R ENORMALIZATION OF S PIN -F LAVOR V AN DER W AALS F ORCES ´ Alvaro Calle Cord´ on Theory Center @ JLab Hadron 2011 Munich, June 17, 2011 In collaboration with Enrique Ruiz Arriola (University of Granada) ´ A LVARO C ALLE C ORD ´ ON (JL AB ) R ENORMALIZATION OF SF V D W F ORCES M UNICH , J UNE 17, 2011 1 / 26
T ABLE OF CONTENTS T ABLE OF CONTENTS 1 I NTRODUCTION 2 S PIN - FLAVOR VAN DER W AALS FORCES 3 P HENOMENOLOGY 4 C ONCLUSIONS AND OUTLOOK ´ A LVARO C ALLE C ORD ´ ON (JL AB ) R ENORMALIZATION OF SF V D W F ORCES M UNICH , J UNE 17, 2011 2 / 26
I NTRODUCTION T ABLE OF CONTENTS 1 I NTRODUCTION 2 S PIN - FLAVOR VAN DER W AALS FORCES 3 P HENOMENOLOGY 4 C ONCLUSIONS AND OUTLOOK ´ A LVARO C ALLE C ORD ´ ON (JL AB ) R ENORMALIZATION OF SF V D W F ORCES M UNICH , J UNE 17, 2011 3 / 26
I NTRODUCTION N UCLEAR POTENTIALS AND SINGULARITIES OPE potential: Yukawa’s meson theory (1935) & Proca, Kemmer, ... (1940’s) g 2 m 3 1 π NN V OPE ( r ) � Y ( m π r ) σ 1 · σ 2 + T ( m π r ) S 12 (ˆ r ) � π = τ 1 · τ 2 M 2 12 4 π N g 2 1 1 π NN r 3 S 12 (ˆ r ) τ 1 · τ 2 → (singular potential) M 2 16 π N OBE models (1960’s): multipions = σ, ρ, ω , ... 1 / r 3 S 12 (ˆ r ) pseudo-scalar mesons ( π, η ) 1 / r 3 L · S scalar mesons ( σ, δ ) 1 / r 3 L · S , 1 / r 3 S 12 (ˆ r ) vector mesons ( ω, ρ ) High quality NN potentials (Nijmegen, Bonn, Paris, ...) (1970’s) OBE + realistic couplings + strong form factors to remove divergences Quark models (1980’s): OGE + confinement → short-range repulsion (form factors) Hybrid models (meson exchanges) → account for medium and long-range attraction EFTs in nuclear physics (1990’s): Chiral NN potential 1 1 1 V LO → ± r 3 , V NLO → ± r 5 , V NNLO , V NLO- ∆ , V NNLO- ∆ → ± r 6 In general, nuclear potentials present singularities ´ A LVARO C ALLE C ORD ´ ON (JL AB ) R ENORMALIZATION OF SF V D W F ORCES M UNICH , J UNE 17, 2011 4 / 26
I NTRODUCTION N UCLEAR POTENTIALS AND SINGULARITIES OPE potential: Yukawa’s meson theory (1935) & Proca, Kemmer, ... (1940’s) g 2 m 3 1 π NN � � V OPE ( r ) Y ( m π r ) σ 1 · σ 2 + T ( m π r ) S 12 (ˆ r ) π = τ 1 · τ 2 M 2 12 4 π N g 2 1 1 π NN r 3 S 12 (ˆ r ) τ 1 · τ 2 → (singular potential) M 2 16 π N OBE models (1960’s): multipions = σ, ρ, ω , ... 1 / r 3 S 12 (ˆ r ) pseudo-scalar mesons ( π, η ) 1 / r 3 L · S scalar mesons ( σ, δ ) 1 / r 3 L · S , 1 / r 3 S 12 (ˆ r ) vector mesons ( ω, ρ ) High quality NN potentials (Nijmegen, Bonn, Paris, ...) (1970’s) OBE + realistic couplings + strong form factors to remove divergences N N N N m m N N N N ´ A LVARO C ALLE C ORD ´ ON (JL AB ) R ENORMALIZATION OF SF V D W F ORCES M UNICH , J UNE 17, 2011 4 / 26
