Meson Baryon Scattering

Transcription

1 Meson Baryon Scattering Aaron Torok Department of Physics, Indiana University May 31, 2010

2 Meson Baryon Scattering in Lattice QCD Calculation of the π + Σ +, and π + Ξ 0 scattering lengths Aaron Torok Department of Physics, Indiana University May 31, 2010

3 OUTLINE Background and motivation Calculating scattering lengths using LQCD Signal to Noise Ratio Meson-Baryon Results Conclusion

4 BACKGROUND AND MOTIVATION We need to calculate in the non-perturbative regime! Number of counterterms in χpt beyond leading order (LO) or next-to-leading order (NLO) in many calculations often exceeds the number of experimental measurements available turn to the lattice Important to recover what is known experimentally to high precision, however, not the main objective calculate what cannot be accessed experimentally, or which can be measured with only limited precision

5 CONTINUUM QCD= = = =

6 = = = = LATTICE QCD

7 LATTICE QCD: SOME CHALLENGES Calculational cost Calculations done at unphysical (larger) light quark masses; Chiral extrapolation convergence m latt π χpt extrapolation Signal to noise ratio (not a problem with pions) > m phys π Discriminating the ground state from excited states Many interesting processes have annihilation diagrams cost

8 STEPS TO CALCULATE THE SCATTERING LENGTH Start with gauge fields, in this case MILC staggered Generate propagators Calculate contractions correlators Fit the correlators to sum of exponentials extract masses, energies Using jackknifed tables of masses and energies calculate the scattering length Using scattering lengths calculated at several quark masses, calculate χpt LECs With LECs that have been fit, extrapolate to physical pion mass

9 GAUGE ENSEMBLES MILC gauge ensembles, staggered, asqtad-improved, coarse (b fm), and fine (b 0.09 fm). The Chroma software system cfgs generated by m l /m s m π (MeV) L(fm) cfgs sources MILC MILC MILC MILC MILC MILC

10 ENERGY EIGENVALUES IN A BOX The exact energy eigenvalue equation for E n : E n E n m 1 m 2 = p 2 n + m p 2 n + m 2 2 m 1 m 2 Energy levels occur at momenta p = 2πj/L (no interaction); The Lüscher formula: p cot δ(p) = 1 ( ) ( ) pl pl πl S, S 2π 2π Λ j j j 2 1 ( pl 2π ) 2 4πΛ j the effective range expansion for pcotδ(p) 1/a, as p 0.

11 CALCULATING E, AND SINGLE PARTICLE MASSES The correlation functions for the meson (φ) and baryon (B) systems are C φ (t) = x φ (t, x) φ(0, 0), C B (t) = x B(t, x) B(0, 0) C φ (t) Be m φt, C B (t) De m Bt, for t, L C φb (p, t) = p =p e ip (x y) φ (t, x) B(t, y) φ(0, 0) B(0, 0) x,y G φb (p, t) C φb(p, t) C φ (t)c B (t) n=0 A n e Ent

12 CALCULATING E, AND SINGLE PARTICLE MASSES We extract, m φ, m B, and E C φ (t) Be m φt, C B (t) De m Bt, for t, L G φb (p, t) C φb(p, t) C φ (t)c B (t) n=0 A n e Ent

13 EFFECTIVE PLOTS m eff φ,b = 1 ( ) Cφ,B (t) log, E eff φb n J C φ,b (t + n J ) = 1 ( ) GφB (t) log. n J G φb (t + n J )

14 LINEAR COMBINATION OF CORRELATORS C SS (t) = Ae m 1t +Be m 2t +..., C SP (t) = Ce m 1t +De m 2t +...,

15 LINEAR COMBINATION OF CORRELATORS C SS (t) = Ae m 1t +Be m 2t +..., C SP (t) = Ce m 1t +De m 2t +..., (C SS (t)) 2 C SP (t) A2 C e m 1t + ( 2ABC A 2 D C 2 ) e m 2t +... m C SS 2 C SP Effective t

16 LINEAR COMBINATION OF CORRELATORS C SS (t) = Ae m 1t +Be m 2t +..., C SP (t) = Ce m 1t +De m 2t +..., (C SS (t)) 2 C SP (t) A2 C e m 1t + ( 2ABC A 2 D C 2 ) e m 2t +... C SS (t) αc SP (t) = (A αc)e m 1t + (B αd)e m 2t +... m C SS 2 C SP Effective t m C SS ΑC SP Effective t

