Including Tetraquark Operators in the Low-Lying Scalar Meson Sectors in Lattice QCD
Daniel Darvish, Ruairí Brett, John Bulava, Jacob Fallica, Andrew Hanlon, Ben Hörz, Colin Morningstar
IIncluding Tetraquark Operators in the Low-Lying ScalarMeson Sectors in Lattice QCD
Daniel Darvish,
1, a)
Ruairí Brett, John Bulava, Jacob Fallica, Andrew Hanlon, Ben Hörz, and Colin Morningstar Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA Department of Physics, The George Washington University, Washington, DC 20052, USA CP3-Origins, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA Helmholtz-Institut Mainz, Johannes Gutenberg-Universität, D 55099 Mainz, Germany Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA a) Corresponding author: [email protected]
Abstract.
Lattice QCD allows us to probe the low-lying hadron spectrum in finite-volume using a basis of single- and multi-hadron interpolating operators. Here we examine the effect of including tetraquark operators on the spectrum in the scalar mesonsectors containing the K ∗ ( ) ( κ ) and the a ( ) in N f = + m π ≈
230 MeV. Preliminary results of additionalfinite-volume states found using tetraquark operators are shown, and possible implications of these states are discussed.
INTRODUCTION
It has been suggested that the light scalar mesons K ∗ ( ) (here referred to as the κ ) and a ( ) could have tetraquarkcontent [1, 2, 3, 4]. To date, there have been a small number of studies investigating tetraquarks on the lattice usinglight quarks. In 2010, Prelovsek et al. investigated the σ and κ as possible tetraquark candidates, but neglected discon-nected diagrams in their calculations [5]. Using tetraquark interpolators, they found an additional light state in boththe σ and κ channels. In 2013, the ETM collaboration examined the a ( ) and κ using four-quark operators [6],though they also neglected disconnected diagrams in their calculations. They found no evidence of an additional statethat can be interpreted as a tetraquark. In 2018, Alexandrou et al. conducted a study of the a ( ) with four-quarkoperators [7], including disconnected contributions. In their study, they found an additional finite-volume state in thesector containing the a ( ) meson, which couples to a diquark-antidiquark interpolating field. Additionally, theyconclude that disconnected diagrams have drastic effects on their results, and thus cannot be neglected.Here we investigate the possible role of tetraquark operators in lattice QCD in the symmetry channels of the κ ( I = , S = P = + J =
0) and a ( ) ( I = S = P = + G = − J = a s a t ≈ . × .
74 fm and a pion mass of approximately 230 MeV. We extract two spectra in each symmetry channel: oneusing a basis of only single- and two-meson operators, and one using a basis that also includes a tetraquark operatorselected from hundreds of tetraquark operators which were tested. We find that including a tetraquark operator yieldsan additional finite-volume state in each symmetry channel. In this work, we use the stochastic LapH method [8] toevaluate all diagrams in our calculations, including all disconnected contributions.
SPECTROSCOPY IN LATTICE QCD
We obtain our finite-volume QCD spectra by calculating discretized path integrals in Euclidean spacetime. Consideran operator O ( t ) that acts on the vacuum state | (cid:105) , creating a state at time t , and a corresponding operator O ( t ) thatannihilates such a state at time t . In imaginary time, we can in principle extract the energy spectrum from the followingtime-ordered correlation function, (cid:104) | T O ( t + t ) O ( t ) | (cid:105) = ∑ n (cid:104) | O ( ) | n (cid:105) (cid:104) n | O ( ) | (cid:105) e − E n t , (1)where | n (cid:105) is the n th ordered energy eigenstate of the theory corresponding to energy E n , and we have shifted thevacuum energy E to be zero. (Henceforth, E will refer to the ground state energy in each symmetry sector.) a r X i v : . [ h e p - l a t ] S e p xpectation values in Euclidean spacetime can be calculated using a path integral over all quark fields ψ , ψ , andall link variables representing the gauge fields, U : (cid:104) O (cid:105) = Z (cid:90) D [ ψ , ψ , U ] O e − S [ ψ , ψ , U ] , Z = (cid:90) D [ ψ , ψ , U ] e − S [ ψ , ψ , U ] . (2)In order to evaluate these integrals on the lattice, we discretize spacetime in the above integral and carefully designoperators that create states having nonzero overlap with the states of interest. We analytically integrate out the quarkfields, and use Monte