Hadrons and Nuclei : Scattering Lattice Summer School Martin Savage Summer 2007 University of Washington

Why Scattering with Lattice QCD ?  Reproducing what is known is important check of lattice QCD, but intrinsically not interesting----not new physics  e.g. NN scattering at physical pion mass  Calculating quantities that cannot be determined (well) any other way is the underlying motivation  e.g. YN scattering, nnn, weak-YN, d ¾ d ­ ( m q )

Present Nuclear Theory fails to Reproduce (a few!) Precise Expts

Courtesy of W. Tornow, data from TUNL, calculations by Present Nuclear Theory fails to Reproduce Precise expts (2)

Worlds YN Data Polinder et al

   p → K     Production)    p →    p  Scattering)    n → K   Production)  p →  p  Scattering) Hyperon-Nucleon Scattering Experiments Kozi Nakai (KEK) (talk given at Hypernuclear 2006 (Mainz))

Beam SCITIC0    Production   p Scattering  Production  p Scattering Hyperon-Nucleon Scattering Experiments Kozi Nakai (KEK) su suu Weak decay Strong scatter

Neutron Stars (1)  Why are we interested in scattering of strange baryons (and mesons) ?  Supernova Remnant ?  neutron stars or black holes, or black holes, ….. kaon condensation, strange baryons ? …..

Neutron Stars (2) Atmosphere Envelope Crust Outer Core Inner Core Homogeneous Matter Lasagna Nuclei + Neutron superfluid n-superfluid and p superconductor Reddy and Page astro-ph/

Neutron Stars (3) : Hyperons in Neutron Matter n   n YN interactions shift the mass of Y in a neutron background H » ® § y § n y n + M ( 0 ) § § y § + :::: ! ³ M ( 0 ) § + ®½ n ´ § y § + :::: M ean ¡ F i e ld

Neutron Stars (4) Vacuum - Masses With Interactions -- meson exchange model WithOUT Interactions We Need QCD Calculations to Improve upon Model Calcs Reddy and Page astro-ph/

Low-Energy Scattering, Phase-Shifts and Scattering Parameters (1) Analytic function of com energy Review :

Low-Energy Scattering, Phase-Shifts and Scattering Parameters (2) k co t ± = ¡ 1 a r k 2 + ::: a = Scattering Length r = Effective Range Can take any valueSize dictated by range of interaction

Low-Energy Scattering, Phase-Shifts and Scattering Parameters (3) k = 0 k = = 0 a ± Ã = 1 ¡ r a

Maiani-Testa no-go Theorem (1)  (s) ? S-matrix elements cannot be extracted from infinite-volume Euclidean-space correlation functions except at kinematic thresholds.

Maiani-Testa no-go Theorem (2) j pp i ou t = S y j pp i i n ou t h pp j pp i i n = i n h pp j S j pp i i n = e i 2 ± ( b e l ow i ne l as t i c t h res h o ld s ) Consider the Euclidean-space correlation function associated with a source J(x) that couples to two protons G E ( t 1 ; t 2 ;q ) = h 0 j © q ( t 1 ) © ¡ q ( t 2 ) J ( 0 )j 0 i J(0) (s-wave to s-wave) © q ( t ) © ¡ q ( t ) Interpolating fields for proton

Maiani-Testa no-go Theorem (3) This dominates at long times unless 2 E q is equal to the minimum value of E n Away from Kinematic threshold…..cannot isolate S-matrix elements from Euclidean space correlators h 0 j Á ( 0 ; 0 )j p i = p Z © q = R d 3 xe ¡ i q ¢ x Á ( x ; t )

Maiani-Testa no-go Theorem (4) In Minkowski space 2 E q P q ( J ; t 2 ) ! 1 2 ( ou t h pp j J ( 0 )j 0 i ¡ i n h pp j J ( 0 )j 0 i)

Luscher (1) Compute something else !!!! Work in finite-volume and look at energy-levels Non-Relativistic QM analysis = Lee + Yang E ( j ) 0 = h 0 ( 0 ) j ^ V j 0 ( j ¡ 1 ) i ; J-th order g.s. energy-shift Unperturbed g.s. wavefunction J-th order contribution to g.s. wavefunction

