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Hyperon-Nucleon Interaction from Chiral EFT

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Hyperon-Nucleon Interaction from Chiral EFT Hyperon-nucleon interaction from chiral EFT Johann Haidenbauer IAS & JCHP, Forschungszentrum Jülich, Germany Hypernuclear Workshop, Newport News, May 28, 2014 Johann Haidenbauer Hyperon-nucleon interaction Outline 1 Introduction 2 YN in chiral effective field theory 3 YN Results 4 Three- and four-body systems Johann Haidenbauer Hyperon-nucleon interaction NN in chiral effective field theory 0 1 10 3 60 1 P P S -10 1 0 0 0 -20 40 -10 Phase Shift [deg] -30 Phase Shift [deg] -20 -40 20 0 50 100 150 200 250 0 50 100 150 200 250 Lab. Energy [MeV] Lab. Energy [MeV] 0 Phase Shift [deg] 50 -5 3 3 0 P P -10 1 40 2 -15 30 -20 -20 -25 20 Phase Shift [deg] Phase Shift [deg] -30 10 -35 150 0 3 0 50 100 150 200 250 0 50 100 150 200 250 S1 Lab. Energy [MeV] Lab. Energy [MeV] 5 4 ε 100 1 3 2 1 0 Phase Shift [deg] 50 -1 Phase Shift [deg] -2 0 50 100 150 200 250 Lab. Energy [MeV] 0 0 50 100 150 200 250 Lab. Energy [MeV] NLO, N2LO, N3LO (E. Epelbaum, W. Glöckle, Ulf-G. Meißner, NPA747 (2005) 362) Johann Haidenbauer Hyperon-nucleon interaction YN in chiral effective field theory Advantages: Power counting systematic improvement by going to higher order Possibility to derive two- and three baryon forces and external current operators in a consistent way Obstacle: YN data base is rather poor about 35 data points, all from the 1960s 10 data points from the KEK-PS E251 collaboration (1999-2005) constraints from hypernuclei no polarization data ) no phase shift analysis ! impose SU(3)f constraints We∗ follow the scheme of S. Weinberg (1990) in complete analogy to the study of NN in χEFT by E. Epelbaum et al. ∗J.H., N. Kaiser, S. Petschauer, U.-G. Meißner, A. Nogga, W. Weise Johann Haidenbauer Hyperon-nucleon interaction Power counting X ν Veff ≡ Veff(Q; g; µ) = (Q=Λ) Vν (Q/µ, g) ν Q ... soft scale (baryon three-momentum, Goldstone boson four-momentum, Goldstone boson mass) Λ ... hard scale g ... pertinent low–energy constants µ ... regularization scale Vν ... function of order one ν ≥ 0 ... chiral power Leading order (LO): ν = 0 a) non-derivative four-baryon contact terms b) one-meson (Goldstone boson) exchange diagrams Next-to-leading order (NLO): ν = 2 a) four-baryon contact terms with two derivatives b) two-meson (Goldstone boson) exchange diagrams Johann Haidenbauer Hyperon-nucleon interaction Contact terms for BB e.g.,LO contact terms for BB: ¯ ¯ 1 ~ 1 ¯ ¯ = Ci NΓi N NΓi N = C BaBb (Γi B) (Γi B) ; L )L i b a 2 ~ 2 ¯ ¯ = Ci Ba (Γi B) Bb (Γi B) ; L a b 3 ~ 3 ¯ ¯ = Ci Ba (Γi B) Bb (Γi B ) L a b 0 pΣ + pΛ Σ+ p 2 6 0 B = Σ− −pΣ + pΛ n 0 2 6 1 Ξ− Ξ0 p2Λ B − − 6 C @ A a, b ... Dirac indices of the particles µ µν µ Γ1 = 1 ; Γ2 = γ ; Γ3 = σ ; Γ4 = γ γ5 ; Γ5 = γ5 ~ Ci , Ci ... low-energy constants Johann Haidenbauer Hyperon-nucleon interaction Contact terms for BB spin-momentum structure of the contact term potential: BB contact terms without derivatives(LO): (0) V = C ; ! + CT ;BB!BB ~σ1 ~σ2 BB!BB S BB BB · BB contact terms with two derivatives(NLO): (2) 2 2 2 2 V = C1~q + C2~k + (C3~q + C4~k )(~σ1 ~σ2) BB!BB · i + C5(~σ1 + ~σ2) (~q ~k) + C6(~q ~σ1)(~q ~σ2) 2 · × · · i + C7(~k ~σ1)(~k ~σ2) + C8(~σ1 ~σ2) (~q ~k) · · 2 − · × note: Ci ! Ci;BB!BB ~q = ~p0 − ~p; ~k = (~p0 + ~p)=2 Johann Haidenbauer Hyperon-nucleon interaction SU(3) symmetry 10 independent spin-isospin channels in NN and YN (for L=0) (NN (I=0), NN (I=1), ΛN, ΣN (I=1/2), ΣN (I=3/2), ΛN ΣN) $ in principle (atLO), 10 low-energy constants ) SU(3) symmetry only5 independent low-energy constants ) SU(3) structure for scattering of two octet baryons: direct product: ∗ 8 8 = 1 8a 8s 10 10 27 ⊗ ⊕ ⊕ ⊕ ⊕ ⊕ CS;i , CT ;i , C1;i , etc., can be expressed by the constants corresponding to the SU(3)f irreducible representations: ∗ C1, C8a , C8s , C10 , C10, C27 Johann Haidenbauer Hyperon-nucleon interaction SU(3) structure of contact terms for BB Channel I V1S0; V3P0;3P1;3P2 V3S1; V3S1−3D1; V1P1 V1P1−3P1 ∗ S = 0 NN ! NN 0 – C10 – NN ! NN 1 C27 – – “ ∗ ” S = −1 ΛN ! ΛN 1 1 `9C27 + C8s ´ 1 C8a + C10 p−1 C8s8a 2 10 2 20 “ ∗ ” ΛN ! ΣN 1 3 `−C27 + C8s ´ 1 −C8a + C10 p3 C8s8a 2 10 2 20 p−1 C8s8a 20 “ ∗ ” ΣN ! ΣN 1 1 `C27 + 9C8s ´ 1 C8a + C10 p3 C8s8a 2 10 2 20 3 27 10 ΣN ! ΣN 2 C C – Number of contact terms: NN: 2 (LO) 7 (NLO) YN : +3 (LO) +11 (NLO) YY : +1 (LO) + 4 (NLO) C1 contributes only to I = 0, S = 2 channels!! ) − Johann Haidenbauer Hyperon-nucleon interaction Pseudoscalar-meson exchange SU(3)f -invariant pseudoscalar-meson-baryon interaction Lagrangian: fi fl gA(1 − α) ¯ µ gAα ¯ µ L = − p Bγ γ5 f@µP; Bg + p Bγ γ5 [@µP; B] 2Fπ 2Fπ f = gA=(2Fπ); gA ' 1:26, Fπ ≈ 93 MeV α = F=(F + D) with gA = F + D 0 0 η 1 pπ + p π+ K + 2 6 B 0 C P = B π− −pπ + pη K 0 C 2 6 @ 0 A K − K¯ − p2η 6 f = f f = p1 (4α − 1)f f = − p1 (1 + 2α)f NNπ NNη8 3 ΛNK 3 f = −(1 − 2α)f f = − p1 (1 + 2α)f f = p1 (4α − 1)f ΞΞπ ΞΞη8 3 ΞΛK 3 f = p2 (1 − α)f f = p2 (1 − α)f f = (1 − 2α)f ΛΣπ 3 ΣΣη8 3 ΣNK f = 2αf f = − p2 (1 − α)f f = −f ΣΣπ ΛΛη8 3 ΞΣK Johann Haidenbauer Hyperon-nucleon interaction Pseudoscalar-meson (boson) exchange One-pseudoscalar-meson exchange (V OBE )[LO] OBE ~σ1 ~q ~σ2 ~q 0 0 = 0 0 VB B !B B fB1B P fB2B P · · 1 2 1 2 − 1 2 ~ 2 + 2 q mP f 0 ... coupling constants B1B1P mP ... mass of the exchanged pseudoscalar meson dynamical breaking of SU(3) symmetry due to the mass splitting of the ps mesons (mπ = 138.0 MeV, mK = 495.7 MeV, mη = 547.3 MeV) taken into account already atLO! Details: (H. Polinder, J.H., U.