S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials J. A. Oller Departamento de F´ ısica Universidad de Murcia 1 in collaboration with D. R. Entem [I] arXiv:1609, next month [II] Long version, in preparation 2nd HSND, Madrid, September 8, 2016 1 Partially funded by MINECO (Spain) and EU, project FPA2013-40483-P
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Overview Lippmann-Schwinger equation 1 New exact equation in NR scattering theory 2 LS equation in the complex plane 3 N/D method with non-perturbative ∆( A ) 4 Regular interactions 5 Singular Interactions 6 T ( A ) in the complex plane 7 Conclusions 8
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation Lippmann-Schwinger equation (LS) Scattering T -matrix T ( z ) , Im ( z ) � = 0 , Two-body scattering T ( z ) = V − V R 0 ( z ) T ( z ) R 0 ( z ) = [ H 0 − z ] − 1 H 0 = − 1 2 µ ∇ 2 H = H 0 + V • Resolvent of H , R ( z ) : R ( z ) =[ H − z ] − 1 R ( z ) = R 0 ( z ) − R 0 ( z ) T ( z ) R 0 ( z )
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation • Spectrum of H : H | ψ p � = E p | ψ p � Continuous spectrum: Povzner’s result | ψ p � = | p � − lim ǫ → 0 + R 0 ( E p + iǫ ) T ( E p + iǫ ) | p � Bound States: Poles in T ( z ) for z ∈ R − • LS in momentum space For definiteness we consider uncoupled spinless case by now: d 3 q 1 � T ( p ′ , p , z ) = V ( p ′ , p ) − (2 π ) 3 V ( p ′ , q ) T ( q , p , z ) q 2 2 µ − z p’ + . . . p POTENTIAL
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation • LS in partial waves � +1 T ℓ ( p ′ , p, z ) =1 d cos θ P ℓ (cos θ ) T ( p ′ , p , z ) 2 − 1 cos θ = ˆ p ′ · ˆ p ∞ � T ( p ′ , p , z ) = (2 ℓ + 1) P ℓ (cos θ ) T ℓ ( p ′ , p, z ) ℓ =0 � ∞ dqq 2 V ℓ ( p ′ , q ) T ℓ ( q, p, z ) T ℓ ( p ′ , p, z ) = V ℓ ( p ′ , p ) + µ q 2 − 2 µz π 2 0 Convention: V ( p ′ , p ) → − V ( p ′ , p ) , T ( p ′ , p , z ) → − T ( p ′ , p , z )
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation • On-shell unitarity (extensively used later) Propagation of real two-body states p ′ = p E p = p 2 2 µ Im T ℓ ( p 2 ) = µp 2 π | T ℓ ( p 2 ) | 2 , p > 0 T ℓ ( p 2 ) = − µp 1 Im 2 π Unitarity cut for p 2 > 0
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation Criterion for Singular Potentials r → 0 αr − γ Potential Ordinary Singular V ( r ) − − − → γ < 2 > 2 γ = 2 α > 0 ¯ α ≤ 0 ¯ α = α + ℓ ( ℓ + 1) ¯ Ordinary/Regular Potentials: Standard quantum mechanical treatment Boundary condition: u (0) = 0 and behavior at ∞ No extra free parameters
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation The One-Pion-Exchange (OPE) potential for the singlet NN interaction ( r > 0) : � 2 e − m π r � g A m π V ( r ) = − τ 1 · τ 2 Yukawa potential 2 f π 4 πr Exchange of a pion between two nucleons π r 1 = r 1 ′
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation In many instances one has singular potentials Multipole expansion ρ ( x ′ ) q ℓm Y ℓm ( θ, φ ) � | x − x ′ | d 3 x ′ = 4 π � Φ( x ) = r ℓ +1 2 ℓ + 1 ℓ,m � ℓm ( θ ′ , φ ′ ) r ′ ℓ ρ ( x ′ ) d 3 x ′ Y ∗ q ℓm =
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation Van der Waals Force (molecular physics) V ( r ) = − 3 α A α B I A I B r 6 2 I A + I B In QFT/EFT we treat composite objects as point like Physical meaning for r → 0 ?: De Broglie length 1 p >> r A ∼ 2 ˚ A
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation Nuclear physics OPE is singular attractive for the Deuteron in NN scattering 2 S +1 L J : 3 P 0 NN partial wave � g A � 2 V ( r ) = m 2 π [ − 4 T ( r ) + Y ( r )] 12 π 2 f π Y ( r ) = e − m π r r T ( r ) = e − m π r � 3 3 � 1 + m π r + r ( m π r ) 2 g 2 π r − 3 Triplet part of the OPE potential V ( r ) − − − → A 4 πf 2 r → 0 Quark Models, pNRQCD, pNRQED, QCD’s EFTs, etc
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation Math: Taking r → 0 for a singular potential • Full range r ∈ ]0 , ∞ [ : Case, PR60,797(1950) • Singular Attractive Potential E Near the origin the solution is the superposition of two oscillatory wave functions One has to fix a relative phase, ϕ ( p ) How to do it?. Mess. r Which are the appropriate boundary conditions? (Orthogonality of wave functions with different energy → dϕ ( p ) /dp = 0 ) Case, PR60,797(’50); Arriola, Pav´ on, PRC72,054002(’05)
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation The potential does not determine uniquely the scattering problem Plesset, PR41,278(1932), Case, PR60,797(1950) • Singular Repulsive Potential E There is only one finite (vanishing) reduced wave function at r = 0 The solution is fixed Pav´ on Valderrama, Ruiz Arriola, r Ann.Phys.323,1037(’08) Typically, the phenomenology is not accurate E.g. this scheme a la Case does not fit well NN phase shifts.
