Roy Steiner-equation analysis of pion nucleon scattering

Transcription

1 Roy Steiner-equation analysis of pion nucleon scattering Jacobo Ruiz de Elvira 1 Martin Hoferichter 2 1 HISKP and Bethe Center for Theoretical Physics, Universität Bonn 2 Institute for Nuclear Theory, University of Washington, Seattle Dr. Klaus Erkelenz Preis 215 Bonn, December 8, 215 MH, JRE, B. Kubis, U.-G. Meißner, PRL 115 (215) 9231, 19231, J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 1

2 From meson-exchange models to chiral forces Pion postulated to explain nuclear force Yukawa 1935, Nobel prize 1949 V(r) e Mπr r V (r) c small mass M π long range 1/M π Intermediate-range of the NN potential: exchange of heavier mesons σ, ρ, ω, φ,... Idea of meson-exchange potentials: fit coefficients of meson-exchange operators to experiment Pioneered by Erkelenz 1974: Bonn potential 1 2 r [fm] π ππ +... Figure courtesy of U.-G. Meißner Phenomenological potentials: CD Bonn, AV18,... describe NN data with χ 2 /dof 1 J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 2

3 From meson-exchange models to chiral forces Phenomenological potentials Beyond single-meson exchange? Hierarchy of multi-nucleon forces? LO 2N force 3N force 4N force Consistency between NN and 3N? NLO Chiral Effective Field Theory (ChEFT) Based on chiral symmetry of QCD 2 N LO Power counting Systematically improvable 3 N LO Same accuracy with less parameters low-energy constants Figure courtesy of E. Epelbaum modern theory of nuclear forces J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 3

4 Consequences of chiral symmetry Chiral symmetry of QCD L QCD = q L i /Dq L + q R i /Dq R q L Mq R q R Mq L 1 4 Ga µνg µν a Expansion in momenta p/λ χ and quark masses m q p 2 scale separation Relates different processes by low-energy theorems Leading order: F and M, determined by F π and M π predict ππ scattering Higher orders: M π ππ scattering scalar radius Nucleon sector: πn coupling g A, m N πn scattering Λ χ,n π J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 4

5 Pion nucleon scattering and nuclear forces πn scattering appears as subprocess in NN and 3N forces long-range part of potential At given chiral order: same low-energy constants as in πn parameter-free prediction reduces number of fit parameters Interpretation: 1π exchange: πn coupling constant 2π exchange: σ, ρ,... 3π exchange: ω,... πn scattering relevant for 2π channel J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 5

6 Scalar content of the nucleon Decomposition of the nucleon mass m N = N β QCD 2g F µνf a a µν + m uūu + m d dd + ms ss + N }{{}}{{} Higgs trace anomaly Mass largely generated by gluon field energy via the trace anomaly of the QCD energy- momentum tensor θ µ µ Contribution from u- and d-quarks σ πn = N ˆm(ūu + dd) N σ πn related to πn scattering via low-energy theorem Cheng, Dashen 1971 Challenges Amplitude in unphysical region: analytic continuation Isoscalar amplitude: chirally suppressed, ππ rescattering strong, isospin breaking large J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 6

7 Why care about σ πn? µ e µ e h h c,b,t u,d,s u,d,s g g Planck 213 Scalar coupling of the nucleon N m q qq N = fq N m N N {p, n} Dark Matter, µ e conversion in nuclei Condensates in nuclear matter WIMP!nucleon cross section (cm 2 ) 1!44 1!45 1!4 1! CP-violating πn couplings, EDMs 1! m (GeV/c 2 ) WIMP LUX 213 J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 7

8 Running coupling of QCD.5 α s (Q).4.3 April 212 τ decays (N 3 LO) Lattice QCD (NNLO) DIS jets (NLO) Heavy Quarkonia (NLO) e + e jets & shapes (res. NNLO) Z pole fit (N 3 LO) pp > jets (NLO) Asymptotic freedom β QCD = µ µ g ( = 11 2n ) f g π +O( g 5) QCD α s(μ Z) =.1184 ± Q [GeV] Bethke 212 Gross, Politzer, Wilczek 1973 (Nobel prize 24) QCD strongly coupled at low energies Perturbation theory fails Need non-perturbative methods J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 8

9 Non-perturbative methods for low-energy QCD Λ χ,n 1 Effective field theories: symmetries, separation of scales ChPT, ChEFT, /πeft, HπEFT, NREFT,... π 2 Dispersion relations: analyticity ( causality), unitarity ( probability conservation), crossing symmetry Cauchy s theorem, analytic structure 3 Lattice: Monte-Carlo simulation solve discretized version of QCD numerically J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 9

