NUCLEAR PHYSICS FROM LATTICE QCD

Twelfth Workshop on Non-Perturbative QCD Paris, June 10-14, 2013 NUCLEAR PHYSICS FROM LATTICE QCD Kostas Orginos

QCD Hadron structure and spectrum Hadronic Interactions Nuclear physics Figure by W. Nazarewicz

Hadron Interactions Goals: Challenge: Compute properties of nuclei from QCD Spectrum and structure Confirm well known experimental observation for two nucleon systems Explore the largely unknown territory of hyper-nuclear physics Provide input for the equation of state for nuclear matter in neutron stars Provide input for understanding the properties of multi-baryon systems -S XX 4 3 X 2 L 1 6 LLHe L 4 L 6 Li 16 He L 3 L 5 L 7 Li 14 Ne 12 F 17 He H L 4 L 6 8 L 9 Li 10 O 15 Ne He 8 N 13 F 18 H 6 C L 7 L 10 Li 11 O 16 Ne B He 9 N 14 F 19 7 C 12 O 17 Ne 5 B L 8 10 N 15 F 20 He 3 Be 8 C 13 O 18 Ne 5 Li 6 B 11 N 16 F 21 4 Be 9 C 14 O 19 Ne 2 Li 7 B 12 N 17 F 22 He 5 Be 10 C 15 O 20 Ne 3 Li 8 B 13 N 18 F 23 1 He 6 Be 11 C 16 O 21 Ne H 4 Li 9 B 14 N 19 F 24 2 He 7 Be 12 C 17 O 22 Ne H 5 Li 10 B 15 N 20 F 25 3 He 8 Be 13 C 18 O 23 Ne F 1 H 6 Li 11 B 16 N 26 21 O n 4 He 9 Be 14 C 24 Ne 19 H 7 Li 12 B 17 N 22 F 27 5 He 10 Be 15 C 20 O 25 Ne N 23 F 28 H 8 Li 13 B 18 6 He 11 Be 16 C 21 O 26 Ne H 9 Li 14 B 19 N 24 F 29 He 7 12 Be 17 C 22 O 27 Ne B 20 N 25 F 30 Li H 10 15 C Be He 18 23 O 28 Ne F B 21 N 26 O 16 C Be 19 24 29 F N B 22 27 O C 25 N 28 O 5 10 N 10 15 20

Scales of the problem Hadronic Scale: 1fm ~ 1x10-13 cm Lattice spacing << 1fm take a=0.1fm Lattice size La >> 1fm take La = 3fm Lattice 32 4 Gauge degrees of freedom: 8x4x32 4 = 3.4x10 7 color dimensions sites

The pion mass is an additional small scale ~1/mπ~1.4fm Single hadron volume corrections e m L ~ 6 fm boxes are needed Two hadron bound state volume corrections e applel Binding momentum κ of the deuteron ~ 45MeV Nuclear energy level splittings are a few MeV Box sizes of about 10 fm will be needed

Bound States Luscher Comm. Math. Phys 104, 177 86 E b = p 2 + m 2 1 + p2 + m 2 2 m 1 m 2 p 2 < 0 E b p 2 2µ = 2 2µ = p κ is the binding momentum and µ the reduced mass Finite volume corrections: E b = 3 A 2 e L cubic group irrep: µl + O e 2 L A + 1

Scattering on the Lattice Luscher Comm. Math. Phys 105, 153 86 Elastic scattering amplitude (s-wave): A(p) = + + +... A(p) = 4π m 1 p cotδ ip At finite volume one can show: ( ) E n =2 p 2 n + m 2 p cot δ(p) = 1 ( p 2 πl S L 2 4π 2 ) S ( η ) j <Λ j 1 j 2 η 4πΛ Small p: p cot δ(p) = 1 a + 1 2 rp2 +... a is the scattering length

Two Nucleon spectrum free nucleons 0.55 0.5 0.45 } 0.4 0.35 free 2 particle spectrum 0.3 0.25 0.2 3fm box M n 24 3 lattice anisotropy factor 3.5 M π =390MeV