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation Two points of view in NN scattering: • Use a finite cutoff Λ fitted to data. Regularization dependence It works phenomenologically Entem,Machleidt, PRC68,041001(R)(’03); Epelbaum,Gloeckle,Meißner, NPA747,362(’05); Epelbaum,Krebs,Meißner, PRL115,122301(’15) • Take r → 0 ( Λ → ∞ ) (Renormalized solutions) Energy-independent boundary condition Arriola, Valderrama, PLB580,149(’04); PRC74,054002(’05); Case, PR60,797(1950) Subtractive renormalization Frederico,Timoteo,Tomio, NPA653,209(’99); Yang,Elster,Phillips, PRC80,044002(’09) Include one/zero counterterm Entem et al. ,PRC77,044006(’08) These three-methods are equivalent. Not phenomenologically successful.
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials Lippmann-Schwinger equation Low-energy EFT paradigm: Contact terms are necessary to reproduce short-distance physics They are allowed by symmetry They are required to make loops finite. Nonrenormalizable QFT/EFT They are expected to be relatively important because of power-counting
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials New exact equation in NR scattering theory New exact equation in NR scattering theory Yukawa potential, 2 g V ( q ) = q 2 + m 2 π Singularity for q 2 = − m 2 π √ 1 S 0 potential: ( 2 S +1 L J ) 8 f π ) 2 g = ( g A m π / g 2 p 2 log(4 p 2 /m 2 V ( p ) = π + 1) Left-hand cut (LHC) discontinuity for On-shell scattering p 2 < − m 2 π / 4 = L Born approximation ∆ 1 π ( p 2 ) = V ( p 2 + i 0 + ) − V ( p 2 − i 0 + ) = Im V ( p 2 + i 0 + ) = gπ 2 p 2 2 i
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials New exact equation in NR scattering theory Full LHC Discontinuity , p 2 = − k 2 < L 2 i ∆( p 2 ) = T ( p 2 + i 0 + ) − T ( p 2 − i 0 + ) ∆( p 2 ) =Im T ( p 2 + i 0 + ) . . . . The LS generates contributions with any number of pions to ∆( p 2 ) , p 2 < L ∆ nπ ( p 2 ) for p 2 < − ( nm 2 π / 2)
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials New exact equation in NR scattering theory How to calculate ∆( A ) ? G.E. Brown, A.D. Jackson “The Nucleon-Nucleon interaction”, North-Holland, 1976. Page 86: “In practice, of course, we do not know the exact form of ∆( p 2 ) for a given potential . . . ” p = ik ± ε , ε = 0 + , p 2 = − k 2 < L T ( ik ± ε, ik ± ε ) = V ( ik ± ε, ik ± ε ) � ∞ + µ dqq 2 V ( ik ± ε, q ) T ( q, ik ± ε ) q 2 + k 2 2 π 2 0 T ( ik ± ε, ik ± ε ) , so calculated, IS PURELY REAL!! You can try to calculate numerically just the once iterated OPE � ∞ µ dqq 2 V ( ik ± ε, q ) V ( q, ik ± ε ) ∈ R q 2 + k 2 2 π 2 0
S -matrix solution of the Lippmann-Schwinger equation for regular and singular potentials New exact equation in NR scattering theory Notation: A = p 2 This is an example of: Not all what you can calculate with a computer is the right answer!!
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