NUCLEAR POTENTIALS IN QCD AND THEIR EXTENSIONS Sinya Aoki, Yukawa - PowerPoint PPT Presentation

NUCLEAR POTENTIALS IN QCD AND THEIR EXTENSIONS Sinya Aoki, Yukawa Institute for Theoretical Physics, Kyoto University Miniworkshop on Lattice QCD Rm 716, CCMS/Physics Building, National Taiwan University, October 18-19, 2013

1. INTRODUCTION POTENTIALS IN QUANTUM FIELD THEORIES? matching between QFT and quantum mechanics ? � QM:scattering via “poptential” QFT scattering matrix element 1 � V + V H 0 − E V + + · · · + · · · Born approximation

More precisely, Scattering in QFT Scattering via “potential” in QM “Inverse scattering problem” A sumilar example virtual state mid-range attraction short-range repulsion NN scatering phase shifts(experiment) potentials which reproduce phase shift in QM

“Nuclear potential” which reproduces the calculate NN phase shift in QCD QCD phase shift Questions Is this “nuclear potential calculated in QCD” ? ambiguity of potentials ? (previous figure) Are potentials observables ? Potential input in QM V ( x ) can be defined V ( x ) same in CM (no “quantum corrections”) in classical mechanics cannot define potential in QM alone ? ? nuclear potential : no correspondence in CM V NN ( x ) “output” from QCD In this talk, I present our proposal for a definition of nuclear potentials in QCD and some results based on it. I also discuss extentions of our definition to more general cases.

Plan 1. Introduction: Potentials in quantum field theories ？ 2. Definition of nuclear potentials in (lattice) QCD (our proposal) 3. Results in lattice QCD 4. Extensions of “potentials” 5. Some applications 6. Conclusion

2. DEFINITION OF NUCLEAR POTENTIALS IN (LATTICE) QCD (OUR PROPOSAL) Aoki, Hatsuda & Ishii, PTP123(2010)89.

conditions satisfied for nuclear potentials reproduce NN scattering phase shift via calculations in QM important for applications to many body problems in nuclear physics calculable in QCD (without using QM calculations) Assumption 1 consider elastic scattering only NN → NN + others ( NN → NN + π , NN + ¯ NN → NN NN, · · · ) � k 2 + m 2 Energy inelastic threshold W k = 2 N < W th = 2 m N + m π Assumption 2 QCD interactions are short-ranged. （ no long-ranged force like Coulomb force. ）

Basic quantity: (Equal time) Nambu-Bethe-Salpeter (NBS) wave functions ϕ k ( r ) = � 0 | N ( x + r , 0) N ( x , 0) | NN, W k � QCD eigenstate QCD vacuum Nucleon operator N ( x ) = ε abc q a ( x ) q b ( x ) q c ( x ): local operator local quarks “a scheme” possible to use smeared quarks but unnecessary possible to use unequal time, but complicated and no new informations properties of NBS wave function. In particular, asymptotic behaviors in large |r|. It is very difficult or almost impossible to calculate NBS wave function “analytically”. However, it is possible to calculate it “numerically” in lattice QCD.

Lippman-Schwinger equation in QFT Ref. Weinberg’s textbook | β � 0 T βα � | α � in = | α � 0 + d β E α � E β + i ε in-state(full) ( H 0 + V ) | α � in = E α | α � in , free H 0 | α � 0 = E α | α � 0 . QCD Hamiltonian in terms of quarks and gluons (with gauge fixing). H 0 + V Free Hamiltonian. No explicit form but “operation” is known. H 0 V “potential” our target ? T βα = 0 � β | V | α � in off-shell T-matrix energy conservation energy T-matrix 0 � β | T | α � 0 = 2 πδ ( E α − E β ) T αβ . conservation (physical observables) off-shell on-shell T-matrix

For simplicity, consider scalars. Complicated for nucleons. NBS wave function center of mass frame Ψ k ( r ) = in � 0 | ϕ 1 ( x + r , 0) ϕ 2 ( x , 0) | k � in e i p · r T ( p ; k ) � d 3 p Z k Z k Ψ k ( r ) = e i k · r + E k − E p + i ε Z p plain wave � k 2 + m 2 � 0 � 0 | ϕ 1 ( x + r , 0) ϕ 2 ( x , 0) | k � 0 energy E k = 2 partial wave spherical harmonics spehrical Bessel e i k · r = 4 π � i l j l ( kr ) Y lm ( Ω r ) Y lm ( Ω k ) � i l Ψ l ( r, k ) Y lm ( Ω r ) Y lm ( Ω k ) Z k Ψ k ( r ) = 4 π lm lm solid angle � T ( p ; k ) = T l ( p, k ) Y lm ( Ω p ) Y lm ( Ω k ) l,m � ∞ p 2 dp j l ( pr ) T l ( p, k ) Z k Ψ l ( r, k ) = j l ( kr ) + 2 π E k − E p + i ε Z p 0 only a pole in T l ( p, k ) no pole on real axes(assumption) denominator contributes r → ∞