Lee + Yang (2) k = 2 ¼ L n n = ( n x ; n y ; n z ) h r j k i = 1 L 3 = 2 e i k ¢ r h r 1 ; r 2 j k ; ¡ k i = 1 L 3 e i k ¢ ( r 1 ¡ r 2 ) P er i o d i c B. C. g i ve r ! r + m L

Lee + Yang (3) : Energies ^ V = ´ ± 3 ( ^ r 1 ¡ ^ r 2 ) ¢E 0 = E ( 1 ) 0 + E ( 2 ) 0 + ::: = ´ L ¡ ´ M 4 ¼ 2 L X n 6 = 0 1 j n j 2 + ::: 3 5

Lee + Yang (4) : Threshold Scattering ^ V = ´ ± 3 ( ^ r 1 ¡ ^ r 2 ) f = ¡ ¹ 2 ¼ Z d 3 r V ( r ) ¡ ¹ 2 ¼ Z d 3 r 1 Z d 3 r 2 V ( r 1 ) G + ( r 1 ; r 2 ) V ( r 2 ) + ::: = ¡ ¹´ 2 ¼ + ¹ 2 ´ 2 ¼ Z d 3 p ( 2 ¼ ) 3 1 j p j 2 + i ² + :::: = a ´ = ¡ 4 ¼a M · 1 ¡ 4 ¼a Z d 3 p ( 2 ¼ ) 3 1 j p j 2 + i ² + ::: ¸

Lee + Yang (4) : Combining ^ V = ´ ± 3 ( ^ r 1 ¡ ^ r 2 ) ¢E 0 = ¡ 4 ¼a ML ³ a ¼ L ´ ¤ j X n 6 = 0 1 j n j 2 ¡ 4 ¼ ¤ j 1 A + ::: 3 5

Luscher (5) : True in QFT !! Below Inelastic thresholds and lattices L >> R UV regulator Measure on lattice S ( x ) = l i m ¤ j ! 1 ¤ j X j 1 j j j 2 ¡ x 2 ¡ 4 ¼ ¤ j pco t ± ( E ) = 1 ¼ L S µ p L 2 ¼ ¶

Luscher (6) : Methodology? (x,t) (y,t) source G ( 0 ; 0 ) 2 ( t ) h G ( 0 ) 1 ( t ) i 2 ! e ¡ ( E 2 ¡ 2 M ) t G ( p 1 = 0 ; p 2 = 0 ) 2 ( t ) = Z d 3 x Z d 3 y G 2 ( y ; t ;x ; t ; 0 ; 0 ) ! A 2 e ¡ E 2 t G ( p = 0 ) 1 ( t ) = Z d 3 x G 1 ( x ; t ; 0 ; 0 ) ! A 1 e ¡ M t

Luscher (7) : Methodology? G ( 0 ; 0 ) 2 ( t ) h G ( 0 ) 1 ( t ) i 2 ! e ¡ ( E 2 ¡ 2 M ) t T = E 2 ¡ 2 M = 2 p q 2 + M 2 ¡ 2 M pco t ± ( T ) = 1 ¼ L S µ q L 2 ¼ ¶

Many-Bosons (1) 3-boson interaction numbers

3-Bosons (1)

I=2   Simplest hadronic scattering process  Physics Wise  Computationally C ( t ) = G ( 0 ; 0 ) 2 ( t ) h G ( 0 ) 1 ( t ) i 2 ! A 2 A 2 1 e ¡ ( E 2 ¡ 2 M ) t l og · C ( t ) C ( t + 1 ) ¸ ! E 2 ( t ) ¡ 2 M = T ( t ) = 2 p q 2 ( t ) + M 2 ¡ 2 M pco t ± ( T ( t )) = 1 ¼ L S µ q ( t ) L 2 ¼ ¶

I=2  [ m ¼ a ¼ + ¼ + ]( t ) m ¼ » 350 M e V e.g. Beane et al, arXiv:

I=2  Beane et al, arXiv: m ¼ a I = 2 ¼¼ = ¡ m 2 ¼ 8 ¼ f 2 ¼ ( 1 + m 2 ¼ 16 ¼ 2 f 2 ¼ " 3 l og µ m 2 ¼ ¹ 2 ¶ ¡ 1 ¡ l I = 2 ¼¼ ( ¹ ) #) (2007)

I=2  Beane et al, arXiv: Ã m ¼ ; f ¼

Lattice m  /f  is the way to go Two-flavor mixed-action Chen, O’Connell, Walker-Loud m ¼ a I = 2 ¼¼ = ¡ m 2 ¼ 8 ¼ f 2 ¼ ( 1 + m 2 ¼ ( 4 ¼ f ¼ ) 2 · 3 l n µ m 2 ¼ ¹ 2 ¶ ¡ 1 ¡ l I = 2 ¼¼ ( ¹ ) ¡ ~ ¢ 4 j u 6 m 4 ¼ ¸ )

I=0 ?  Computationally expensive Need to compute N ~ Volume propagators I = 2 I = 0 u u d d u u

I=0  Point to all propagators all to all propagators

CP-PACS : Phase-shift (extrapolation) CP-PACS 12 3 X X X 48 L = 2.5 fm n f = 2 (2002)

 – Scattering phase-shift CP-PACS

K  and KK Scattering Lattice QCD + Chiral Symmetry b = fm (S. Beane, P. Bedaque, T. Luu, K. Orginos, E. Pallante, A.Parreno, mjs) K +  + K +

Baryon Potentials from LQCD Why bother with the scattering amplitude…just calculate the potential, and use that in the Schrodinger equation !!! h 0 j ^ O 1 ( x ; t 0 ) i ® ^ O 1 ( y ; t 0 ) j ¯ j à 0 i = Z ( S ; I ) NN (j r j)h 0 j N ( x ; t 0 ) i ® N ( y ; t 0 ) j ¯ j à 0 i + ::: ^ O 1 ( x ; t ) i ® = ² a b c q i ; c ® ¡ q a ; T C ° 5 ¿ 2 q b ¢ ( x ; t ) G NN ( x ; y ; t ) = h 0 j ^ O 1 ( x ; t ) i ® ^ O 1 ( y ; t ) j ¯ J ( 0 )j 0 i = X n h 0 j ^ O 1 ( x ; 0 ) i ® ^ O 1 ( y ; 0 ) j ¯ j à n ih à n j J ( 0 )j 0 i e ¡ E n t 2 E n

Baryon Potentials from LQCD (2) Potential between two infinitely massive mesons is well-defined e.g. B-mesons in the HQ limit – E = V(R) r is a constant of the motion U E ( r ) = E ¹ r 2 G NN G NN 1 2 ¹ r 2 G NN + U E ( r ) G NN = EG NN trivially 1 ] Energy-dependent potential… i.e. each different energy requires a different potential ….not as useful as it first sounds !!! 2 ] NOT unique … only constrained to reproduce ONE quantity ….  ( E 0 )

Q Q QQ Direct Exchange BB t-channel Potentials... Insight into NN ? B B BB ¼¼¼ B -meson h as I = 1 2 ; s l = 1 2

BB (2) (W. Detmold, K.Orginos, mjs, 2007) Quenched : a = 0.1 fm

Periodic Boundary Conditions and Images

BB (3) (W. Detmold, K.Orginos, mjs )

Tensor Force between BB ? Deuteron

Quenched Potential : Hairpins  No strong anomaly ´ 0 i sa l soapseu d o- G o ld s t one b oson G ´´ ( q 2 ) = i ( q 2 ¡ m 2 ¼ + i ² ) + i ( M 2 0 ¡ ® © q 2 ) ( q 2 ¡ m 2 ¼ + i ² ) 2 V ( Q ) ( r ) = 1 8 ¼ f 2 ¾ 1 ¢ r ¾ 2 ¢ r µ g 2 A ¿ 1 ¢ ¿ 2 r + g ¡ ® © r ¡ g 2 0 M 2 0 ¡ ® © m 2 ¼ 2 m ¼ ¶ e ¡ m ¼ r Dominates at long-distances

Nucleons on the Lattice ~ ¾ x = p h G 2 i ¡ h G i 2 h G i ! e ( M N ¡ 3 2 m ¼ ) t 3 ¼ Mesons are easy, Nucleons are hard and two nucleons are even harder !!! G.P. Lepage, Tasi 1989

Large Scattering Lengths are OK ! Require : L >> r 0 but ANY a  M  = 350 MeV 2.5 fm lattices...  YESTERDAY !!