10 From Cauchy s theorem to dispersion relations Cauchy s theorem f(s) = 1 ds f(s ) 2πi Ω s s J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 1

11 From Cauchy s theorem to dispersion relations Cauchy s theorem f(s) = 1 ds f(s ) 2πi Ω s s J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 1

12 From Cauchy s theorem to dispersion relations Dispersion relation f(s) = 1 π analyticity cuts ds Im f(s ) s s J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 1

13 From Cauchy s theorem to dispersion relations Dispersion relation f(s) = 1 π analyticity Subtractions f(s) = f()+ s π cuts ds Im f(s ) s s cuts ds Im f(s ) s (s s) J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 1

14 From Cauchy s theorem to dispersion relations Dispersion relation f(s) = 1 π analyticity Subtractions f(s) = f()+ s π cuts ds Im f(s ) s s cuts ds Im f(s ) s (s s) Imaginary part from Cutkosky rules forward direction: optical theorem Unitarity for partial waves Im f(s) = ρ(s) f(s) 2 f(s) f(s) J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8, 215 1

15 ππ Roy equations Roy equations = Dispersion relations + partial-wave expansion + crossing symmetry + unitarity Coupled system of integral equations for partial waves t I J(s) Roy 1971 t I J(s) = k I J(s)+ 2 I = J = 4Mπ 2 ds K II JJ (s, s )Im t I J (s ) J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8,

16 ππ Roy equations Roy equations = Dispersion relations + partial-wave expansion + crossing symmetry + unitarity σ π(s) = Coupled system of integral equations for partial waves t I J(s) Roy M2 π s t I J(s) }{{} e 2iδI J (s) 1 2iσπ(s) = k I J(s)+ 2 I = J = 4Mπ 2 ds K II JJ (s, s ) Im t I J (s ) }{{} 1 σπ(s) sin2 δ I J (s ) t I J t I J J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8,

17 ππ Roy equations Roy equations = Dispersion relations + partial-wave expansion + crossing symmetry + unitarity σ π(s) = Coupled system of integral equations for partial waves t I J(s) Roy M2 π s t I J(s) }{{} e 2iδI J (s) 1 2iσπ(s) = k I J(s) }{{} δ I ai I = J = 4Mπ 2 t I J t I J free parameters a, a 2 ds K II JJ (s, s ) Im t I J (s ) }{{} 1 σπ(s) sin2 δ I J (s ) J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8,

18 ππ Roy equations Roy equations = Dispersion relations + partial-wave expansion + crossing symmetry + unitarity σ π(s) = Coupled system of integral equations for partial waves t I J(s) Roy M2 π s t I J(s) }{{} e 2iδI J (s) 1 2iσπ(s) = k I J(s) }{{} δ I ai I = J = 4Mπ 2 ds K II JJ (s, s ) }{{} 1 δ JJ δ ll π s + K II s iǫ JJ (s,s ) Im t I J (s ) }{{} 1 σπ(s) sin2 δ I J (s ) t I J t I J free parameters a, a 2 Res Ims 4M 2 π Self-consistency condition for phase shifts J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8,

19 Matching to ChPT Roy equations: ππ phase shifts in terms of a, a 2 Ananthanarayan et al. 21 Matching of two-loop ChPT and Roy equations Colangelo, Gasser, Leutwyler 21 Match low-energy polynomials l 1, l 2 as by-product Scattering lengths in terms of quark-mass LECs l 3, l 4 a =.198± fm 2 r 2 S π.17 l 3 =.22±.5 a 2 =.392±.3.66 fm 2 r 2 S π.4 l 3 =.444±.1 J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8,

20 Matching to ChPT Roy equations: ππ phase shifts in terms of a, a 2 Ananthanarayan et al. 21 Matching of two-loop ChPT and Roy equations Colangelo, Gasser, Leutwyler 21 Match low-energy polynomials l 1, l 2 as by-product Scattering lengths in terms of quark-mass LECs l 3, l 4 a =.198± fm 2 r 2 S π.17 l 3 =.22±.5 a 2 =.392±.3.66 fm 2 r 2 S π.4 l 3 =.444±.1 Prediction tested in K e4 and K 3π decays NA48/2 21 a =.221±.47 stat ±.4 syst a 2 =.429±.44 stat ±.28 syst J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8,