Carlo methods to integrate over the link variables, using gauge field configurations generatedby the Hadron Spectrum Collaboration [9, 10, 11].By calculating Eq. (1) on the lattice, we can, in practice, only fit the lowest one or two energies. At small t , there isa high signal-to-noise ratio in the Monte Carlo determination of the correlator, but there are many excited states thatcontribute. It would be better if we could devise a way to extract more than just the lowest one or two states. In orderto do this, we construct an N × N matrix of correlators, C i j ( t ) ≡ (cid:104) | T O i ( t + t ) O j ( t ) | (cid:105) , (3)using a large basis of single- and multi-hadron interpolating operators. As a preliminary step, we rescale Eq. (3) inorder to compensate for varying normalizations between the different operators: C i j ( t ) ≡ C i j ( t ) (cid:112) C ii ( τ N ) C j j ( τ N ) , (4)where the normalization time τ N is taken at some early time, e.g. τ N <
4. We will also assume that for sufficientlylarge t , and for sufficiently small Z ( n ) i ≡ (cid:104) | O i | n (cid:105) , we can well approximate the correlation matrix by, C i j ( t ) ≈ N − ∑ n = Z ( n ) i Z ( n ) ∗ j e − E n t . (5)With our truncated C ( t ) , we then solve the following generalized eigenvalue problem: C ( t ) v n ( t , τ ) = λ n ( t , τ ) C ( τ ) v n ( t , τ ) , t > τ . (6)It is shown in Ref. [12] that for τ ≥ t / λ n ( t , τ ) = e − E n ( t − τ ) (cid:16) + O (cid:16) e − ( E N − E n ) t (cid:17)(cid:17) . (7)(In practice, we find that the condition τ ≥ t / N energiesof the spectrum by fitting λ n ( t ) to single- or two-exponentials. In practice, the excited state contamination in λ ( t , τ ) scales such that it is prudent to discard the highest few levels, or rather, to use more operators in the basis than levelswe wish to determine. OPERATOR CONSTRUCTION
In our operator bases, we include single- and two- meson operators, as well as tetraquark operators. We make use ofsmeared, gauge-covariantly displaced quark fields, and stout-smeared link variables (introduced in Ref. [13]) in ouroperator construction. For example, an elemental meson operator of definite-momentum p at time t can be written asfollows, Φ AB αβ ; i jk ( p , t ) = ∑ x e − i p · ( x + ( d α + d β )) δ ab q Aa α i ( x , t ) q Bb β jk ( x , t ) , (8)where capital Latin indices denote flavor, lowercase Latin a and b denote color, Greek indices are Dirac spin, andlowercase Latin i , j , k , denote quark displacement. The vectors d α and d β are quark displacement vectors, and areresent to ensure proper transformation under G -parity. To form the final operators out of our elemental operators,we project the elemental operators onto various symmetry channels according to isospin, parity, G -parity, octahedrallittle group, etc. That is, to form a meson operator M l ( t ) that transforms irreducibly under all symmetries of interest(labeled by the compound index l ) at time t , we must take a linear combination of our elemental meson operators, M l ( t ) = c ( l ) αβ Φ AB αβ ( p , t ) . To form a two-meson operator O l ( t ) , we would follow a similar procedure and project theproduct of two final meson operators M al a ( t ) M bl a ( t ) onto a final symmetry channel l : O l ( t ) = c ( l ) l a l b M al a ( t ) M bl a ( t ) .In order to construct a tetraquark operator, we must consider the various ways to construct a color-singlet four-quarkobject out of four quark fields. As seen in Ref. [14], the Clebsch-Gordon decompositions show that the only way toconstruct a color-singlet is by using two quarks and two antiquarks, and that doing so yields two linearly independentcolor singlet objects: 3 ⊗ ⊗ ⊗ = ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ , ⊗ ⊗ ⊗ = ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ , ⊗ ⊗ ⊗ = ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ . (9)There are 81 basis vectors formed by the quark fields, p ∗ α ( x ) q ∗ β ( x ) r γ ( x ) s µ ( x ) , where each r , s transforms as a colorvector in the fundamental 3 irrep, and so, p ∗ , q ∗ transform in the 3 irrep. We need two linearly independent andgauge-invariant combinations of these to exhaust all possible tetraquark operators. It is easy to see that the followingcombinations are both linearly independent and gauge-invariant (and thus form our elemental tetraquark operators): T S = (cid:0) δ αγ δ β µ + δ αµ δ βγ (cid:1) p ∗ α ( x ) q ∗ β ( x ) r γ ( x ) s µ ( x ) T A = (cid:0) δ αγ δ β µ − δ αµ δ βγ (cid:1) p ∗ α ( x ) q ∗ β ( x ) r γ ( x ) s µ ( x ) . (10)These elemental tetraquark operators are combinations of two gauge-invariant quark-antiquark constituents. The indi-vidual constituents are not mesons since they separately do not have well-defined quantum numbers. In other words,we project the entire elemental tetraquark operator onto relevant symmetry channels, rather than each individualquark-antiquark operator.While we chose only a handful of tetraquark operators for our final analysis, we designed hundreds of operatorswith differing flavor structures and displacements. We tested these operators by individually adding them to a basisof single- and multi-meson operators to see if an additional level was found. Most of the operators did not yield anadditional level, but we found particular operators that did. In the κ channel, we tested the following flavor structures: suss , suuu , sudu . We found that only operators with the suss flavor structure yielded an additional finite-volume state.We tested both single-site and quadruple displacements, and found operators of both types that yielded additionalfinite-volume states. The quadruply-displaced operators came at a higher computational cost and offered no improve-ments, and so were excluded from the final operator sets. In the a ( ) channel, we tested the following flavorstructures: uudu , ssdu , dudu . We found that only operators with the uudu flavor structure yielded an additional finite-volume state. We only tested single-site operators in the a ( ) channel, after finding no improvement with otherdisplacement types in the κ channel. We also constructed operator bases that included several tetraquark operators,and found that the number of additional levels in the energy range we examined was unchanged. LATTICE SPECTRA RESULTS (PRELIMINARY) κ Channel
We summarize results obtained by fitting a spectrum in the κ at-rest symmetry channel for two operator bases: oneincluding only single-meson and two-meson operators, and one including single-meson, two-meson, and tetraquarkoperators. Figure 1 shows the spectrum with and without the inclusion of a tetraquark operator in the basis. Thetetraquark operator is of the flavor structure suss , is of the antisymmetric form in Eq. (10), and has no quark displace-ment. We found that single-site ( d α = d β =
0) tetraquark operators resulted in better (less noisy) correlator signalsthan displaced operators. We see that including a tetraquark operator yields an additional finite-volume state in therange of ( . − . ) m K , which is not present when only single- and two-meson operators are used. Additionally,a plot of the overlap factors for the tetraquark operator (Figure 2) shows significant overlap onto this extra state (level3 in the plot). This suggests that there is a finite-volume state in our lattice spectrum that shares quantum numbers . . . . . . . . . E / m K K (0) π (0) K (1) π (1) K (0) η (0) K (2) π (2) K (1) η (1) κ channel Without tetraquark operatorWith tetraquark operatorNon-interacting
FIGURE 1.
The first five and six levels of the spectrum in the κ at-rest symmetry channel. On the left: the spectrum obtainedusing a basis with no tetraquark operators. In the middle: the spectrum obtained using one tetraquark operator. On the right:non-interacting levels shown for reference, where ( d ) denotes particles with squared momentum ( π d / L ) . n Z ( n ) FIGURE 2.
The overlap factors for the tetraquark operator used to produce the extra level in the κ symmetry channel. with the κ resonance, and that has tetraquark content. Whether or not this is evidence of the κ resonance havingtetraquark content, however, will have to wait for future scattering studies using Lüscher’s method. a ( ) Channel
We summarize results obtained by fitting a spectrum in the a ( ) at-rest symmetry channel for again for two operatorbases as in the κ channel. Figure 3 shows the spectrum with and without the inclusion of a tetraquark operator in the . . . . . E / m K π (0) η (0) K (0) K (0) π (1) η (1) K (1) K (1) π (2) η (2) K (2) K (2) a channel Without tetraquark operatorWith tetraquark operatorNon-interacting
FIGURE 3.
The first six and seven levels of the spectrum in the a ( ) at-rest symmetry channel. On the left: the spectrumobtained using a basis with no tetraquark operators. In the middle: the spectrum obtained using one tetraquark operator. On theright: non-interacting levels shown for reference, where ( d ) denotes particles with squared momentum ( π d / L ) . n Z ( n ) FIGURE 4.
