Lattice QCD investigation of heavy-light four-quark systems

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collaboration with Pedro Bicudo, Krzysztof

1 Lattice QCD investigation of heavy-light four-quark systems Antje Peters Goethe-Universität Frankfurt am Main in collaboration with Pedro Bicudo, Krzysztof Cichy, Luka Leskovec, Stefan Meinel and Marc Wagner Lunch Club Seminar, Justus-Liebig-Universität Gießen

2 Study of four-quark states Motivation A number of mesons observed in particle detectors (LHCb, Belle) is not well understood. E.g. charged charmonium- and bottomonium-like states (Z c ± and Z b ± ) They include b b or c c, but are also charged: must be 4-quark states Different four-quark/tetraquark structures possible, two examples: Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

Meson spectroscopy on the lattice The static-light approach Computation of 4-quark states very difficult If 2 quarks are heavy and 2 quarks are light: Treat degrees of freedom independently

3 Meson spectroscopy on the lattice The static-light approach Computation of 4-quark states very difficult If 2 quarks are heavy and 2 quarks are light: Treat degrees of freedom independently (Born-Oppenheimer approximation [M. Born, R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann.Phys. 389, Nr. 20, 1927]). 1 Lattice computation of the potential of two static quarks in the presence of two light quarks, i.e. potential can be interpreted as the potential between two B mesons 2 Solve Schrödinger s equation to check whether potentials are sufficiently attractive to host a bound state. Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

Lattice QCD Meson spectroscopy on the lattice QCD: Theory to describe quarks and gluons and the forces between them No analytical solution for low energy observables No pertubative approach on the

4 Lattice QCD Meson spectroscopy on the lattice QCD: Theory to describe quarks and gluons and the forces between them No analytical solution for low energy observables No pertubative approach on the potential Lattice QCD: continuum 4-dimensional lattice discretize quark and gluon action (many different possibilities) use path integral formalism and compute vacuum expectation values numerically very often: necessity to use non-physically high quark masses [ April 28, 2015] Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

5 Meson spectroscopy on the lattice Hadron spectroscopy I A hadron is described by its isospin (flavor content) I, total angular momentum J, parity P and charge conjugation C. If two static quarks are involved, the distance r between them also characterizes the hadronic system. Application of a suitable operator O on the vacuum generates field excitations which are similar to the hadron of interest. Here: Use two static quarks and two quarks of finite mass Static quarks: no spin, no contribution to total angular momentum and isospin BB: Q Qqq with Q = b and q {u, d, s, c} O BB = (CΓ) AB (C Γ) CD Qa C ( x, t)q (1)a A( x, t) Q b }{{} D( y, t)q (2)b B( y, t) }{{} B meson at x B meson at y Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

6 Meson spectroscopy on the lattice Hadron spectroscopy II Obtain the correlation function in time for each separation r = x y of the static quarks via C r (t) = Ω O r (t)o r (0) Ω Requires O(months) of computing time on high performance computers. For large t one finds the potential V (r) of the hadronic state O(t): lim Ω t O r (t)o r (0) Ω exp( V (r)t) Get a value of the potential for each quark separation r and obtain the full potential: resulting potential V r/a Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

7 BB systems BB systems in the static-light approach P. Bicudo, K. Cichy, A. Peters, B. Wagenbach and M. Wagner, Phys. Rev. D 92 (2015) no.1, doi: /physrevd [arxiv: [hep-lat]]. P. Bicudo, K. Cichy, A. Peters and M. Wagner, Phys. Rev. D 93 (2016) no.3, doi: /physrevd [arxiv: [hep-lat]]. related to previous papers: C. Michael et al. [UKQCD Collaboration], Phys. Rev. D 60 (1999) doi: /physrevd [hep-lat/ ]. M. S. Cook and H. R. Fiebig, hep-lat/ T. Doi, T. T. Takahashi and H. Suganuma, AIP Conf. Proc. 842 (2006) 246 doi: / [hep-lat/ ]. W. Detmold, K. Orginos and M. J. Savage, Phys. Rev. D 76 (2007) doi: /physrevd [hep-lat/ [HEP-LAT]]. M. Wagner [ETM and Y Collaborations], PoS LATTICE 2010 (2010) 162 [arxiv: [hep-lat]]. G. Bali et al. [QCDSF Collaboration], PoS LATTICE 2010 (2010) 142 [arxiv: [hep-lat]]. Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

screening of the antiquark-antiquark interaction due to light quarks (stronger, the more massive the light quarks)

