Progress on the H dibaryon from N f = 2 + 1 CLS ensembles Andrew Hanlon Helmholtz-Institut Mainz, Johannes Gutenberg-Universit¨ at In collaboration with: Jeremy Green, Parikshit Junnarkar, Hartmut Wittig International Molecule-type Workshop Frontiers in Lattice QCD and related topics Yukawa Institute for Theoretical Physics, Kyoto University April 15-26, 2019
Outline • Motivation for studying the H dibaryon • Interpolating operators in Lattice QCD • Overview of N f = 2 CLS ensemble results from the Mainz group [arXiv:1805.03966] • Distillation vs. point sources • Finite-volume analysis using the L¨ uscher formalism • Preliminary results on N f = 2 + 1 CLS ensembles • Larger basis of operators • Use of spin-1 baryon-baryon operators • Future work • L¨ uscher analysis with multiple partial waves and/or decay channels, using the TwoHadronsInBox code (NPB 924 , 477 (2017)) • SU (3) broken ensembles Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 1 / 36
Motivation • In 1977, Jaffe predicts deeply bound dibaryon ( E B ≈ 80 MeV) with quark content uuddss , J P = 0 + , I = 0 • Conclusive experimental evidence for such a state is still lacking • Upper bound of ≈ 7 MeV on binding energy at 90% confidence level • Early quenched lattice calculations disagree on existence of a bound state • More recent results with dynamical quarks from NPLQCD and HAL QCD disagree on the binding energy for m π ≈ 800 MeV • Relatively simple dibaryon system Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 2 / 36
SU (3) Flavor Structure • The H dibaryon lies in the 1 -dimensional irrep of SU (3) F • Can form singlet from two octet baryons 8 ⊗ 8 = ( 1 ⊕ 8 ⊕ 27 ) S ⊕ ( 8 ⊕ 10 ⊕ 10 ) A • Upon SU (3) symmetry breaking, 8 and 27 mix with 1 • Construct linear combinations of ΛΛ, ΣΣ, and N Ξ operators to obtain BB 1 , BB 8 , and BB 27 • Can study other interesting dibaryon systems: • The dineutron lives in the 27 irrep • The deuteron lives in the 10 irrep (with J P = 1 + ) Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 3 / 36
Interpolating Operators • Two-baryon operators: • Momentum-projected octet baryon operators � e − i p · x ǫ abc ( s a C γ 5 P + t b ) r c B α ( p , t )[ rst ] = α x • Can form spin-zero and spin-one operators [ B 1 B 2 ] 0 ( p 1 , p 2 ) = B (1) ( p 1 ) C γ 5 P + B (2) ( p 2 ) [ B 1 B 2 ] i ( p 1 , p 2 ) = B (1) ( p 1 ) C γ i P + B (2) ( p 2 ) • Hexaquark operators inspired by Jaffe’s bag model prediction: [ rstuvw ] = ǫ ijk ǫ lmn ( s i C γ 5 P + t j )( v l C γ 5 P + w m )( r k C γ 5 P + u n ) • Can form singlet H 1 and 27-plet H 27 flavor combinations Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 4 / 36
Energies from Lattice QCD • In principal, can extract energies from two-point correlations � C ( t ) = � 0 | O ( t + t 0 ) O † ( t 0 ) | 0 � = | � 0 | O | n � | 2 e − E n t n =0 • Define the effective energy � C ( t + ∆ t ) � E eff ( t ) ≡ − 1 ∆ t ln C ( t ) • For large times, can extract the ground state t →∞ E eff ( t ) = E 0 lim • To better extract ground state, need operators with low overlap onto excited states Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 5 / 36
Ground State for Singlet Channel on E 1 ( SU (3) Symmetric) • Legend indicates sink operators • Point-to-all propagators used • Hexaquark operators noisier and slower ground-state saturation 1.6 1.40 H 1 ,N , H 1 ,M E1, singlet H 1 ,N , BB 1 ,N, 0 BB 1 ,N, 0 , BB 1 ,N, 1 1.38 BB 1 ,N, 0 1.5 1.36 aE eff 1.4 1.34 1.32 1.3 1.30 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.8 1.0 1.2 1.4 t [fm] t [fm] Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 6 / 36
Adding Distillation to the Mix • Use of point sources requires local operators at the source • Leads to non-Hermitian correlator matrices � H ( t ) H † (0) � � BB ( t ) H † (0) � • Add use of timeslice-to-all method: Distillation! Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 7 / 36
Distillation y , t ) = S ( t ) ( � • Smearing of quark fields, ˜ q ( � y , � x ) q ( � x , t ), in interpolating operators reduces excited state contamination • A particular smearing kernel, Laplacian-Heaviside (LapH) smearing, turns out to be particularly useful N LapH � S ( t ) y ) = Θ( σ s + ∆ ( t ) x , t ) υ ( k ) υ ( k ) y , t ) ∗ ab ( � x , � a ( � b ( � ab ( x , y )) ≈ k =1 • Smearing of quark fields results in smearing of quark propagator S M − 1 S = V ( V † M − 1 V ) V † where the columns of V are the eigenvectors of ∆ • Only need the elements of the much smaller matrix (perambulators) ab ( x , y ) υ ( k ′ ) τ kk ′ ( t , t ′ ) = V † M − 1 V = υ ( k ) a ( x ) ∗ M − 1 ( y ) b Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 8 / 36
