Lattice QCD for Nuclear Physics Saul D. Cohen (for NPLQCD Collaboration) International Workshop on Lattice QCD 2013 October 18 S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 1 / 38
Outline Lattice QCD for Nuclear Spectroscopy 1 Quarkonia in Nuclei 2 Nuclear Sigma Term 3 Nuclear Equation of State 4 S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 2 / 38
Nuclear LQCD Section 1 Lattice QCD for Nuclear Spectroscopy S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 3 / 38
Nuclear LQCD NPLQCD Collaboration S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 4 / 38
Nuclear LQCD Strategy The Trouble with Nucleons Nucleons are more complicated than mesons because... Noise Signal diminishes at large t relative to noise Excited-state contamination Nearby excited state Roper N (1440) Hard to extrapolate in pion mass ∆ resonance nearby; multiple expansions, poor convergence Requires large volume and high statistics Ensembles are not always generated with nuclear physics in mind Quark contractions Naively scale like N u ! N d ! N s ! For a typical nucleus: ((3 A / 2)!) 2 , α : 518400, 12 C: 4 × 10 31 S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 5 / 38
Nuclear LQCD Strategy Signal-to-Noise Ratio Why is noise such a problem for nucleons? Recall that variance is σ 2 O = � O 2 � − � O � 2 . For a nucleon correlator, our operator is a correlator: O ∝ qqq ( t ) ¯ q ¯ q ¯ q (0) What you want What you get u u u u Π Π p p u u u u d d u u Π Π u u u u p u u p d d Π Π d d d d ∼ e − M N t / e − 3 M π t / 2 ∼ e − ( M N − 3 S / N ∼ e − M N t / e − 2 M N t / 2 ∼ const 2 M π ) t S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 6 / 38
Nuclear LQCD Strategy The Golden Window Things don’t always have to go wrong Although the exponential behavior is known, the coefficients are not. Suppose there is suppression of the overlap of the � N † N � onto the 3 π state. A 2 σ 2 = Z 0 e − 2 AM N t + ( M π L ) 3 Z 1 e − (2( A − 1) M N +3 M π ) t + . . . 1.0 For Z n ≪ Z n +1 , 0.8 � ( M π L ) 3 � � 0 � 0 n 2 t gold ∼ 2 M N − 3 M π ln A 2 0.6 � 0 � 0 b t E Momentum-projected 0.4 operators cover full volume; nn � 0 pions only form if N and N † 0.2 n coincide spatially. 0.0 [PRD80,074501 (2009)] 0 10 20 30 40 50 60 t � b t S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 7 / 38
Nuclear LQCD Strategy Excited-State Contamination Trade-off: We must remain at small t to avoid noise, but we must wait for large t for excited states to die away Use large t , defeat noise with enormous statistics Craft sources that improve overlap with ground state or reduce overlap with noise or both Directly address excited states in analysis Multistate fitting Variational method or matrix Prony S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 8 / 38
Nuclear LQCD Strategy Contractions Building a multibaryon operator Need to tame squared-factorial behavior Symmetries give substantial improvements: Triton: 2880 → 93, 4 He: 518400 → 1107 a w a 1 , a 2 ,..., a n q ( a 1 ) q ( a 2 ) . . . q ( a n ) Consider A = � � Use total antisymmetry: w a 1 , a 2 ,..., a n A = � N i ǫ i 1 , i 2 ,..., i n q ( a i 1 ) q ( a i 2 ) . . . q ( a i n ) k =1 ˜ � � k Constrain wavefunction to include only 3-quark baryon subblocks B a 1 , a 2 , a 3 = � N B w c 1 , c 2 , c 3 i ǫ i 1 , i 2 , i 3 S ( c i 1 , a 1 ) S ( c i 2 , a 2 ) S ( c i 3 , a 3 ) k =1 ˜ � � k [PRD87,114512 (2013)] S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 9 / 38
Nuclear LQCD Strategy Contractions Assembling blocks Use recursion and save intermediate blocks Connect each baryon in all possible ways to the quarks at the source. With M terms in the baryon side and N on the quark side Final cost: M × N × N u ! N d ! N s ! 2 A − N Σ / Λ [PRD80,074501 (2009)] S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 10 / 38
Nuclear LQCD Results H Dibaryon Is it bound? Long predicted by Jaffe to be bound-state of ΛΛ [PRL38,195 (1977)] Experiments inconclusive NPLQCD Calculation [PRD85,054511 (2011)] Anisotropic 2+1-flavor 390-MeV O ( a )-improved Wilson-clover fermions a s = 0 . 1227 fm, a s / a t ≈ 3 . 5 2 volumes: 3 fm and 4 fm Very high statistics: 178 × 2215 (3 fm), 174 × 739 (4 fm) S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 11 / 38
Nuclear LQCD Results H Dibaryon Is it bound? Long predicted by Jaffe to be bound-state of ΛΛ [PRL38,195 (1977)] Experiments inconclusive NPLQCD Calculation [PRD85,054511 (2011)] 0.015 0.02 0.010 0.010 � k � 2 � s.l.u. � 2 � k � 2 � s.l.u. � 2 0.005 0 0 � 0.005 � 0.010 � 0.010 � 0.02 � 0.015 0 10 20 30 40 0 10 20 30 40 t � t.l.u. � t � t.l.u. � S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 11 / 38