8 BB systems The BB system - Expectations small separations of the static antiquarks: interaction due to 1-gluon exchange bound state: static Q Q pair in a color triplet (attractive) antidiquark large separations of the static antiquarks: screening of the antiquark-antiquark interaction due to light quarks (stronger, the more massive the light quarks) basically 2 static-light mesons Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

9 BB systems The BB four-quark operator O BB = (CΓ) AB (C Γ) CD Q C a ( x, t)q (1)a A( x, t) Q b }{{} D( y, t)q (2)b B( y, t) }{{} B meson at x B meson at y different choice of Γ corresponds to different potentials (different couplings of B, B, B 0 and B 1 ) use of gauge configurations generated by the ETMC corresponding to 3 different pion masses 2 attractive ground state channels: spin scalar isosinglet with I (J P ) = 0(1 + ) spin vector isotriplet I (J P ) {0(1 + ), 1(1 + ), 1(2 + )} Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

10 Fit an ansatz to the potentials: V (r) = BB systems α r }{{} e ( r d ) 2 }{{} colour screening Coulomb like qq = (ud-du)/ 2 (scalar) qq = (s (1) s (2) -s (1) s (2) )/ 2 (scalar) qq = (c (1) c (2) -c (1) c (2) )/ 2 (scalar) scalar isosinglet U(r) in GeV r in fm U(r) in GeV r in fm U(r) in GeV r in fm u/d s c vector isotriplet U(r) in GeV qq = uu, (ud+du)/ 2, dd (vector) r in fm U(r) in GeV qq = ss (vector) r in fm Observation: Potentials show more promise for binding the less massive the light quarks are! Strongest attraction for scalar isosinglet qq = ud du 2. Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

11 BB systems Solve Schrödinger s equation solve Schrödinger s equation for the radial component of the two b quarks: ( ) 1 d 2 2µ dr 2 + V (r) R(r) = E B R(r), ψ(r) = R(r)/r lowest eigenvalue E B < 0 binding (4-quark bound state), E B > 0 no binding (2 mesons) Perform a large number of fits varying... the temporal separation at which the lattice potential is read off the correlation function the range at which the fit to the potential is performed Binding for light isosinglet channel only! Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

12 BB systems Most promising channel for a bound state: scalar ud b b remember: V (r) = α r e ( r d ) 2 Extrapolation to the physical quark mass: Binding increases For the BB four-quark state with quantum numbers I (J P ) = 0(1 + ) we find E B = MeV at the physical point binding with more than 2σ confidence level Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

13 BB systems with NRQCD Study of BB systems by means of NRQCD A. Peters, P. Bicudo, L. Leskovec, S. Meinel and M. Wagner, Lattice QCD study of heavy-heavy-light-light tetraquark candidates, arxiv: [hep-lat]. Work in progress. Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

14 BB systems with NRQCD The I (J P ) = 0(1 + ) ud b b state with NRQCD strong evidence for a ud b b bound state in the I (J P ) = 0(1 + ) channel from static-light approach to be confirmed with b-quarks of finite mass instead of static quarks search for a bound state with nonrelativistic QCD (NRQCD) positions of b quarks not fixed computation of V (r) not possible but direct computation of mass of lowest BB state in I (J P ) = 0(1 + ) channel possible n f = dynamical sea quarks, m π = 336 MeV [arxiv: ] The NRQCD correlator O BB : e i p1( x1 x 1 ) e i p2( x2 x 2 ) δ x1, x 2 δ x 1, x 2 x 1, x 2, x 1, x 2 ( bγ 5 d( x 1 ) bγ i u( x 2 ) bγ 5 u( x 1 ) bγ i d( x 2 ) ) ( dγ 5 b( x 1)ūγ i b( x 2) ūγ 5 b( x 1) dγ i b( x 2) ) Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

