Coupled-channel Bethe-Salpeter

Transcription

1 Coupled-channel Bethe-Salpeter approach to pion-nucleon scattering Maxim Mai 01/11/2012

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3 ChPT vs. uchpt + chiral symmetry + crossing symmetry + counterterm renormalization low-energy (ρ, ) perturbative unitarity fails in vicinity of resonances e.g. Λ(1405) [Kaiser 2001] [MM et al. 2009]... threshold behaviour + chiral symmetry + energy range extended 1.65 GeV + elastic unitarity no crossing symmetry? renormalization difficult resonances 3/17

4 Market situation (selectively) Lippmann-Schwinger iterated (pseudo-)potentials from (LO and NLO) chiral Lagrangian [Kaiser, Siegel, Weise 1995] N/D approach with expl. resonances [Meißner, Oller 2000] BSE iterated Weinberg Tomozawa contact term (LO chiral Lagrangian): ON-shell factorization of the BSE [Inoue, Oset, Vacas 2001] OFF-shell calculation (with some NLO dependence) [Nieves, Arriola 2001] OFF-shell calculation within gauge invariant framework for meson-photoproduction [Borasoy, Bruns, Meißner, Nißler 2007] 4/17

5 Method (meson-baryon interaction) Chiral perturbation theory: model independent framework to analyse hadronic processes at low-energies [Gasser, Leutwyler (1981)] LO chiral Lagrangian: L φb = L (1) φb +L(2) φb +... L (1) φb = B(iγ µ D µ m 0 )B + D/F 2 Bγ µ γ 5 [u µ,b] ± WT contact term dominates the s-wave interaction u- and s-channel Born graphs (omitted) 5/17

6 Method (meson-baryon interaction #2) NLO corrections are sizable, e.g. a (3/2) πn = O(q3 ) [MM, Bruns, Kubis, Meißner (2009)] L (2) φb =b D/F B[ χ ] [ +,B ± + b 0 BB χ + + b 1/2 B u [ µ, u µ,b ] ] { + b 3 B u { µ, u µ,b }} + b 4 BB u µu µ + ib 5/6 Bσ µν[ [ uµ,u ν ],B ] + ib 7 Bσ µν u µ u νb + i b ( 8/9 Bγ µ[ u [ µ, u [ ν, D ν,b ]] ] + Bγ µ[ D [ ν, u ν, [ u ]] ] ) µ,b 2m 0 + i b ( 10 Bγ µ{ u { µ, u [ ν, D ν,b ]}} + Bγ µ[ D { ν, u ν, { u }}] ) µ,b 2m 0 + i b ( 11 2 Bγ µ[ D ] ν,b u µu ν + Bγ µ B [ ] ν [ D ν] ) ν,u µ u + uµ Dν,u 2m 0 chiral potential for φ(q 1 )B(p q 1 ) φ(q 2 )B(p q 2 ): V(/q 2,/q 1 ;p) =A WT (/q 1 + /q 2 )+A 14 (q 1 q 2 )+A 57 [/q 1, /q 2 ] +A M +A 811 ( / q 2 (q 1 p)+ /q 1 (q 2 p) ) 6/17

7 Method (iteration) Bethe Salpeter equation: d d k T(/q 2,/q 1 ;p) = V(/q 2,/q 1 ;p)+ (2π) d V(/q 2,/k;p) 1 i k 2 M 2 +iǫ /p /k m+iǫ T(/k,/q 1 ;p) = + Loop integration UV divergencies: strict chiral expansion - counterterm renormalization (order by order) iterated bubble sum - many counterterms!!! 7/17

8 Method (renormalization) shift loop divergencies into the kernel: 1 T ˆ V 1 G loop modification 1 T δˆ V 1 δ G δ technically: 1. dim-reg. one-loop diagramms 2. omit loop divergencies 3. scale dependence remains [Borasoy, Bruns, Meißner, Nißler (2007)] subtraction constants: {log(µ π ),log(µ η ),log(µ K )} and 14 LECs to be fixed 8/17