Nuclear LQCD Results H Dibaryon Is it bound? Long predicted by Jaffe to be bound-state of ΛΛ [PRL38,195 (1977)] Experiments inconclusive NPLQCD Calculation [PRD85,054511 (2011)] 50 n f � 2 � 1 40 n f � 3 30 B H � MeV � 20 10 0 � 10 � 20 with HALQCD 0 200 400 600 800 m Π � MeV � [PRL106,162002 (2011)] S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 11 / 38
Nuclear LQCD Results d Deuteron Is it bound? Very weakly bound Expect strong finite-volume effects NPLQCD Calculation [PRD85,054511 (2011)] 0.03 0.02 0.02 0.01 � k � 2 � s.l.u. � 2 � k 2 � s.l.u. � 2 0.01 0 0 � 0.01 � 0.01 � 0.02 � 0.03 � 0.02 0 10 20 30 40 0 10 20 30 40 t � t.l.u. � t � t.l.u. � S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 12 / 38
Nuclear LQCD Results d Deuteron Is it bound? Very weakly bound Expect strong finite-volume effects NPLQCD Calculation [PRD85,054511 (2011)] 25 n f � 2 � 1 n f � 0 20 Experiment B d � MeV � 15 10 5 � 0 � 5 0 200 400 600 800 with PACS-CS m Π � MeV � [PRD84,054506 (2011)] S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 12 / 38
Nuclear LQCD Results SU(3)-Symmetric QCD Move to the SU(3) symmetric point: M π ≈ 800 MeV NPLQCD Calculation [PRD87,034506 (2012)] Isotropic 2+1-flavor 800-MeV O ( a )-improved Wilson-clover fermions a s = 0 . 145 fm 3 volumes: 3.4 fm, 4.5 fm and 6.7 fm Very high statistics: 72 × 3822 (3.4 fm), 48 × 3050 (4.5 fm), 54 × 1905 (6.7 fm) S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 13 / 38
Nuclear LQCD Results SU(3)-Symmetric QCD 0 s = − 1 s = − 2 s = 0 1 + 0 + 0 + − 20 1 + 1 + − 40 + + 1 1 + 3 2 2 2 − 60 + 1 2 0 + − 80 + − B [MeV] 3 2 − 100 0 + 0 + 0 + − 120 − 140 0 + − 160 2-body 3-body − 180 4-body − 200 3 3 3 4 4 3 He 4 He d nn nΣ Λ H Λ He Σ He Λ He H - dib nΞ ΛΛ He S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 14 / 38
Nuclear LQCD Results SU(3)-Symmetric QCD H -dibaryon deeply bound: 0 s = 0 1 + s = − 1 s = − 2 0 + 0 + − 20 1 + B H = 74 . 6(3 . 3)(3 . 3)(0 . 8) MeV 1 + − 40 Deuteron clearly bound: + + 1 1 3 + 2 2 − 60 2 + 1 2 0 + B d = 19 . 5(3 . 1)(0 . 2) MeV − 80 + − B [MeV] 3 2 − 100 more bound than quenched 0 + 0 + 0 + − 120 Small d - nn splitting − 140 0 + other splittings larger − 160 2-body − 180 3-body 4-body α also more bound than 0f − 200 d 3 He 4 He Λ H 3 3 Λ He Σ He 3 Λ He 4 H - dib nΞ ΛΛ He 4 nn nΣ B α = 107(12)(21)(1) MeV S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 14 / 38
Quarkonia in Nuclei Section 2 Quarkonia in Nuclei S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 15 / 38
Quarkonia in Nuclei Models and Experiment Unique Probe of QCD Effects Heavy quarkonia share no valence quarks with nuclei Normally dominant quark c c exchange suppressed to Ψ c Ψ c second order Dominated by two-gluon u u exchange p p u u (color van der Waals) d d Color Stark effect: Chromoelectric field induces dipoles in neutral hadrons that interact S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 16 / 38
Quarkonia in Nuclei Models and Experiment Unique Probe of QCD Effects Heavy quarkonia share no valence quarks with nuclei Normally dominant quark exchange suppressed to second order Dominated by two-gluon exchange c c Ψ c Ψ c (color van der Waals) D D Color Stark effect: u u Chromoelectric field induces p p u u dipoles in neutral hadrons d d that interact S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 16 / 38
Quarkonia in Nuclei Models and Experiment Unique Probe of QCD Effects Heavy quarkonia share no valence quarks with nuclei Normally dominant quark c c exchange suppressed to Ψ c Ψ c second order Dominated by two-gluon u u exchange p p u u (color van der Waals) d d Color Stark effect: Chromoelectric field induces dipoles in neutral hadrons that interact S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 16 / 38
Quarkonia in Nuclei Models and Experiment Model History Brodsky et al. [PRL64,1011 (1990)] Z , A noted features of pp scattering near p open-charm threshold n n No Pauli blocking; no quark-exchange n p p p p η c h : 19 MeV, η c 9 Be: 407 MeV(!) n n n n n Wasson [PRL67,2237 (1991)] points out the nucleus is not pointlike p p p p n Charm binding saturates for large A n n η c h : 0.8 MeV, η c 208 Pb: 27 MeV p S. D. Cohen (U Washington) NPLQCD 2013 Oct 18 17 / 38
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