Multi-Channel Systems in a Finite Volume

Transcription

1 INT-12-2b Multi-Channel Systems in a Finite olume RaúL Briceño In collaboration with: Zohreh Davoudi (arxiv:

2 Motivation: Scalar Sector The light scalar spectrum Its nature remains puzzling Pertinent examples: f 0 (980, M f0 = 990 ± 20 Me, I G (J PC =0 + (0 ++ a 0 (980, M a0 = 980 ± 20 Me, I G (J PC =1 (0 ++ Possible interpretations: Tetraquark states K K molecular states Desired Ab Initio calculation: {ππ, 4π, 6π,K K,ηη} K K = Exp. R. L. Jaffe (1977. Weinstein and Isgur (1982.

3 Motivation: Baryonic Sector Strongly-attractive iso-singlet NK Kaon condensation (a KN I=0 suffers from large systematic errors Multi-channels: {Σπ, KN, Σππ, Λη,...} Resonances: Λ(1405, Λ(1520,... Non-perturbative, model-dependent,... Desired Ab Initio calculation: {Σπ, KN, Σππ, Λη,...} Σππ NK Kaplan and Nelson (1986. Σπ = Exp.

4 LQCD & EFT in F (I LQCD: numerical evaluation of QCD LQCD artifacts: finite Euclidian spacetime, a 0, m π 140 Me,... Maiani-Testa theorem Luscher s Method: S-matrix elements from F effects E ππ (L δ ππ Periodic finite volume, Discretized momenta: a 0, T p = 2πn L a a 0 L Maiani and M. Testa (1990 Luscher (1991

5 LQCD & EFT in F (II Coupled channels (CC: [eg. ] E n (L {δ ππ, δ K K, ɛ ππ K K} Previous CC work: QM 2-body scattering, NR EFT, Relativistic EFT, UChPT CC more parameters : Twisted BC, asymmetric lattices,..., boosted systems,... Boosted systems: More measurements/reduce systematics Reduce F artifacts for bound states Work presented: ππ K K ππ Model independent, relativistic EFT multi-coupled channels with non-zero momenta He, Feng, and Liu (2005 Lage, Meissner, and Rusetsky (

6 Review: Uncoupled System in a Moving Frame Obtain poles in the F four-point correlation (M = K + K K + K K K + = (1PI diagrams K = (s 2PI diagrams Kim, Sachrajda, and Sharpe (2005.

7 Power-Law vs. Exponential F Effects Kinematically allowed regime: 0 E 2 P 2 (4m 2 p-regime: m πl 2π 1 Dressed propagator ( = Bethe-Salpeter kernel: ( K = = ( = ( K + O + O ( e m π L L m π L ( e m π L mπ L Power-law corrections Gasser and Leutwyler (1987. Bedaque, Sato and Walker-Loud (2006

8 Power-Law vs. Exponential F Effects Kinematically allowed regime: 0 E 2 P 2 (4m 2 p-regime: m πl 2π 1 Dressed propagator ( = Bethe-Salpeter kernel: ( K = = ( = ( K + O + O ( e m π L L m π L ( e m π L mπ L Power-law corrections particles on-shell explore the boundaries of the volume Gasser and Leutwyler (1987. Bedaque, Sato and Walker-Loud (2006

9 Generic Loop P =(E,P K K n = symmetry factor δg (k i, k f G (k i, k f G (k i, k f = n ( 1 L 3 l dl 3 (2π 3 nk(k i, lk(l, k f 2ω l [(E ω l 2 ω 2 Pl + iɛ] k i,k f C.M. coordinates ω 2 l = m 2 + l 2 ω 2 Pl = m 2 +(P-l 2 Kim, Sachrajda, and Sharpe (2005.

10 Generic Loop K = + K K K K K on-shell G (k i, k f =G (k i, k f +δg (k i, k f Angular momentum decomposition: (δg l,m;l,m = i ( KδG K l,m;l,m (δg l1,m 1 ;l 2,m 2 = i q n 8πE δ l1,l 2 δ m1,m 2 i 4π q l,m 4π q l c P lm(q 2 dω Y l 1 m 1 Y lmy l2 m 2 where c P lm(q 2 Z d lm[1; (q L/2π 2 ] partial wave mixing! Kim, Sachrajda, and Sharpe (2005.

11 F Scattering Amplitude and Quantization Condition (M = K + K K + K K K + = K + K K + K K K + } M ( + K K K K K K K + K K + K K K + } M ( } δg } M +

12 F Scattering Amplitude and Quantization Condition (M = K + K K + K K K + = + M M M + M M M + = im 1 1+δG M Quantization condition (QC: det ( 1+δG M =0 In practice: det ( 1+δG M l max =0 l max =0 A1-cubic irrep, : q cot(φ 4πc P (q QC can be written as: cot(φ = cot(δ δ + φ = mπ E δ Kim, Sachrajda, and Sharpe (2005. Rummukainen and Gottlieb (1995