Signal to Noise ratio for correlation functions C(t) = N(t) N(0) Ee M N t var(c(t)) = N N(t)N N(0) Ae 2M N t + Be 3m t StoN = C(t) var(c(t)) = Ae (M N 3/2m )t The signal to noise ratio drops exponentially with time The signal to noise ratio drops exponentially with decreasing pion mass For two nucleons: StoN(2N) = StoN(1N) 2

10 3 proton proton proton 10 2 Signal to Noise 10 1 10 0 10 1 10 2 20 0 20 40 60 80 100 120 140 t Signal to Noise 32 3 x 256 M π =390MeV anisotropy factor 3.5 NPLQCD data

Challenges for Nuclear physics New scales that are much smaller than characteristic QCD scale appear The spectrum is complex and more difficult to extract from euclidean correlators Construction of multi-quark correlations functions may be computationally expensive Monte-Carlo evaluation of correlation functions converges slowly

Interpolating fields Most general multi-baryon interpolating field N h = X a w a 1,a 2 a n q h q(a 1 ) q(a 2 ) q(a nq ) The indices a are composite including space, spin, color and flavor that can take N possible values The goal is to calculate the tensors w The tensors w are completely antisymmetric Number of terms in the sum are N! (N n q )!

Imposing the anti-symmetry: N h = XN w k=1 w (a 1,a 2 a n q ),k h X i i 1,i 2,,i n q q(ai1 ) q(a i2 ) q(a in q ) Reduced weights Totally anti-symmetric tensor 1,2,3,4,,n q =1 Total number of reduced weights: N! n q!(n n q )!

Hadronic Interpolating field N h = XM w k=1 W (b 1,b 2 b A ) h X i i 1,i 2,,i A B(bi1 ) B(b i2 ) B(b ia ) hadronic reduced weights baryon composite interpolating field B(b) = N X B(b) k=1 w (a 1,a 2,a 3 ),k b X i 1,i 2,i3 q(a i1 ) q(a i2 ) q(a i3 ) i Basak et.al. PhysRevD.72.074501 (2005)

Calculation of weights Compute the hadronic weights Replace baryons by quark interpolating fields Perform Grassmann reductions Read off the reduced weights for the quark interpolating fields Computations done in: algebra (C++) N h = XM w k=1 W (b 1,b 2 b A ) h X i i 1,i 2,,i A B(bi1 ) B(b i2 ) B(b ia ) N h = XN w k=1 w (a 1,a 2 a n q ),k h X i i 1,i 2,,i n q q(ai1 ) q(a i2 ) q(a in q )

Interpolating fields Label A s I J Local SU(3) irreps int. field size N 1 0 1/2 1/2 + 8 9 1-1 0 1/2 + 8 12 1-1 1 1/2 + 8 9 1-2 1/2 1/2 + 8 9 d 2 0 0 1 + 10 21 nn 2 0 1 0 + 27 21 n 2-1 1/2 0 + 27 96 n 2-1 1/2 1 + 8 A, 10 48, 75 n 2-1 3/2 0 + 27 42 n 2-1 3/2 1 + 10 27 n 2-2 0 1 + 8 A 96 n 2-2 1 1 + 8 A, 10, 10 52,66,75 H 2-2 0 0 + 1, 27 90,132 3 H, 3 He 3 0 1/2 1/2 + 35 9 3 H(1/2+ ) 3-1 0 1/2 + 35 66 3 H(3/2+ ) 3-1 0 3/2 + 10 30 3 He,3 H, nn 3-1 1 1/2 + 27, 35 30,45 3 He 3-1 1 3/2+ 27 21 He 4 0 0 0 + 28 1 4 He, 4 H 4-1 1/2 0+ 28 6 4 He 4-2 1 0+ 27, 28 15, 18 0 pnn 5-3 0 3/2 + 10 +... 1 NPLQCD arxiv:1206.5219

Contraction methods quark to hadronic interpolating fields quark to quark interpolating fields