17 SIGNAL TO NOISE RATIO for the pion, signal Ae mπt, σ 2 Be 2mπt for the proton, signal Ce mpt, σ 2 De 3mπt s/n(t): pion constant s/n(t): proton e (mp 3 2 mπ)t

18 MESON BARYON The are six elastic MB scattering processes without annihilation diagrams that we calculated on the Lattice Particles Isospin Quark Content π + Σ + 2 uuu ds π + Ξ 0 3/2 uu dss K + p 1 uuud s K + n 0 and 1 uudd s K 0 Σ + 3/2 uu dss K 0 Ξ 0 1 u dsss

19 MESON BARYON The are six elastic MB scattering processes without annihilation diagrams that we calculated on the Lattice Particles Isospin Quark Content π + Σ + 2 uuu ds π + Ξ 0 3/2 uu dss K + p 1 uuud s K + n 0 and 1 uudd s K 0 Σ + 3/2 uu dss K 0 Ξ 0 1 u dsss coupled channel, same valence quarks uu dss

20 COUPLED CHANNEL The π + Ξ 0 and K 0 Σ + have the same quark content, and constitute a coupled channel m K +m Σ =1.231 m π +m Ξ =1.124 n J = 2 K 0 Σ + Total E m m π +m Ξ =1.124 n J = 2 π + Ξ 0 Total E m010 be 1.2 be t/b t/b We extracted a π + Ξ 0, but not a K 0 Σ +

21 CALCULATED MESON AND BARYON MASSES AND E

22 CALCULATED MESON AND BARYON MASSES AND E π + m010 Ξ 0 m χ 2 per d.o.f = bm = ± n J = χ 2 per d.o.f = bm = ± n J = 2 bm bm t/b t/b

23 CALCULATED MESON AND BARYON MASSES AND E π + m010 Ξ 0 m χ 2 per d.o.f = bm = ± n J = χ 2 per d.o.f = bm = ± n J = 2 bm bm t/b π + Σ + E m010 χ 2 per d.o.f = b E = ± n J = t/b π + Ξ 0 E m010 χ 2 per d.o.f = 0.62 b E = ± n J = 2 b E b E t/b t/b

24 LATTICE CALCULATION OF 1/a p cot δ(p) = 1 ( ) ( ) pl pl πl S, S 2π 2π Λ j j j 2 1 ( pl 2π ) 2 4πΛ j pcot mπ pcot m Π vs. E m Π E m Π pcotδ/m π π + Σ + π + Ξ 0 K + p K + n K + Ξ 0 pcotδ/m π versus E/m π m E/m π

25 LATTICE CALCULATION OF 1/a p cot δ(p) = 1 ( ) ( ) pl pl πl S, S 2π 2π Λ j j j 2 1 ( pl 2π ) 2 4πΛ j pcot mπ pcot m Π vs. E m Π pcotδ/m π π + Σ + π + Ξ 0 K + p K + n K + Ξ 0 pcotδ/m π versus E/m π m E m Π E/m π

26 LATTICE CALCULATION OF 1/a For comparison, including π + π + and K + K + on the plot: pcotδ/m π versus E/m π m010 pcotδ/m π π + Σ + π + Ξ 0 K + p K + n K + Ξ 0 π + π + K + K E/m π

27 CHIRAL EXTRAPOLATION: SU(3) HBχPT TO NNLO a π + Σ + = 1 4π a π + Ξ 0 = 1 4π a K + p = 1 4π a K + n = 1 4π a K 0 Ξ 0 = 1 4π m Σ m π + m Σ m Ξ m π + m Ξ m N m K + m N m N m K + m N m Ξ m K + m Ξ» 2mπ f 2 π» mπ f 2 π» 2m K f 2 K» m K f 2 K» 2m K f 2 K + 2m2 π C fπ Y π + Σ +(µ) + 8h 123(µ) m3 π fπ 2 + m2 π C fπ Y π + Ξ 0(µ) + 8h 1(µ) m3 π fπ 2 + 2m2 K C fk Y K + p(µ) + 8h 123 (µ) m3 K fk 2 + m2 K C fk Y K + n(µ) + 8h 1 (µ) m3 K fk 2 + 2m2 K C fk Y K 0 Ξ 0(µ) + 8h 123(µ) m3 K fk 2 1 Liu and Zhu, hep-ph/ , hep-ph/