15 BB systems with NRQCD The I (J P ) = 0(1 + ) ud b b state with NRQCD - preliminary results compute masses of B, B and BB if m BB < m B + m B : bound state in I (J P ) = 0(1 + ) channel but ae (eff) BB * -mol E (eff) BB * -mol E B + E B * = t/a Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

BB systems with NRQCD Two four-quark operators For I (J P ) = 0(1 + ) there is another operator to be included to the analysis: O B B Build 2x2 correlation matrix using O BB and O B B Lowest lying

16 BB systems with NRQCD Two four-quark operators For I (J P ) = 0(1 + ) there is another operator to be included to the analysis: O B B Build 2x2 correlation matrix using O BB and O B B Lowest lying state lies below previous m B + m B threshold! 1.3 ae (eff) BB * -mol 1.2 E (eff) BB * -mol E B + E B * t/a (m B + m B ) 140 MeV, i.e. strong indication that mass of the four-quark ud b b is smaller than the sum of B and B m (2x2) BB Qualitative confirmation of static-light result Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

17 B B systems B B systems in the static-light approach A. Peters, P. Bicudo, K. Cichy and M. Wagner, Investigation of B B four-quark systems using lattice QCD, arxiv: [hep-lat]. Work in progress. Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

18 The B B system B B systems There are several different structures possible for the experimentally relevant case of I (J P ) = 1(1 + ) (i.e. Z + b ), e.g. A loosely bound B B pair, i.e. a mesonic molecule B B A bound 4-quark state made of a diquark and an antidiquark diquark antidiquark Two mesons: a B meson and a far separated B meson B B A bottomonium state and a far separated π +, i.e. Q Q + π QQ pion lightest state is Q Q + π large separations of static quarks: 2 mesons B and B Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

19 B B systems To separate these structures, we... 1 implement a correlation matrix C jk (t) = Ω O j (t)o k(0) Ω O 0 O B B = Γ AB ΓCD Qa C ( x)q a A( x) q b B( y)q b D( y) large t A 0 jk exp ( V 0 (r)t)+a 1 jk exp ( V 1 (r)t)+... O 1 O Q Q+π = Γ AB Q a A( x)u ab ( x, t; y, t)q b B( y) z q c C ( z) (γ 5 ) CD q c D( z) C(t) = 2 extract the potentials with the Generalized Eigenvalue Problem (GEP). Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

20 Preliminary results I B B systems Va r/a Potentials obtained Q Q + π: ground state (blue) GEP with 2x2 matrix C(t): first excited state GEP with 2x2 matrix C(t): ground state QQ - first excited state of the 2x2 matrix: free of contributions of Q Q + π (red) Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

21 Preliminary results II B B systems Binding energy Identical analysis as in the BB case (Schrödinger s equation) yields for quantum numbers I (J P ) = 1(1 + ) (i.e. Z + b ): E B = ( 58 ± 71)MeV vague indication for a ūd bb bound state Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

22 B B systems Summary BB systems BB systems are experimentally hard to prepare, but theoretically easier to investigate For a BB system with light quarks qq = ud and static b quarks with quantum numbers I (J P ) = 0(1 + ) we find E B = MeV binding with more than 2σ confidence level This result is supported by preliminary computations with 4 quarks of finite mass. next steps: repeat computations with smaller lattice spacing, investigate resonances Summary B B systems B B systems are experimentally rather easy to access, but theoretically more challenging. Candidate for a binding B B state with I (J P ) = 1(1 + ) (i.e. Z + b ) is currently investigated, we find E B = ( 58 ± 71) MeV next steps: analysis of possible bound state (binding energy, structure, life time ) Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 21