21 Hadronic atoms: constraints for πn 3s 3p 3d ã + = a + + { 4(M 2 1 π M 2 } π ) 4π(1+Mπ/mp) F π 2 c 1 2e 2 f 1 2s 2p πh/πd: bound state of π and p/d, spectrum sensitive to thresholdπn amplitude πh level shift π p π p πd level shift isoscalar π N π N πh width π p π n Combined analysis of πh and πd a + a + + = (7.5±3.1) 1 3 M 1 π a a + = (86.±.9) 1 3 M 1 π ǫ1s 1s } Γ1s xxxxxxxxxx xxxxxxxxxx xx xxxxxxxxxx xxxx xxxxxxxxxx xxxxx xxxxxxxxxx xxxxxxx xxxxxxxxxx xxxxxxxxx xxxxxxxxxx xxxxxxxxxxx xxxxxxx xxxxxxxxxx xxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxx xxxxxxx xxxxxxxxxx xxxxxxxxxxx xxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxxx xxxxxxxxxx xxxxxxxxx xxxxxxxxxx xxxxxxx xxxxxxxxxx xxxxx xxxxxxxxxx xxx xxxxxxxxxx x xxxxxxxxxx Baru et al. 211 J. Ruiz de Elvira & M. Hoferichter (HISKP & INT) Roy Steiner-equation analysis of πn scattering Bonn, December 8,

22 Imf J ± (t) = σπ t f J ± (t)tl J (t) t I J f I J Roy Steiner equations for πn: differences to ππ Roy equations Key differences compared to ππ Roy equations Crossing: coupling between πn πn (s-channel) and ππ NN (t -channel) need a different kind of dispersion relations [Hite, Steiner 1973], [Büttiker et al. 24] Unitarity in t-channel, e.g. in single-channel approximation Watson s theorem: phase of f± J (t) equals δ IJ [Watson 1954] solution in terms of Omnès function [Muskhelishvili 1953, Omnès 1958] Large pseudo-physical region in t -channel K K intermediate states for s-wave in the region of the f (98) M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

23 Roy Steiner equations for πn: flow of information Limited range of validity s sm = 1.38 GeV t tm = 2. GeV Input/Constraints S- and P-waves above matching point s > s m (t > t m) Inelasticities Higher waves (D-, F-, ) Scattering lengths from hadronic atoms Baru et al. 211 Output S- and P-wave phase-shifts at low energies s < s m (t < t m) Subthreshold parameters Pion-nucleon σ-term Nucleon form factor spectral functions ChPT LECs M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

24 Solving t-channel: P, D and F waves up to NN 16 9 Im f 1 +(t) [GeV 1 ] Im f 1 (t) [GeV 2 ] Im f 2 +(t) [GeV 3 ] Im f 2 (t) [GeV 4 ] Im f 3 +(t) [GeV 5 ] Im f 3 (t) [GeV 6 ] t [GeV] t [GeV] M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

25 Solving t-channel: S-wave results RS solution in general consistent with KH8 results M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

26 Solving s-channel: strategy Parameterize S and P waves up to W < W m Using SAID partial waves as starting point Impose as constraints the hadronic atom scattering lengths Introduce as many subtractions as necessary to match d.o.f [Gasser, Wanders 1999] Minimize difference between LHS and the RHS on a grid of points W j χ 2 = N l,i s,± j=1 F [f Is l± ](W j ) right hand side of RS-equations ( ) 2 Re f Is l± (W j ) F [f Is l± ](W j ) Re f Is l± (W j ) Parametrization and subthreshold parameters are the fitting parameters M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

27 Solving s-channel: results s 11 (s) p 13 (s) s 31 (s) p 33 (s) ssssblue/red sssss ssssslhs/rhs sssssafter the fit ssssgray/black sssss ssssslhs/rhs sssssbefore the fit p 11 (s) s (GeV) p 31 (s) s (GeV) sssssnotation: L 2Is 2J M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

28 Uncertainties: s-channel partial waves P 11 S 11 S W [GeV] P 13 P P W [GeV] M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th, 215 2

29 Uncertainties: imaginary part t-channel partial waves Im f +(t) [GeV] Im f 1 +(t) [GeV 1 ] Im f 1 (t) [GeV 2 ] Im f 2 +(t) [GeV 3 ] Im f 2 (t) [GeV 4 ] t [GeV] t [GeV] M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

30 Threshold parameters Threshold parameters defined as: Re f I l± (s) = q2l {a I l± + bi l± q2 + } Extracted from hyperbolic sum rules RS KH8 a + + [1 3 M 1 π ].9 ± ± 1.7 a + [1 3 M 1 π ] 85.4 ± ± 1.7 a + 1+ [1 3 M 3 π ] ± ± 1.3 a 1+ [1 3 M 3 π ] 8.3 ± ± 1. a + 1 [1 3 M 3 π ] 5.9 ± ± 1.3 a 1 [1 3 M 3 π ] 9.9 ± ± 1. b + + [1 3 M 3 π ] 45. ± ± 6.7 b + [1 3 M 3 π ] 4.9 ± ± 6. Reasonable agreement with KH8 but for the scattering lengths Disagreement in the scattering lengths in 4σ M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