spherical Neumann � ∞ p 2 dp j l ( pr ) T l ( p, k ) Z k E k � E p + i ε � � kE k 4 T l ( k, k ) [ n l ( kr ) + ij l ( hr )] 2 π Z p 0 on-shell T-matrix Unitarity of S-matrix + elastic condition E k < E th T l ( k, k ) = − 4 e i δ l ( k ) sin δ l ( k ) kE k final result Ψ l ( r, k ) � e i δ l ( k ) [ j l ( kr ) cos δ l ( k ) + n l ( kr ) sin δ l ( k )] � e i δ l ( k ) sin( kr � l π / 2 + δ l ( k )) kr Lin et al., 2001; CP-PACS, 2004/2005; Ishizuka, 2009 Aoki, Hatsuda & Ishii, 2010 NBS wave function =scattering wave in QM at large |r|. “phase shift”= phase of S-matrix due to its unitarity phase of S-matrix =phase shift (in QFT) potentials from this NBS wave function ?

| r | → ∞ no interaction (under our assumption on short-ranged force) � e i δ l ( k ) no interaction sin( kr � l π / 2 + δ l ( k )) kr 2 particle energy in finite box with length L. � interaction range k 2 + m 2 E = 2 k = 2 π L n ( n ∈ Z 3 ) k is discrete due to PBC free (no interaction) L Luescher’s finite volume method Due to interactions, k is different from free case informations of interactions 2 ex. scalar, L=0 √ π LZ 00 (1; q 2 ) k cot δ 0 ( k ) = generalized zeta function k = | k | 1 � � k 2 + m 2 Z 00 ( s ; q 2 ) = ( n 2 − q 2 ) − s E = 2 q = kL √ 4 π integer for free case n ∈ Z 3 2 π Our proposal: extract informations of interactions directly at short distance (?)

Our proposal reduced mass µ = m N / 2 Step 1 define non-local but energy-independent potential as � d 3 y U ( x , y ) � k ( y ) [ � k − H 0 ] � k ( x ) = H 0 = −∇ 2 � k = k 2 non-relativistic kinetic energy non-relativistic free Hamiltonian 2 µ 2 µ “proof of existence” Using NBS wave functions below inelastic threshold, we can construct elastic W k ,W k � ≤ W th k , k � � † � [ � k − H 0 ] � k ( x ) � − 1 U ( x , y ) = k � ( y ) inner product k , k � η k , k � = ( ϕ k , ϕ k � ) inverse of Indeed, it is easy to see � � d 3 y U ( x , y ) � p ( y ) = [ � k − H 0 ] � k ( x ) � − 1 ∀ W p ≤ W th k , k � � k � , p = [ � p − H 0 ] � p ( x ) k , k � elastic

If we solve Schroedinger equation with this potential, we obtain NBS wave function as a solution, which gives correct the “phase shift” by construction. Finite size effect to the potential is small if the box is large enough. non-local potential which satisfies the Schroedinger equation is NOT unique. there exist many potentials different only for inelastic scatterings. No non-relativistic approximation is used. Equal NBS WF has no time. time dependence can be derived from Lorentz transformation, but no new information on scattering. Schroedinger equation is convenient for nuclear physics calculations. In practice, it is almost impossible to calculate non-local potential from the above construction. We will introduce some “approximation”.

Step 2 velocity(derivative) expansion U ( x , y ) = V ( x , r ) δ 3 ( x � y ) V ( x , ∇ ) = V 0 ( r ) + V σ ( r )( σ 1 · σ 2 ) + V T ( r ) S 12 + V LS ( r ) L · S + O ( ∇ 2 ) NLO NNLO LO LO LO spin tensor operator S 12 = 3 r 2 ( σ 1 · x )( σ 2 · x ) − ( σ 1 · σ 2 ) To calcilate non-local potentil in terms of this expansion is a part of “our scheme”. V LO ( x ) = [ � k − H 0 ] � k ( x ) Ex ： Leading order � k ( x ) Step 3 obtain binding energies and scattering phase shifts, by solving the Schroedinger equation in infinite volume with the potential obtained with this expansion. Physical observables have some errors due to the truncation of the expansion. It is possible to estimate these errors.

Comments “potentials” are NOT physical observables, and therefore depend on their definition (scheme). Phase shifts and binding energies do NOT depend on the scheme. analogy:running coupling in QCD. Scheme dependent. Cross section is an observable. choice of nucleon operator 、 definition of non-local potential, derivative expansion “good” scheme ? fast convergence of the derivative expansion （ convenient ）。 analogy: fast convergence of perturbative expansion (“good” running coupling). it is best if local potential is exact. energy-independent “local” potential ？（ ”inverse scattering problem” ） Potential is useful to understand “physics”, though it is not observable. analogy:asymptotic freedom. attractions at long and intermediate distances, repulsive core Our proposal: give a method to extract observables in QCD via potential.

3. RESULTS IN LATTICE QCD