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QUARKS, GLUONS AND NUCLEAR FORCES Paulo Bedaque University of Maryland, College Park strong nuclear force: binds neutrons and protons into nuclei Quantum Chromodynamics (QCD) What do we know ? 1) NN phase shifts 1S 0 neutron-proton What do we know ? 2) Several potentials that fit them pion exchange all kinds of things … What do we know ? 3) These potentials explain a lot but not everything • NNn, NNg, couplings • NNN forces • NY forces • ... few % on nd ~5% of nuclei binding strangeness in neutron stars Can we understand the nuclear forces (and NNN, NNn, …) from first principles ? LATTICE QCD PATH INTEGRALS iS1 e Probability | eiS1 eiS1 e iS2 |2 Quantum mechanics reduced to quadratures x(t ) x(0) operators iS [ x(t )] x(t ) x(0) Dx ( t ) e iS [ x(t )] Dx ( t ) e numbers iS [ x ( t )] Dx ( t ) e is as well (or ill) defined as dx ei x Imaginary time (t it): just like stat mech probability distribution x(t ) x(0) Dx(t ) Z1 e S [ x (t )] x(t ) x(0) 1 N N x (t ) x (0) i 1 i i But I don’t live in imaginary time ! What can I do with imaginary time correlators ? 0 | x(t ) x(0) | 0 0 | e Ht x(0)e Ht x(0) | 0 0 | x | n e( E E )t n | x | 0 n 0 n e( E E0 )t | 0 | x | 1 |2 t 1 lowest energy state w/ some overlap Typical paths xi (t ) xi (0) N 1 x (t ) x (0) i N i 1 i PATH INTEGRALS FOR FIELDS iS1 e iS1 e Quantum Chromodynamics Q = spinor, 3 colors, 6 flavors = quarks U = SU(3) matrix = gluons QCD reduced to quadratures qg 5q( x ) qg 5q(0) 1 Z 1 Z SG [U ]q ( DU m ) q DUDqDq e qg 5q( x ) qg 5q(0) SG [U ] DU e det( DU m) tr[ 1 g 1 g ] 5 DU m DU m 5 probability distribution for Ui qg 5q( x ) qg 5q(0) 1 Z 1 N SG [U ] DU e det( DU m) tr[ N tr[ i 1 1 1 g5 g 5] DU m DU m i i algorithm 1. find {Ui} 2. compute 1/(DUi+m) 3. compute observable 1 g 1 g ] 5 DU m DU m 5 Scattering through finite volumes: the Luscher method (Marinari, Hamber, Parisi, Rebbi) one particle Periodic boundary conditions: box is a torus Energy levels at En 2 n m 2 L 2 Scattering through finite volumes: the Luscher method (Marinari, Hamber, Parisi, Rebbi) two particles 1 M EL2 M E cot (E ) S L 4 2 known function Learn about the deuteron in boxes smaller than the deuteron 0 | N (t , k ) N (t , k ) N † (0, k ) N † (0, k ) | 0 0 | N (0, k ) N (0, k )e Ht | n n | N † (0, k ) N † (0, k ) | 0 n e t E2 N t | NN at rest | N † (0, k ) N † (0, k ) | 0 |2 The difference between E2N and EN is our signal phase shift The time to try it is now • Pion masses small enough for chiral extrapolation • No quenching • Volumes ~ (3 fm)3 • Improved actions • Good chiral symmetry • Software resources S. Beane, T. Luu, K. Orginos, E. Pallante, A. Parreno, M. Savage, A. Walker-Loud, … Gold platted scattering observable: I=2 pp K(e4) CP-PACS m2 m a2 8 f2 3m2 m2 1 log l ( ) 2 2 2 16 f Improved statistics K(e4) CP-PACS m2 m a2 8 f2 3m2 m2 1 log l ( ) 2 2 2 16 f Nucleon-nucleon Nucleon-nucleon “natural” |a| < 1 fm for 350 < m < 600 MeV a=5.4 fm or 20 fm for m=138 MeV is indeed fine tuned Chiral “extrapolation” • no anchor at mp= 0 • wild behavior of the scattering length with mq The crucial problem is the large statistical errors signal: C (t ) q6 (t ) q 6 (0) e 2 Mt 2 baryons error: 2 (t ) q6 (t )q 6 (t ) q6 (0)q 6 (0) e 6 m t 6 pions signal noise 1 (2 M N 3m )t e N signal noise 1 (2 M N 3m )t e N If the minimum pion energy was larger m, the signal would be better (-z) = -(z) ? Parity orbifold (P.B. +Walker-Loud) parity reversed ( z) ( z) minimum pion energy is E L 2 m2 Parity orbifold: pinhole these points are related by parity ( x, y, z) ( x, y, z) minimum pion energy is E 3 L 2 m2 ? Summary • Lattice QCD calculation of hadron interactions are doable • Meson-meson scattering can be computed with few % precision • There is a serious noise problem in baryonbaryon channels, new ideas are needed • New ideas exist ! We’ll find out how they work really soon weighted fit: lpp = 3.3(6)(3) different weigths mp a2 = -0.0426 (6)(3)(18) lpp 1-loop – 2-loop w/o counterterm K(e4): mp a2 = -0.0454(31)(10)(8) theoretical cPT predicts discretization errors (a2) ~ 1% (D. O’Connel, A. Walker-Loud, R. V. Water, J. Chen) Finite volume (e-mpL) ~ 1% (P.B. & I. Sato) Extracting physics from euclidean space : energies are "easy" 0 | (t , k 0) † (0, k 0) | 0 e Ht 0 | (0, 0)| n n | † (0, 0) | 0 n e t some operator with quantum numbers of the pion, made of quarks and gluons, for instance: q (0, p)g 5 a q(0, p) m t 0 | (0, 0)| | † (0, 0) | 0 lowest energy state with the quantum numbers of the pion Solution 2: Aharonov-Bohm effect add a background magnetic potential coupled to baryon number with zero curl q ( L) q (0) A 3L L zˆ q( L) ei / 3q(0) A0 zˆ N ( L) N (0) A or or N ( L) ei N (0) A0 no coupling to local operators !