I NTRODUCTION N UCLEAR POTENTIALS AND SINGULARITIES OPE potential: Yukawa’s meson theory (1935) & Proca, Kemmer, ... (1940’s) g 2 m 3 1 π NN V OPE ( r ) � Y ( m π r ) σ 1 · σ 2 + T ( m π r ) S 12 (ˆ r ) � π = τ 1 · τ 2 M 2 12 4 π N g 2 1 1 π NN r 3 S 12 (ˆ r ) τ 1 · τ 2 → (singular potential) M 2 16 π N OBE models (1960’s): multipions = σ, ρ, ω , ... 1 / r 3 S 12 (ˆ r ) pseudo-scalar mesons ( π, η ) 1 / r 3 L · S scalar mesons ( σ, δ ) 1 / r 3 L · S , 1 / r 3 S 12 (ˆ r ) vector mesons ( ω, ρ ) High quality NN potentials (Nijmegen, Bonn, Paris, ...) (1970’s) OBE + realistic couplings + strong form factors to remove divergences Quark models (1980’s): OGE + confinement → short-range repulsion (form factors) Hybrid models (meson exchanges) → account for medium and long-range attraction EFTs in nuclear physics (1990’s): Chiral NN potential 1 1 1 V LO → ± r 3 , V NLO → ± r 5 , V NNLO , V NLO- ∆ , V NNLO- ∆ → ± r 6 In general, nuclear potentials present singularities ´ A LVARO C ALLE C ORD ´ ON (JL AB ) R ENORMALIZATION OF SF V D W F ORCES M UNICH , J UNE 17, 2011 4 / 26
I NTRODUCTION N UCLEAR POTENTIALS AND SINGULARITIES OPE potential: Yukawa’s meson theory (1935) & Proca, Kemmer, ... (1940’s) g 2 m 3 1 π NN V OPE ( r ) � Y ( m π r ) σ 1 · σ 2 + T ( m π r ) S 12 (ˆ r ) � π = τ 1 · τ 2 M 2 12 4 π N g 2 1 1 π NN r 3 S 12 (ˆ r ) τ 1 · τ 2 → (singular potential) M 2 16 π N OBE models (1960’s): multipions = σ, ρ, ω , ... 1 / r 3 S 12 (ˆ r ) pseudo-scalar mesons ( π, η ) 1 / r 3 L · S scalar mesons ( σ, δ ) 1 / r 3 L · S , 1 / r 3 S 12 (ˆ r ) vector mesons ( ω, ρ ) High quality NN potentials (Nijmegen, Bonn, Paris, ...) (1970’s) OBE + realistic couplings + strong form factors to remove divergences Quark models (1980’s): OGE + confinement → short-range repulsion (form factors) Hybrid models (meson exchanges) → account for medium and long-range attraction EFTs in nuclear physics (1990’s): Chiral NN potential 1 1 1 V LO → ± r 3 , V NLO → ± r 5 , V NNLO , V NLO- ∆ , V NNLO- ∆ → ± r 6 In general, nuclear potentials present singularities ´ A LVARO C ALLE C ORD ´ ON (JL AB ) R ENORMALIZATION OF SF V D W F ORCES M UNICH , J UNE 17, 2011 4 / 26
I NTRODUCTION S INGULAR POTENTIALS AT SHORT DISTANCES By definition a singular potential satisfies lim r → 0 r 2 | U ( r ) | > ∞ with U ( r ) = 2 µ V ( r ) � n � R n Two-body scattering problem (for S-wave) with U ( r ) = ± 1 and n ≥ 2, r R 2 n − u ′′ ( r ) + U ( r ) u ( r ) = k 2 u ( r ) At short distances ( r → 0) the WKB approximation is applicable λ ′ ( r ) = d | p ( r ) | = d � n 2 − n R n 2 1 1 � r ≪ ≪ 1 ⇒ dr dr � k 2 − U ( r ) 2 � � 6 R 6 e.g. for a vdW potential U ( r ) = − 1 the applicability condition reads r � R 6 . r R 2 6 Semiclassical (WKB) short-distance wave functions Orthogonality condition between different energy states: � ∞ − � u ′ 0 ( r ) � ∞ k ( r ) u 0 ( r ) − u k ( r ) u ′ = k 2 u k ( r ) u 0 ( r ) dr = sin ( ϕ k − ϕ 0 ) = 0 0 0 imply the short-distance phase to be common to all eigenfunctions. ´ A LVARO C ALLE C ORD ´ ON (JL AB ) R ENORMALIZATION OF SF V D W F ORCES M UNICH , J UNE 17, 2011 5 / 26