Quarks to Quarks [N h 1 (t) N h 2 (0)] = Z X j DqD q e S QCD N 0 w X XN w k 0 =1 k=1 w 0(a0 1,a0 2 a0 nq ),k0 h w (a 1,a 2 a n q ),k h X j 1,j 2,,j n q i 1,i 2,,i n q q(a 0 ) q(a0 jn q j 2 )q(a 0 j 1 ) q(a i1 ) q(a i2 ) q(a ) in q i = e S eff X j N 0 w X k 0 =1 XN w k=1 w 0(a0 1,a0 2 a0 nq ),k0 h w (a 1,a 2 a n q ),k h X j 1,j 2,,j n q i 1,i 2,,i n q S(a 0 j1 ; a i1 )S(a 0 j 2 ; a i2 ) S(a 0 j ; a n i ) q n q i Define the matrix: G(j, i) (a0 1,a0 2 a0 nq );(a 1,a 2 a n q ) = S(a 0 j; a i )

Quarks to Quarks The matrix: G(j, i) (a0 1,a0 2 a0 nq );(a 1,a 2 a n q ) = S(a 0 j; a i ) The Correlation function: [N h 1 (t) N h 2 (0)] = N 0 w X XN w k 0 =1 k=1 w 0(a0 1,a0 2 a0 nq ),k0 h w (a 1,a 2 a n q ),k h G (a0 1,a0 2 a0 nq );(a 1,a 2 a n q ) Total momentum projection is implicit in the above

~P single point source Quarks to Quarks Naive Cost: Actual Cost: n u!n d!n s! NM n 3 un 3 dn 3 s MN M terms N terms Loop over all source and sink terms Compute the determinant for each flavor Cost is polynomial in quark number

H-dibaryon R. L. Jaffe, Phys. Rev. Lett. 38, 195 (1977) Proposed by R. Jaffe 1977 Perturbative color-spin interactions are attractive for (uuddss) Diquark picture of scalar diquarks (ud)(ds)(su) S=-2, B=2, J p =0 + Experimental searches of the H have not found it BNL RHIC (+model): Excludes the region [-95, 0] MeV KEK: Resonance near threshold A. L. Trattner, PhD Thesis, LBL, UMI-32-54109 (2006). C. J. Yoon et al., Phys. Rev. C 75, 022201 (2007). Several Lattice QCD calculations have been addressing the existence of a bound H

NPLQCD: lattice set up Anisotropic 2+1 clover fermion lattices a ~ 0.125fm (anisotropy of ~ 3.5) Hadron Spectrum/JLAB pion mass ~ 390 MeV Volumes 16 3 x 128, 20 3 x 128, 24 3 x 128, 32 3 x 256 largest box 4fm Smeared source - 3 sink interpolating fields Interpolating fields have the structure of s-wave Λ-Λ system I=0, S=-2, A1, positive parity

H-dibaryon: Towards the physical point [ S. Beane et.al. arxiv:1103.2821 Mod. Phys. Lett. A26: 2587, 2011] 50 H-dibaryon: Is bound at heavy quark masses. May be unbound at the physical point B H HMeVL 40 30 20 10 0 NPLQCD n f =2+1-10 HALQCD n f =3-20 0.0 0.2 0.4 0.6 0.8 m 2 p HGeV 2 L HALQCD:Phys.Rev.Lett.106:162002,2011 ChiPT studies indicate the same trend: P. Shanahan et.al. arxiv:1106.2851 J. Haidenbauer, Ulf-G. Meisner arxiv:1109.3590 NPLQCD

H-dibaryon: Towards the physical point [ S. Beane et.al. arxiv:1103.2821 Mod. Phys. Lett. A26: 2587, 2011] 50 H-dibaryon: Is bound at heavy quark masses. May be unbound at the physical point B H HMeVL 40 30 20 10 0 NPLQCD n f =2+1-10 HALQCD n f =3-20 0.0 0.2 0.4 0.6 0.8 m p HGeVL HALQCD:Phys.Rev.Lett.106:162002,2011 ChiPT studies indicate the same trend: P. Shanahan et.al. arxiv:1106.2851 J. Haidenbauer, Ulf-G. Meisner arxiv:1109.3590 NPLQCD