23 Backup Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 8

24 Meson content of a four-quark state One can extend the meson content of the four-quark states by application of the following light quark projectors on O: Parity projectors: P P=+ = 1+γ 0 and P 2 P= = 1 γ 0 2 Spin projectors: P jz = = 1+iγ 0γ 3 γ 5 and P 2 jz = = 1 iγ 0γ 3 γ 5 2 P = : S state, meson in the ground state P = +: P state, first excitation : light quark angular momentum An example for O B B : Γ = γ 5 ˆ= + S S + S S + P P + P P Γ = γ 0γ 5 ˆ= S S S S + P P + P P Therefore a B B state with Γ = γ 5 γ 0γ 5 only contains S mesons. Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 8

25 Lattice Setups BB and B B in the static-light approach: Ens. β lattice aµ m π [MeV] a [fm] L [fm] confs B (26) B (26) B (26) Gauge configurations generated by ETMC BB with NRQCD: Ens. β lattice am u,d am s m π [MeV] a [fm] L [fm] confs C (17) Gauge configurations generated by RBC and UKQCD collaborations Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 8

26 Different attractive BB channels spin scalar isosinglet: qq spin j z = 0 antisymmetric flavour qq {(ud du)/ 2, (s 1 s 2 s 2 s 1 )/ 2, (c 1 c 2 c 2 c 1 )/ 2)} I (J P ) = 0(1 + ) spin vector isotriplet: qq spin j z = 1 symmetric flavour qq {uu, (ud + du)/ 2, dd, ss, cc} I (J P ) {1(0 + ), 1(1 + ), 1(2 + )} Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 8

27 Perform a large number of fits varying... the temporal separation at which the lattice potential is read of the correlation function the range at which the fit to the potential is performed qq = (ud-du)/ 2 qq= uu, (ud+du)/ 2, dd qq = (s (1) s (2) -s (1) s (2) )/ 2 qq = ss r min=3a r min=2a scalar vector r min=3a r min=2a scalar vector MeV MeV α MeV α MeV MeV MeV MeV MeV d in fm d in fm qq = (c (1) c (2) -c (1) c (2) )/ r min=3a r min=2a scalar 0.1 α MeV -20 MeV -60 MeV Binding for light isosinglet channel only! MeV d in fm Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 8

28 Extrapolation to the physical pion mass Example plots for a t-range [4a...9a] av(r) r/a=5 r/a=4 r/a=3 r/a=2 linear fit, χ 2 /dof=0.49 linear fit, χ 2 /dof=0.19 linear fit, χ 2 /dof=1.00 linear fit, χ 2 /dof= m π in MeV 2 Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 8

29 The GEP 1 Build a matrix C(t) of correlation functions C ij (t) 2 Solve the GEP: C jk (t)v (n) k (t, t 0 ) = λ (n) (t, t 0 )C jk (t 0 )v (n) k (t, t 0 ) 3 And find: m (n) eff (t, t 1 0) = lim t a log λ (n) (t, t 0 ) λ (n) (t + a, t 0 ) Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 8

30 Choice of Γ for the B B system The choice of the matrix Γ is constrained by the quantum numbers of the O Q Q+π Only taking into account O B B we find the strongest attraction for Γ = γ 5 γ 0 γ 5 Antje Peters (Goethe University Frankfurt) Heavy-light four-quark systems November 16, / 8

arxiv: v1 [hep-lat] 3 Aug 2015

arxiv: v1 [hep-lat] 3 Aug 2015 Exploring possibly existing qq b b tetraquark states with qq = ud,ss,cc arxiv:158.343v1 [hep-lat] 3 Aug 215 a, Pedro Bicudo b, Krzysztof Cichy a,c, Björn Wagenbach a, Marc Wagner a a Goethe-Universität