I NTRODUCTION S INGULAR POTENTIALS AT SHORT DISTANCES By definition a singular potential satisfies lim r → 0 r 2 | U ( r ) | > ∞ with U ( r ) = 2 µ V ( r ) � n � R n Two-body scattering problem (for S-wave) with U ( r ) = ± 1 and n ≥ 2, r R 2 n − u ′′ ( r ) + U ( r ) u ( r ) = k 2 u ( r ) Semiclassical (WKB) short-distance wave functions �� r � C � k 2 − U ( r ′ ) d r ′ + ϕ k U ( r ) < k 2 u WKB ( r ) = , sin k � k 2 − U ( r ) 4 r 0 � r � � C � U ( r ) > k 2 u WKB ( r ) = U ( r ′ ) − k 2 d r ′ , U ( r ) − k 2 exp − k � 4 r 0 with ϕ k a short-distances phase which may depend on the energy. Orthogonality condition between different energy states: � ∞ � u ′ k ( r ) u 0 ( r ) − u k ( r ) u ′ 0 ( r ) � ∞ = k 2 u k ( r ) u 0 ( r ) dr = sin ( ϕ k − ϕ 0 ) = 0 − 0 0 imply the short-distance phase to be common to all eigenfunctions. ´ A LVARO C ALLE C ORD ´ ON (JL AB ) R ENORMALIZATION OF SF V D W F ORCES M UNICH , J UNE 17, 2011 5 / 26
I NTRODUCTION S INGULAR POTENTIALS AT SHORT DISTANCES By definition a singular potential satisfies lim r → 0 r 2 | U ( r ) | > ∞ with U ( r ) = 2 µ V ( r ) � n � R n Two-body scattering problem (for S-wave) with U ( r ) = ± 1 and n ≥ 2, r R 2 n − u ′′ ( r ) + U ( r ) u ( r ) = k 2 u ( r ) Semiclassical (WKB) short-distance wave functions � R n � r � R n � n � � � n � n / 4 2 − 1 U ( r ) → − 1 2 u k ( r ) → C , sin − + ϕ k R 2 r R n n − 2 r n � R n � r � n � 2 − 1 � � n � n / 4 � R n U ( r ) → + 1 2 u k ( r ) → C , exp − R 2 r R n n − 2 r n with ϕ k a short-distances phase which may depend on the energy. Orthogonality condition between different energy states: � ∞ − � u ′ k ( r ) u 0 ( r ) − u k ( r ) u ′ 0 ( r ) � ∞ = k 2 u k ( r ) u 0 ( r ) dr = sin ( ϕ k − ϕ 0 ) = 0 0 0 imply the short-distance phase to be common to all eigenfunctions. ´ A LVARO C ALLE C ORD ´ ON (JL AB ) R ENORMALIZATION OF SF V D W F ORCES M UNICH , J UNE 17, 2011 5 / 26
I NTRODUCTION S INGULAR POTENTIALS AT SHORT DISTANCES By definition a singular potential satisfies lim r → 0 r 2 | U ( r ) | > ∞ with U ( r ) = 2 µ V ( r ) � n � R n Two-body scattering problem (for S-wave) with U ( r ) = ± 1 and n ≥ 2, r R 2 n − u ′′ ( r ) + U ( r ) u ( r ) = k 2 u ( r ) Semiclassical (WKB) short-distance wave functions � R n � r � R n � n � � � n � n / 4 2 − 1 U ( r ) → − 1 2 u k ( r ) → C , sin − + ϕ k R 2 r R n n − 2 r n � R n � r � n � 2 − 1 � � n � n / 4 � R n U ( r ) → + 1 2 u k ( r ) → C , exp − R 2 r R n n − 2 r n with ϕ k a short-distances phase which may depend on the energy. Orthogonality condition between different energy states: � ∞ − � u ′ k ( r ) u 0 ( r ) − u k ( r ) u ′ 0 ( r ) � ∞ = k 2 u k ( r ) u 0 ( r ) dr = sin ( ϕ k − ϕ 0 ) = 0 0 0 imply the short-distance phase to be common to all eigenfunctions. ´ A LVARO C ALLE C ORD ´ ON (JL AB ) R ENORMALIZATION OF SF V D W F ORCES M UNICH , J UNE 17, 2011 5 / 26
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