The overlap factors for the tetraquark operator used to produce the extra level in the a ( ) symmetry channel. basis. The tetraquark operator is of the flavor structure uudu , is also of the antisymmetric form in (10), and againhas no quark displacement. We again found that using single-site tetraquark operators resulted in better correlatorsignals than displaced operators. We see an extra level appear in the range of ( . − . ) m K when we includea tetraquark operator. Again, overlap factors are shown for the tetraquark operator, and significant overlaps with theadditional level (level 3) can be seen in Figure 4. This suggests there is a finite-volume state in our lattice spectrumthat shares quantum numbers with the a ( ) resonance, and that has tetraquark content. As in the κ -channel case,evidence for or against the a ( ) having tetraquark content will have to wait for future scattering studies done byapplying Lüscher’s method. ONCLUSIONS
We have presented results detailing the effect of including tetraquark operators on determining the lattice spectrum ineach of the κ and a ( ) symmetry channels of N f = + m π ≈
230 MeV. We find that the inclusionof a tetraquark operator in the κ channel yields an additional finite-volume state in the range of ( . − . ) m K ,and the inclusion of a tetraquark operator in the a ( ) channel yields an additional finite-volume state in the rangeof ( . − . ) m K . To address the issue of tetraquark content of the κ and a ( ) resonances, future studiesemploying Lüscher’s method will be required. ACKNOWLEDGMENTS
This work was supported by the U.S. National Science Foundation under award PHY-1613449. Computing resourceswere provided by the Extreme Science and Engineering Discovery Environment (XSEDE) under grant number TG-MCA07S017.
REFERENCES
1. R. L. Jaffe, “Exotica,”
Proceedings, 6th International Conference on Hyperons, charm and beauty hadrons (BEACH 2004): Chicago, USA,June 27-July 3, 2004 , Phys. Rept. , 1–45 (2005), [,191(2004)], arXiv:hep-ph/0409065 [hep-ph].2. C. Amsler and N. A. Törnqvist, “Mesons beyond the naive quark model,” Physics Reports , 61 – 117 (2004).3. F. E. Close and N. A. Törnqvist, “Scalar mesons above and below 1 GeV,” Journal of Physics G: Nuclear and Particle Physics , R249–R267(2002).4. L. Maiani, F. Piccinini, A. D. Polosa, and V. Riquer, “New look at scalar mesons,” Phys. Rev. Lett. , 212002 (2004).5. S. Prelovsek, T. Draper, C. B. Lang, M. Limmer, K.-F. Liu, N. Mathur, and D. Mohler, “Lattice study of light scalar tetraquarks with I = , , , : Are σ and κ tetraquarks?” Physical Review D , 094507 (2010).6. C. Alexandrou, J. O. Daldrop, M. Dalla Brida, M. Gravina, L. Scorzato, C. Urbach, and M. Wagner, “Lattice investigation of the scalar mesons a ( ) and κ using four-quark operators,” JHEP04 , 137 (2013).7. C. Alexandrou, J. Berlin, M. Dalla Brida, J. Finkenrath, T. Leontiou, and M. Wagner, “Lattice QCD investigation of the structure of the a ( ) meson,” Physical Review D (2018), 10.1103/PhysRevD.97.034506.8. C. Morningstar, J. Bulava, J. Foley, K. J. Juge, D. Lenkner, M. Peardon, and C. H. Wong, “Improved stochastic estimation of quark propagationwith Laplacian Heaviside smearing in lattice QCD,” Phys. Rev. D83 , 114505 (2011), arXiv:1104.3870 [hep-lat].9. R. G. Edwards, B. Joó, and H.-W. Lin, “Tuning for three flavors of anisotropic clover fermions with stout-link smearing,” Phys. Rev. D ,054501 (2008).10. J. M. Bulava, R. G. Edwards, E. Engelson, J. Foley, B. Joó, A. Lichtl, H.-W. Lin, N. Mathur, C. Morningstar, D. G. Richards, and S. J.Wallace, “Excited state nucleon spectrum with two flavors of dynamical fermions,” Phys. Rev. D , 034505 (2009).11. M. Clark, A. Kennedy, and Z. Sroczynski, “Exact 2+1 flavour RHMC simulations,” Nuclear Physics B - Proceedings Supplements , 835– 837 (2005), LATTICE 2004.12. M. Lüscher and U. Wolff, “How to calculate the elastic scattering matrix in two-dimensional quantum field theories by numerical simulation,”Nuclear Physics B , 222 – 252 (1990).13. C. Morningstar and M. Peardon, “Analytic smearing of SU ( ) link variables in lattice QCD,” Phys. Rev. D , 054501 (2004).14. A. D. Hanlon, “The ρ meson spectrum and k ππ