Two baryon bound states [ S. Beane et.al. arxiv:1108.2889 submitted to Phys.Rev.D] V. G. J. Stoks and T. A. Rijken Phys. Rev. C 59, 3009 (1999) [arxiv:nucl-th/9901028 G. A. Miller, arxiv:nucl-th/0607006 J. Haidenbauer, Ulf-G. Meisner Phys.Lett.B684,275-280(2010) arxiv:0907.1395 n f =0: Yamazaki, Kuramashi, Ukawa Phys.Rev. D84 (2011) 054506 arxiv: 1105.1418 gauge fields 2+1 flavors (JLab) anisotropic clover m π ~ 390MeV NPLQCD

Avoiding the noise Work with heavy quarks StoN = C(t) var(c(t)) = Ae (M N 3/2m )t

Lattice Setup Isotropic Clover Wilson with LW gauge action Stout smeared (1-level) Tadpole improved SU(3) symmetric point Defined using m π /m Ω Lattice spacing 0.145fm Set using Υ spectroscopy Large volumes NPLQCD arxiv:1206.5219 6000 configurations, 200 correlation functions per configuration 24 3 x 48 32 3 x 48 48 3 x 64 3.5fm 4.5fm 7.0fm computer time: XSEDE/NERSC

Nuclear spectrum NPLQCD

Nucleon Phase shifts Luscher Comm. Math. Phys 105, 153 86 Elastic scattering amplitude (s-wave): A(p) = + + +... A(p) = 4π m 1 p cotδ ip At finite volume one can show: ( ) E n =2 p 2 n + m 2 p cot δ(p) = 1 ( p 2 πl S L 2 4π 2 ) S ( η ) j <Λ j 1 j 2 η 4πΛ Small p: p cot δ(p) = 1 a + 1 2 rp2 +... a is the scattering length

E(p) =2 p 2 + m 2 2m Two Body spectrum in a box... p cot (p) =S( p2 L 2 4 2 ) p cot (p) = 1 a + r2 p 2 + E(p) 0 E -1 k back to back momentum k single point source

nucleon-nucleon phase shifts NPLQCD 200 150 200 L=24,»P»=0 L=32,»P»=0 L=24,»P»=1 L=32 150,»P»=1 Levinson's Theorem Experimental L=24,»P»=0 L=32,»P»=0 L=24,»P»=1 Levinson's Theorem Experimental experiment HdegreesL d H3 S1 L 100 50 0 HdegreesL d H1 S0 L 100 50 0-50 -50-100 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 kêm p kêm p lattice QCD spin triplet spin singlet M π =800MeV degenerate up down and strange quarks

Correlators for large nucleii 4 He HSPL 40 20 log 10 CHtL 0-20 -40 0 10 20 30 40 têa

Correlators for large nucleii 8 Be HSPL 60 40 20 log 10 CHtL 0-20 -40-60 0 10 20 30 40 têa

Correlators for large nucleii 100 12 C HSPL 50 log 10 CHtL 0-50 -100 0 10 20 30 40 têa

Correlators for large nucleii 150 16 O HSPL 100 50 log 10 CHtL 0-50 -100-150 0 10 20 30 40 têa

Correlators for large nucleii 28 Si HSPL 200 100 log 10 CHtL 0-100 -200 0 10 20 30 40 têa

Expected Carbon spectrum in the 32 3 box 16 15.8 15.6 15.4 15.2 15 14.8 14.6 14.4 14.2 14 12 N 6 d 4 3 He 3 4 He 4N+2 4 He 8N+ 4 He 3N+3 3 He

Conclusions We have a systematic way of constructing all possible interpolating fields Using heavy quarks noise is reduced Special care needs to be given to the selection of interpolating fields Minimize number of terms in the interpolating field and optimize the signal NPLQCD: Presented results for the spectrum of nuclei with A<5 and S>-3... and nucleon-nucleon phase shifts NPLQCD arxiv: 1206.5219,1301.5790 We have an algorithm for quark contractions in polynomial time for A>5 Future work must focus on the improved sampling methods to reduce statistical errors at light quark masses