region. Further, these states can be observed in different elec- tric charge channels (different Iz) at about the same energy, which is a useful experimental signature. The lightest 1−+ is predicted near 1.4 GeV which is close to the observed π(1400), suggesting this state has a molecular resonance Fig. 3. Tetraquark I = 1 singlet (molecule) and octet (atom) schemes spectra. 6 Table 1 Selected tetraquark molecular exotic states. For qq̄ pairs, Si and Li, i = A,B, are the total spin and orbital angular momentum and LI is the orbital angular momentum between the pairs. The variables not shown are assumed to be 0. Units are GeV. (uū)1(uū)1 I=2 I=1(1x1) I=1(1x0) I=0(1x1≃ 0x0) 1−+ (LI = 1) - 1.42 1.42 - 1−+ (LA = SA = 1) 1.80 1.80 1.80 1.80 1−+ (LA = SB = 1) 1.91 1.91 1.91 1.91 0−− (LI = SA = 1) 1.35 1.36 1.36 1.36 structure. The computed mass for 1−+ states with more exotic octet color configurations are all above 2 GeV. This is consistent with model predictions [29] for exotic hybrid meson (qq̄g) 1−+ states also lying above 2 GeV due to repulsive color octet quark interactions. Finally, for any JPC state, including the 1−+, the computed masses (not shown) in both the triplet and the sextet diquark color representations are all heavier than in the singlet representation and comparable to the octet scheme results. Our predictions for the lightest exotic molecules are given in Table 1. It is interesting to document the current quark mass dependence of our results. This is illustrated in Fig. 4, where the calculated rest mass corresponding to 1 10 100 1000 m q (MeV) 0 0 1000 1000 2000 2000 3000 3000 4000 4000 E (M eV ) (m+M) TDA (m+M) RPA M 4q (1-1) M 4q (3-3) 2 Meson thresholds Tetraquark and two meson masses Quark mass dependence Fig. 4. Dotted lines, from bottom to top, are the ππ, πη, πη′,πηc observed thresholds, respectively. The two other plots are the pseudoscalar mπ+Mqq̄ model predictions in the (chiral respecting) RPA and (chiral violating) TDA. The squares and diamonds correspond respectively to predicted scalar isoscalar tetraquark masses in the color singlet-singlet and triplet-triplet schemes. Note the two-meson calculation [19] has σ = 0.18 Gev2, αs = 0, whereas our results use σ = 0.135 Gev2, αs = 0.4, but the difference among both sets is known to be small for the ground state. 7 Table 2 Predicted masses of hidden-charm exotic 1−+ mesons (molecular configuration). mc = 1.2 GeV mq Mcc̄qq̄ cc̄qq̄ 0 4.04 GeV cc̄qq̄ 110 MeV 4.10 GeV cc̄qq̄ 150 MeV 4.15 GeV different four quark systems (tetraquarks and two mesons) is displayed with one qq̄ pair being held fixed at a low 1-5 MeV while the other pair varies from 1 to 1300 MeV. Note the predicted cc̄uū tetraquark mass is relatively closer to the two meson decay threshold indicating a narrow decay width to the πηc and ηηc channels (more so for bb̄, not displayed). Also note a quark-meson exchange model [37] claims some J = 1 partners will be absolutely stable. The lowest-lying hidden-charm exotics are predicted in Table 2 and can be searched for in hidden charm decays (p-wave ηcπ0, ηcη for example). Figure 5 displays the sensitivity of the scalar tetraquark mass to the variational parameters for different color schemes. The color singlet-singlet configuration, at 1.28 GeV, is clearly the lightest, in agreement with previous predictions [38]. Similar results are obtained for other JPC configurations. This suggests that the lightest tetraquark states, including the π1(1400) should it exist, are best interpreted as molecules of color singlets (the “extraordinary hadrons” [39]) and directly excitable via meson rescattering. 0.0 0.3 0.6 0.9 1.2 1.5 1.8 0.00 0.15 0.30 0.45 0.60 αααα A = αααα B = 0.65 (singlet) αααα A = αααα B = 0.75 (triplet) αααα A = αααα B = 0.65 (sextet) αααα A = αααα B = 0.6 (octet) M as s - M si ng le t ( G eV ) αααα I Fig. 5. Tetraquark mass of the ground state isoscalar 0++ as a function of the intercluster variational parameter, for various color configurations. Note the more exotic color configurations are above 1.7 GeV. 8 Fig. 6. Hybrid (left) and tetraquark (right) meson probability density contour plots. The left horizontal axis is half the momentum of the constituent gluon, q+ = g/2. The left vertical axis is the angle between the qq̄ relative momentum q− and the hybrid total angular momenta J . The right horizontal axis corresponds