 M  = 350 MeV 2.5 fm lattices...  TODAY !! NN on the Lattice ~E L 2 Deuteron 1st continuum 2nd continuum (S. Beane, P. Bedaque, A.Parreno, mjs) Effective Field Theory Calculation at Finite-Volume

NN Correlators 1S01S0 3 S D 1 G Fully-Dynamical QCD Domain-Wall Valence on rooted-Staggered Sea  C n Exp [-E n t] n C 0 Exp [-E 0 t]

NN Scattering (S. Beane, P. Bedaque, K. Orginos, mjs ; PRL97, (2006)) 1 S 0 : pp, pn, nn 3 S D 1 : pn : deuteron a ~ 1/m  Scale-Invariance

NN is Fine-Tuned a 1 S 0 n p = ¡ 23 : 710 § 0 : 030 f m ; r 1 S 0 n p = + 2 : 73 § 0 : 03 a 3 S 1 n p = + 5 : 432 § 0 : 005 f m ; r 3 S 1 n p = + 1 : 73 § 0 : 02 a >> r

Toy-Model r = 0 a = + 1 ~E L 2

Hyperon-N Interactions (S.Beane, P.Bedaque, T.Luu, K.Orginos, E.Pallante, A.Parreno, mjs, 2006) |k| = 261 MeV |k| = 179 MeV |k| = 255 MeV|k| = 169 MeV

 Interactions --- I=0, J=0, s=2 -ve Energy Shift ? M  = 590 MeV

Luscher Relation -- revisited pco t ± ( E ) = 1 ¼ L S µ p L 2 ¼ ¶ V = 0 ! a = r = :::: = 0 S ( ´ ) = 1 Non-interacting particles k = 2 ¼ L n n = ( n x ; n y ; n z ) T < 0T > 0 Bound-state or Scattering state ?

Bound states are also described !! ° + pco t ± j p 2 = ¡ ° 2 = 0 E ¡ 1 = ¡ ° 2 M · ° L 1 1 ¡ 2 ° ( pco t ± ) 0 e ¡ ° L + ::: ¸ pco t ± ( E ) = 1 ¼ L S µ p L 2 ¼ ¶ (S. Beane, P. Bedaque, A.Parreno, mjs)

Bound States vs Scattering States (1) L = 200 b 1/r = M  = 350 MeVa = 10, 5, 1, -1, -5, -10 fm () P L 2  2 p cot  ´ = b = fm Bound States

Bound States vs Scattering States (3) L = 20 b 1/r = M  = 350 MeVa = 10, 5, 1, -1, -5, -10 fm () P L 2  2 p cot  ´ << ´ 0 ! mos t pro b a bl ya b oun d s t a t e ´ 0 ´ =

NN Resource Requirements with Current Algorithms NN Scattering Length fixed at 2 fm for demonstrative purposes Domain-Wall Propagator Generation ONLY !! Does not include time for lattice generation

Contractions  Usually  Lattice generation > propagator generation > contractions Not so for nuclear physics Need high statistics.. Many propagators per lattice Large number of quarks in initial and final states con t rac t i ons » u !d! s ! = ( A + Z ) ! ( 2 A ¡ Z ) ! S ! P ro t on: N con t : = U : N con t : =

nnn  Important for neutron rich nuclei  Lattice calc. is not as easy as   Cannot have all at rest as they are fermions

Closing Remarks on Scattering  Two hadron scattering can be studied with lattice QCD by studying the energy eigenstates at finite-volume  Simplest system well-understood  Baryon-baryon systems still very primitive.. a lot of room for improvement and contribution

The END