9 Method (solution of the BSE) 1. start with the chiral potential V(/q 2,/q 1 ;p) =A WT (/q 1 + /q 2 )+A 14 (q 1 q 2 )+A 57 [/q 1, /q 2 ] +A M +A 811 ( / q 2 (q 1 p)+ /q 1 (q 2 p) ) 2. number of structures is limited to 20: ( ℵ := /q 1, /p /q 1, /q 2/p /q 1, /q 2 /q 1, /p /q 1 (q 2 p), /q 1 (q 2 p), /q 2 (q 1 p), /q 2 /q 1, (q 1 p)(q 2 p), /p(q 1 p)(q 2 p), (q 1 p), /p(q 1 p), (q 2 q 1 ), /p(q 2 q 1 ), /q 2/p, /q 2, /p(q 2 p), (q 2 p), ) 1, /p T(/q 2,/q 1 ;p) = 20 i=1 ℵ i(/q 2,/q 1 ;p) T i (s) 3. pull T i (s) out of the BSE and solve X ik T k (s) = V i 9/17

10 Fit compare πn channel: open channels: {pπ 0 ;nπ + ;pη;λk + ;Σ 0 K + ;Σ + K 0 } data: Re(S11 ), Im(S 11 ), Re(S 31 ) and Im(S 31 ) from GWU [Arndt, Briscoe, Strakovsky, Workman 2006] fit range: 1080 < s < 1560 MeV errors: f 0+ ( s < 1280 MeV) = f 0+ ( s > 1280 MeV) = (ππn) χ 2 dof minimization by MINUIT(migrad) on HISKP cluster 10/17

11 Fit (result) best fit: χ 2 dof = (scales in GeV, LECs in GeV 1 ) log(µ π) = b 0 = b 1 = b 5 = b 8 = log(µ K ) = b D = b 2 = b 6 = b 9 = log(µ η) = b F = b 3 = b 7 = b 10 = S 11 fits better than S 31, e.g. χ 2 dof (S 11) = b 4 = b 11 = s-wave scattering lengths agree roughly with those of direct GWU extraction: a 1/ M π ( ±0.0022) a 3/ M π ( ±0.0017) 11/17

12 Fit (high-energy behaviour #1) 0,8 0,6 0,4 0,2 Im S31 0 πp ηp ΛK ΣK -0,2 Re S31-0, Wcms [MeV] S 31 (1620) is not dynamically generated! 12/17

13 Fit (high-energy behaviour #2) S 11 (1535) is reproduced nicely 1 0,8 0,6 0,4 Re S11 0,2 0 πp Im S11 ηp ΛK ΣK -0,2-0, Wcms [MeV] what s about higher energies? 13/17

14 Fit (high-energy behaviour #2) S 11 (1535) is reproduced nicely 1 0,8 0,6 0,4 Re S11 0,2 0 πp Im S11 ηp ΛK ΣK -0,2-0, Wcms [MeV] a second structure is obtained S 11 (1650)!? 14/17

15 S 11 : 2. Riemann sheet Ηp Ηp K K K K Re s Im s 15/17

16 Pole structure - analysis sheet ( ): W 1535 = ( i)GeV sheet ( ): W 1650 = ( i)GeV Nieves(2001) Doering(2009) i i i i Arndt(2006) Cutkosky(1980) i i i i Hoehler(1979) Manley(1992) i i i i structure of resonances: T ON ij (s) g ig j s s R S 11 (1535) : g ΛK + 2 > g pη 2 > g Σ + K 0 2 g nπ + 2 > g Σ 0 K + 2 g pπ 0 2 S 11 (1650) : g Σ + K 0 2 > g pη 2 > g Σ 0 K + 2 g nπ + 2 > g pπ 0 2 g ΛK /17

17 Summary & Outlook coupled channel BSE solved with the full off-shell dependence.. with all local terms of second chiral order S 31 (1620) is not generated dynamically S 11 (1535) is generated dynamically S 11 (1650) is at right position without (!) fitting include data for KΣ and also ηn/ππn channels improve error analysis extend to photoproduction (gauge invariance) 17/17