-G. Meißner, NPA 779 (2006) 244; PLB 653 (2007) 29) Johann Haidenbauer Hyperon-nucleon interaction Two-pseudoscalar-meson exchange diagrams Two-pseudoscalar-meson exchange diagrams (V TBE )[NLO] Σ N Σ N Σ N Σ N π π π π u u u u u u u u π π u u π uπ u Σ N Σ N Σ N Σ N Σ N Σ N Σ N η π K u u Λ u u Ξ u uΣ ··· uη u u π u uK u Σ N Σ N Σ N ) J.H., S. Petschauer, N. Kaiser, U.-G. Meißner, A. Nogga, W. Weise, NPA 915 (2013) 24 Johann Haidenbauer Hyperon-nucleon interaction Coupled channels Lippmann-Schwinger Equation ν0ν;J 0 ν0ν;J 0 Tρ0ρ (p ; p) = Vρ0ρ (p ; p) Z 1 00 002 X dp p ν0ν00;J 0 00 2µν00 ν00ν;J 00 + V 0 00 (p ; p ) T 00 (p ; p) (2π)3 ρ ρ p2 − p002 + iη ρ ρ ρ00,ν00 0 ρ0;ρ = ΛN, ΣN LS equation is solved for particle channels (in momentum space) Coulomb interaction is included via the Vincent-Phatak method The potential in theLS equation is cut off with the regulator function: ν0ν;J 0 Λ 0 ν0ν;J 0 Λ Λ −(p=Λ)4 Vρ0ρ (p ; p) f (p )Vρ0ρ (p ; p)f (p); f (p) = e ! consider values Λ = 450 - 700 MeV [500 - 650 MeV] Johann Haidenbauer Hyperon-nucleon interaction NLO Results Pseudoscalar-meson exchange All one- and two-pseudoscalar-meson exchange diagrams are included SU(3) symmetry is broken by using the physical masses of the pion, kaon, and eta SU(3) breaking in the coupling constants is ignored Fπ = FK = Fη = F0 = 93 MeV; gA = 1.26 o assume that η η8 (i.e. θP = 0 and fBBη 0) ≡ 1 ≡ assume that α = F=(F + D) = 2/5 (semi-leptonic decays ) α ≈ 0:364) Correction to V OBE due to baryon mass differences are ignored (A fit with two-pion-meson exchange diagrams is possible!) (A fit with physical values for Fπ, FK , Fη is possible!) Johann Haidenbauer Hyperon-nucleon interaction Contact terms SU(3) symmetry is assumed (at NLO SU(3) breaking corrections to theLO contact terms arise!) 10 contact terms in S-waves no SU(3) constraints from the NN sector are imposed! 3 3 12 contact terms in P-waves and in S1– D1 SU(3) constraints from the NN sector are imposed! 1 3 1 contact term in P1– P1 (singlet-triplet mixing) is set to zero contact terms in S-waves: can be fairly well fixed from data some correlations between NLO andLO LECs Johann Haidenbauer Hyperon-nucleon interaction Contact terms inP-waves contact terms in P-waves are much less constrained use SU(3) and fix some (5) LECs from NN the others (7) are fixed from “bulk” properties: (1) σΛp 10 mb at plab 700 900 MeV/c ≈ ≈ − (2) dσ=dΩ − at plab 135 160 MeV/c Σ p!Λn ≈ − Other (future) options: consider matter properties: use spin-orbit splitting of the Λ single particle levels in nuclei Consider the Scheerbaum factor SΛ calculated in nuclear matter to relate the strength of the Λ-nucleus spin-orbit potential to the two body ΛN interaction (R.R.
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