13 Coupled Channels (I Above the inelastic threshold: ππ K K ππ S-matrix for N=2 channels : S 2 = ( e i2δ I cos 2ɛ ie i(δ I +δ II sin 2ɛ ie i(δ I +δ II sin 2ɛ e i2δ II cos 2ɛ, Channels: I,II Scattering amplitude: M ( MI,I M I,II M II,I M II,II S-wave-three parameters : δ (0 I, δ (0 II, ɛ(0

14 Coupled Channels (II Finite volume scattering amplitude for I to I: = Full scattering matrix ( ( 0 0 ( + ( ( = + } ik Quantization condition: det ( 1+δG M = det } ig ( 1+δG I M I,I δg I M I,II δg II M II,I 1+δG II M II,II = im 1 1+δG M =0 M. T. Hansen and S. R. Sharpe(2012, arxiv: PRD[hep-lat]

15 ππ K K Numerical complications: Disconnected diagrams Ō I=0 ππ O I=0 ππ + Ōππ I=0 Oππ I=0 Physical m π Theoretical complications: ππππ threshold 560 Me << 2M K Not a problem for m π > 300 Me

16 ππ K K [m π 310Me, m K 530Me] det ( 1+δG M l max =0 =0 Center of Mass Spectrum ππππ En [Me] K K L [fm] Thanks to M. T. Hansen.

17 Coupled Channels (III A1-cubic irrep and l max =0: cos 2 ɛ cos (φ 1 + δ 1 φ 2 δ 2 = cos (φ 1 + δ 1 + φ 2 + δ 2 limit, channels decouple: ɛ 0 δ I + φ I = mπ, δ II + φ II = m π Generalization to arbitrary N: M M I,I M I,II... M I,N M I,II M II,II.... δg δgi δgii.... M N,I M N,N 0 δg N Quantization condition: det ( 1+δG M =0

18 Weak-Matrix Elements Multi-hadron matrix elements power-law volume dependence (2001 Lellouch & Luscher: (LL-factor K ππ (2012 Hansen & Sharpe: D {ππ, KK} Extension to 2 2 (e.g. In the absence of 1-Body Kπ ππ 1 S 0 3 S 1 mixing in the NN sector: l max =0 p + p d + e + + ν e ν + d ν + p + n ν e + d n + n + e +

19 Weak-Matrix Elements: Relativistic Case Starting point: M I,II 2 e i2(δ I +δ II ( M I,I + 1 δg I ( M II,II + 1 δg II =0 LL- trick Weak interactions lift degeneracy: E = M I,II qi CM momenta and pseudo phase: = q i M I,II δi (qi =δ i(q i q i M I,II Generalized LL factor: ( ( 8πE M I,II 2 = { q 2 I q II 0 8πE 0 n I qi n II qii requires high precision phase shift determination } (φ I(qI +δ I(q I (φ II(qII+δ II(q II M I,II 2 } 2 2 boosted L.L. factor caution!

20 NN Weak Matrix Elements Axial vector current: A µ=3 = 1 2 (ūγ 3 γ 5 u dγ 3 γ 5 d 1-Body + 2-Body NN current 2-Body ~ dominant uncertainty in deuteron break-up (2004 Detmold & Savage: background field 1 S 0 3 S 1 coupled channels reaction in F 5-point correlation functions 1-body current: p p n e + p n ν e p p n p e + ν e

21 NN Weak Matrix Elements Below pion production: EF T ( π Strong interactions: 1 S 1 S S 1 S 1 2-Body~ π L 1,A 1-Body~ g A Weak interaction: l ( W 1 S 3 0 S 1 = + ( Loops: I 0 (E,P J 0 (E,P power-law corrections full NR prop

22 Effective 1 S 0 1 S 0 vertex Renormalization Conditions = + + = D 1 Infinite volume 1 S 0 1 S 0 scattering: = = id 1 1 D 1 I 0 im1 S 0 Infinite volume 1 S 0 3 S 1 transition amplitude: = im1 S 0 3 S 1 Finite olume : D 1 D 1 D 1 + δd 1 Evaluate poles : i (M1 S 0 = id 1 1 D 1 I 0

23 Quantization Condition and F Matrix Element Poles for 1 S 0 1 S 0 F scattering amplitude: ( M1 S 0 3 S 1 M1 S 0 M3 S 1 g A W 3 δj 0 2 (M1 S δi 0 ( M3 S δi 0 =0 where δj 0 = J 0 J 0, δi 0 = I 0 I 0 LL- trick δe = M 1 S 0 3 S 1 = Z A NN; 1 S 0 A µ=3 NN; 3 S 1 ( M 1 S 0 3 S 1 ga W 3 δj 0 e i2φ ( δi = ( 2π q ( φ + δ 3 S 1 ( φ + δ 1 S 0 M 1 S 0 3 S 1 2 g A = Z A N A µ=3 N = Z A N A µ=3 N + O ( e Lm π

24 Summary & Extension Strongly multi-coupled system More parameters to extract Boosted systems Possible to study {ππ, KK}, {ππ, KK, ηη},... Weakly multi-coupled system 2 2 boosted L.L. factor (no 1-Body oper. 1 S 0 3 S 1 mixing in the NN sector Extension: Exponential corrections in the NN sector due to pions Isoscalar NN electroweak matrix elements

25 Thank You!

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