28 CHIRAL EXTRAPOLATION: SU(3) HBχPT TO NNLO tree-level (Weinberg), and the LEC s at NLO and NNLO 1 a π + Σ + = 1 4π a π + Ξ 0 = 1 4π a K + p = 1 4π a K + n = 1 4π a K 0 Ξ 0 = 1 4π m Σ m π + m Σ m Ξ m π + m Ξ m N m K + m N m N m K + m N m Ξ m K + m Ξ» 2mπ f 2 π» mπ f 2 π» 2m K f 2 K» m K f 2 K» 2m K f 2 K + 2m2 π C fπ Y π + Σ +(µ) + 8h 123(µ) m3 π fπ 2 + m2 π C fπ Y π + Ξ 0(µ) + 8h 1(µ) m3 π fπ 2 + 2m2 K C fk Y K + p(µ) + 8h 123 (µ) m3 K fk 2 + m2 K C fk Y K + n(µ) + 8h 1 (µ) m3 K fk 2 + 2m2 K C fk Y K 0 Ξ 0(µ) + 8h 123(µ) m3 K fk 2 1 Liu and Zhu, hep-ph/ , hep-ph/

29 CHIRAL EXTRAPOLATION: SU(3) HBχPT TO NNLO tree-level (Weinberg), and the LEC s at NLO and NNLO 1 a π + Σ + = 1 4π a π + Ξ 0 = 1 4π a K + p = 1 4π a K + n = 1 4π a K 0 Ξ 0 = 1 4π m Σ m π + m Σ m Ξ m π + m Ξ m N m K + m N m N m K + m N m Ξ m K + m Ξ Y φb are the loop functions.» 2mπ f 2 π» mπ f 2 π» 2m K f 2 K» m K f 2 K» 2m K f 2 K + 2m2 π C fπ Y π + Σ +(µ) + 8h 123(µ) m3 π fπ 2 + m2 π C fπ Y π + Ξ 0(µ) + 8h 1(µ) m3 π fπ 2 + 2m2 K C fk Y K + p(µ) + 8h 123 (µ) m3 K fk 2 + m2 K C fk Y K + n(µ) + 8h 1 (µ) m3 K fk 2 + 2m2 K C fk Y K 0 Ξ 0(µ) + 8h 123(µ) m3 K fk 2 1 Liu and Zhu, hep-ph/ , hep-ph/

30 CHIRAL EXTRAPOLATION: SU(3) HBχPT TO NNLO π + Σ +, K + p, and K 0 Ξ 0 Γ (1) LO 2πaf φ m «φ m φ Γ (1) NLO 2πaf 2 φ Γ (1) NNLO 2πaf2 φ 1 + m φ m φ m B m φ m B 1 + m φ m B «= 1 = 1 C 1 m φ «+ f2 φ 2m φ Y φb (Λ χ) = 1 C 1 m φ 4h 123 (Λ χ)m 2 φ π + Σ + SU(3) K + p SU(3) K 0 Ξ 0 SU(3) Γ 0 Γ -0.5 Γ NLO NNLO m π (MeV) NLO NNLO m K (MeV) NLO NNLO m K (MeV)

31 CHIRAL EXTRAPOLATION: SU(3) HBχPT TO NNLO π + Ξ 0, and K + n Γ (2) NNLO 4πaf2 φ m φ Γ (2) LO 4πaf φ m «φ m φ Γ (2) NLO 4πaf 2 φ 1 + m φ m B m φ m B 1 + m φ m B «= 1 = 1 C 01 m φ «+ f2 φ m φ Y φb (Λ χ) = 1 C 01 m φ 8h 1 (Λ χ)m 2 φ. Γ π + Ξ 0 SU(3) Γ K + n SU(3) NLO NNLO m π (MeV) NLO NNLO m K (MeV)

32 CHIRAL EXTRAPOLATION: SU(2) χpt TO NNLO a π + Σ + and a π + Ξ0 are given by 2 a π + Σ + = 1 4π a π + Ξ 0 = 1 4π m Σ m π + m Σ m Ξ m π + m Ξ» 2mπ f 2 π» mπ f 2 π + 2m2 π C fπ 2 π + Σ + + m3 π log mπ π 2 fπ 4 + m2 π C fπ 2 π + Ξ 0 + µ + 2m3 π h π + Σ +(µ) f 2 π m3 π log mπ 2π 2 fπ 4 µ + m3 π h fπ 2 π + Ξ 0(µ) 2 Mai, et. al., arxiv: v1[hep-ph]