Distillation vs. Smeared Point Sources • Ensemble E1, ground state in singlet channel • Better quality data with less inversions Distillation Point-to-all 0 ( E eff − 2 m Λ , eff ) [MeV] -50 -100 -150 -200 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t [fm] Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 9 / 36
L¨ uscher Quantization Condition • What do finite-volume energies say about the real world? • Avoided level crossings contain information about the scattering process in infinite volume • More generally, the L¨ uscher quantization condition can be used to constrain scattering amplitudes from finite-volume energies det[1 + F ( P ) ( S − 1)] = 0 F ( P ) are known functions of finite-volume energy Credit: K. Rummukainen and S. A. Gottlieb, Nucl. Phys. B450, 397 (1995) Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 10 / 36
Finite Volume Analysis - L¨ uscher Method • S-wave scattering phase shift: 2 p 2 = 1 q = pL 00 (1 , q 2 ) , 4( E 2 − P 2 ) − m 2 √ π L γ Z P p cot δ 0 ( p ) = 2 π , Λ • Perform fit with effective 0 . 1 range expansion a ∆ E = 0 . 0062 ( 34 ) 0 . 0 • Pole below threshold − 0 . 1 ( p / m π ) cot δ indicates a bound state − 0 . 2 1 [ 000 ] − 0 . 3 A ∝ [ 000 ] ∗ p cot δ 0 ( p ) − ip [ 001 ] − 0 . 4 [ 011 ] � [ 111 ] − p 2 − 0 . 5 = ⇒ p cot δ 0 ( p ) = − − 0 . 20 − 0 . 15 − 0 . 10 − 0 . 05 0 . 00 0 . 05 0 . 10 ( p / m π ) 2 Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 11 / 36
Comparison to Other Collaborations • Green are SU (3)-symmetric, and blue are SU (3) broken 80 HAL QCD NPLQCD 60 This work, distillation This work, FV-analysis ∆ E [MeV] 40 20 0 -20 0 200 400 600 800 1000 1200 m π [MeV] Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 12 / 36
Extending to a larger basis of operators • Previous two-flavor project used a small basis of spin-0 operators in the trivial irreps (i.e. A + 1 , A 1 ) • Latest study now includes spin-1 operators and a much larger set of irreps. • For instance, the T + 1 operators can be used to study the deuteron: � 1 , i = 1 [ B 1 B 2 ] ( a )( n ) [ B 1 B 2 ] ( a ) ( p , − p ) T + i N p | p 2 = n � i ) − 1 [ B 1 B 2 ] ( a ) 1 , i = [ B 1 B 2 ] ( a ) [ B 1 B 2 ] ( a ) (ˆ i , − ˆ (ˆ j , − ˆ j ) T + i i 3 j • A need for checking the transformation properties of this large set of new operators was needed Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 13 / 36
Rotational Properties of Operators • Python package using SymPy libary to determine rotation properties • Can very simply construct needed operators: u = QuarkField.create(’u’) a = ColorIdx(’a’) i = DiracIdx(’i’) ... Delta = Eijk(a,b,c) * u[a,i] * u[b,j] * u[c,k] • Project to definite momentum, and determine Little Group delta_ops = Operator(Delta, P([0,0,1])) delta_op_rep = OperatorRepresentation(*delta_ops) delta_op_rep.lgIrrepOccurences() # output: 6 G1 + 4 G2 • Supports multi-particle operators, and constructing octet baryons Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 14 / 36
Some Details of the Python Package • The representation matrix W ij ( R ) ( R ∈ G ) for a given basis of operators O i can be found via U R O i U † R = O j W ji ( R ) • Much can be uncovered from W ij ( R ) • Is W irreducible? | 2 = g G ⇐ � � � | χ W ( R ) ⇒ W is irreducible R ∈G • How many times does the irrep Γ occur in W ? Γ = 1 � � ∗ χ n W � � � χ Γ( R ) W ( R ) g G R ∈G • Apply group-theoretical projections = d Λ � Γ (Λ) P Λ λ λλ ( R ) W ji ( R ) ij g G R ∈G • Perform tests for rotations between equivalent momentum frames Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 15 / 36
CLS Ensembles Used for Larger basis of Operators • Beginning extensions to CLS ensembles with N f = 2 + 1 O ( a )-improved Wilson fermions • Initial results for the SU (3)-symmetric point, m π = m K = m η ≈ 420 MeV • U103 - β = 3 . 40, 24 3 × 128, N LapH = 20, N cfg = 5721 • H101 - β = 3 . 40, 32 3 × 96, N LapH = 48, N cfg = 2016 • B450 - β = 3 . 46, 32 3 × 64, N LapH = 32, N cfg = 1612 • Need high statistics to overcome signal-to-noise problem • Try to make N LapH as small as possible Andrew Hanlon Progress on the H dibaryon from Nf = 2 + 1 CLS ensembles 16 / 36
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