to the in- tracluster momentum qI and the angle between this momentum and J for the tetraquark. Angles in radians, momenta in GeV. For further dynamical insight, Fig. 6 details contour plots of the probability densities using the hybrid (left) and tetraquark (right) variational wavefunc- tions. Note the depletion of the wavefunction at low-momentum, reflecting the rising confining potential at large distance, and the significantly different par- ton momentum distributions between hybrids and tetraquark systems. This difference in momentum distributions will produce distinct decay signatures which can be utilized to identify hybrids and tetraquarks as we now discuss. It has been proposed [40] that 1−+ exotic hybrid mesons decay preferentially to a meson pair in a relative s-wave, where one of the mesons is a p-wave (axial) meson. However that prediction was based on the Flux Tube model and not known to hold exactly in any limit. We have recently [29] detailed a less model dependent decay signature based upon the Franck-Condon principle of molecular physics, which predicts that the momentum distribution of decay products parallels the internal momentum distribution of the parent meson. This is an exact statement in the infinitely heavy quark limit, however it is also useful for physical quarks. Our previous application [29] was designed to distinguish qq̄ from qq̄g mesons and is now generalized to the four-body problem. For two and three-body systems, in the cm partons’ momenta are restricted to a plane, however for tetraquarks there are off-plane degrees of freedom which can be exploited as an identification signature (see the cartoon in Fig. 7). One observable sensitive to this is an exclusive measurement of decay to four mesons. By then determining their momenta pi=1,2,3,4 in the cm of the exotic candidate, kinematic cuts can be applied for every two and 9 Fig. 7. Differences between internal in-plane three-body hybrid (top left) and off– plane tetraquark (top right) meson decays. From the Frank-Condon principle the internal momentum distribution is reflected in the momentum distribution of the final products yielding an off-plane final state momentum observable, Π, that is very different for qq̄g (bottom left) and qqq̄q̄ (bottom right) mesons. three meson groups to eliminate intermediate two and three-body resonances that could confuse the analysis. From the pure (and much reduced) four-body decay sample count, one can construct the off-plane correlator, Π, for any three momenta which is the volume of the parallelepiped they form in momentum space, Π(p1, p2, p3, p4) = ((p1 × p2)·p3) 2 √ |p1 × p2||p2 × p3||p1 × p3||p1 × p4||p2 × p4||p3 × p4| , which has been normalized by the area of the six faces. This dimensionless off-plane correlator is useful for the very different scales involved in light and heavy quark physics, positive definite and invariant under permutation of the four momenta. In the cm system only three momenta are linearly independent. Taking them equal and along the edges of a cube yields Π = 8−1/4 = 0.59, with a more general maximum value close to 0.707. Therefore the value of the correlator event by event will be a random variable distributed between 0 and 0.707 and, with sufficient statistics, one could deduce the internal structure of the decaying resonance (bottom right plot of Fig. 7). This could then be compared to four-meson decays of established conventional and hybrid meson benchmarks, as distinguished using the procedure outlined in Ref. [29], which would be radically different (their Π volumes collapse to 0 in the heavy quark limit). Finally, we are currently calculating tetraquark decay widths to com- pare with both conventional and hybrid meson decays to two and multi-meson final states. These will also reflect hadronic structure differences and therefore aid exotic identification. Results will be reported in a future communication. 10 In conclusion, we submit that the observed 1−+ exotics below 2 GeV are not color exotic hadrons but rather somewhat more conventional tetraquark molecular resonances involving color singlet qq̄ pairs. Our results also agree with lattice and other models predicting that color exotics (hybrid mesons, octet-octet, sextet-sextet and triplet-triplet tetraquarks) will have masses near and above 2 GeV. 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