33 CHIRAL EXTRAPOLATION: SU(2) χpt TO NNLO a π + Σ + and a π + Ξ0 are given by 2 tree-level (Weinberg), and the LEC s at NLO and NNLO a π + Σ + = 1 4π a π + Ξ 0 = 1 4π m Σ m π + m Σ m Ξ m π + m Ξ» 2mπ f 2 π» mπ f 2 π + 2m2 π C fπ 2 π + Σ + + m3 π log mπ π 2 fπ 4 + m2 π C fπ 2 π + Ξ 0 + µ + 2m3 π h π + Σ +(µ) f 2 π m3 π log mπ 2π 2 fπ 4 µ + m3 π h fπ 2 π + Ξ 0(µ) 2 Mai, et. al., arxiv: v1[hep-ph]

34 CHIRAL EXTRAPOLATION: SU(2) χpt TO NNLO Γ LO 1 Γ NLO 1 C π + Bm π Γ NNLO 1 C π + Bm π h π + B(Λ χ)m 2 π where B is either Σ + or Ξ π + Σ + SU(2) 1.5 π + Ξ 0 SU(2) Γ NLO NNLO m π (MeV) Γ NLO NNLO m π (MeV)

35 CHIRAL EXTRAPOLATION: 1.5 π + Σ + SU(3) 1.5 π + Ξ 0 SU(3) Γ Γ NLO NNLO m π (MeV) π + Σ + SU(2) -0.5 NLO NNLO m π (MeV) Γ Γ NLO NNLO m π (MeV) π + Ξ 0 SU(2) -1.5 NLO NNLO m π (MeV)

36 RESULTS extrapolating using SU(2) χpt, the scattering lengths are a π + Σ + = ± fm ( fm) a π + Ξ0 = ± fm ( fm) where the error encompasses statistical and systematic uncertainties

37 RESULTS As pointed out by Mai, et. al., if f π f, where f is the decay constant in the chiral limit, chiral log is removed. a π + Σ + = 1 2π a π + Ξ 0 = 1 4π m Σ m π + m Σ m Ξ m π + m Ξ» mπ + m2 π f 2 f C 2 π + Σ + + m3 π f 2 h π + Σ +, h π + Σ + = 4 f 2 lr 4 + h π + Σ +» mπ f 2 + m2 π f 2 C π + Ξ 0 + m3 π f 2 h π + Ξ 0, h π + Ξ 0 = 4 f 2 lr 4 + h π + Ξ 0

38 0 m π a π+π MA χ - PT (One Loop) χ - PT (Tree Level) CP-PACS (2004) (n f = 2) E 865 (2003) NPLQCD m K+ a K+K χ-pt (Tree Level) MILC coarse (b /= 0) MILC fine (b /= 0) physical point extrapolated with MAχ-PT m π / f π a π + Σ + SU(2) m K+ / f K+ a π + Ξ 0 SU(2) a π + Σ + (fm) -0.4 a π + Ξ 0 (fm) m π (MeV) m π (MeV)

39 THE FUTURE HIGHER STATISTICS

40 THE FUTURE HIGHER STATISTICS Speak softly and carry a big stick... Theodore Roosevelt

41 THE FUTURE HIGHER STATISTICS Speak softly and carry a big stick... Theodore Roosevelt and the lesser known second part:

42 THE FUTURE HIGHER STATISTICS Speak softly and carry a big stick... Theodore Roosevelt and the lesser known second part: then use it to beat down the exponentially-increasing noise!

43 HIGH STATISTICS CALCULATIONS: Recent studies of single baryon, two-baryon, and 3-baryons at one quark mass and one volume 3 stats increased by factor of 10 3 see recent papers High Statistics Anisotropic...NPLQCD collaboration

44 HIGH STATISTICS CALCULATIONS: Recent studies of single baryon, two-baryon, and 3-baryons at one quark mass and one volume 3 stats increased by factor of M t bt M t bt 3 see recent papers High Statistics Anisotropic...NPLQCD collaboration

45 CONCLUSION first dynamical LQCD calculation of meson-baryon scattering prediction of the π + Σ + and π + Ξ 0 scattering lengths using SU(2) χpt No convergence for kaon-baryon processes using SU(3) χpt

46 ACKNOWLEDGEMENTS Thanks to the MENU 2010 Local Organizing Committee, and the College of William & Mary. Thanks to the NPLQCD collaboration, who are: Silas Beane, William Detmold, Huey-Wen Lin, Tom Luu, Kostas Orginos, Assumpta Parreño, Martin Savage, and Andre Walker-Loud