31 Results for the sigma-term σ πn = F 2 π ( ) d + + 2M2 πd D σ R subthreshold parameters output of the Roy Steiner equations d + = 1.36(3)M 1 π [KH: 1.46(1)M 1 π ] d + 1 = 1.16(2)M 3 π [KH: 1.14(2)M 3 π ] D σ = (1.8 ±.2) MeV [MH at al. 212], R 2 MeV [Bernard, Kaiser, Meißner 1996] Isospin breaking in the CD theorem shifts σ πn by +3. MeV Final results: σ πn = (59.1 ± 1.9 RS ± 3. LET ) MeV=(59.1 ± 3.5) MeV σ πn depends linearly on the scattering lengths [MH, JRE, Kubis, Meißner] σ πn = I s c Is a Is + KH input σ πn = 46 MeV to be compared with σ πn = 45 MeV [Gasser, Leutwyler, Socher, Sainio 1988] compare also σ πn (64 ± 8) MeV [Pavan et al. 22] M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

32 Comparison with lattice σ πn results Recent lattice determination of σ πn from the BMW collaboration σ πn = 38(3)(3)MeV [Durr. et al. 215] The linear dependence of σ πn on the scattering lengths introduces an additional constraint 5 ] ã + [ 1 3 M 1 π level shift of πh level shift of πd width of πh -2 BMW a [ ] 1 3 Mπ 1 Fully inconsistent with the hadronic atom phenomenology M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

33 Matching to Chiral Perturbation Theory Matching to ChPT at the subthreshold point: Chiral expansion expected to work best at subthreshold point Maximal distance from threshold singularities πn amplitude can be expanded as polynomial Preferred choice for NN scattering due to proximity of relevant kinematic regions Express the subthreshold parameters in terms of the LECs to O(p 4 ) d + = 2M2 π (2 c 1 c 3 ) F 2 π + g2 a ( ) { 3 + 8g 2 a M 3 π + M 4 16ē14 64πFπ 4 π 2c } 1 c 3 Fπ 2 16π 2 Fπ 4 Chiral πn amplitude to O(p 4 ) depends on 13 low-energy constants Roy Steiner system contains 1 subtraction constants Calculate remaining 3 from sum rules Invert the system to solve for LECs M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

34 Chiral low-energy constants NLO N 2 LO N 3 LO c 1 [ GeV 1 ].74 ± ± ±.3 c 2 [ GeV 1 ] 1.81 ± ± ±.3 c 3 [ GeV 1 ] 3.61 ± ± ±.6 c 4 [ GeV 1 ] 2.17 ± ± ±.4 d 1 + d 2 [ GeV 2 ] 1.4 ± ±.8 d 3 [ GeV 2 ].48 ± ±.1 d 5 [ GeV 2 ].14 ±.5.59 ±.5 d 14 d 15 [ GeV 2 ] 1.9 ± ±.12 ē 14 [ GeV 3 ].89 ±.4 ē 15 [ GeV 3 ].97 ±.6 ē 16 [ GeV 3 ] 2.61 ±.3 ē 17 [ GeV 3 ].1 ±.6 ē 18 [ GeV 3 ] 4.2 ±.5 Subthreshold errors tiny, chiral expansion dominates uncertainty d i at N 3 LO increase by an order of magnitude due to terms proportional to g 2 A (c 3 c 4 ) = 16 GeV 1 mimic loop diagrams with degrees of freedom What s going on with chiral convergence? look at convergence of threshold parameters with LECs fixed at subthreshold point M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

35 Convergence of the chiral series NLO N 2 LO N 3 LO RS a + + [1 3 Mπ 1 ] ± 1.4 a + [1 3 M π 1 ] ±.9 a + 1+ [1 3 M π 3 ] ± 1.7 a 1+ [1 3 Mπ 3 ] ± 1.1 a + 1 [1 3 Mπ 3 ] ± 1.9 a 1 [1 3 M π 3 ] ± 1.2 b + + [1 3 M π 3 ] ± 1. b + [1 3 M π 3 ] ±.8 N 3 LO results bad due to large Delta loops Conclusion: lessons for few-nucleon applications 1 either include the to reduce the size of the loop corrections work in progress or use LECs from subthreshold kinematics 2 error estimates: consider chiral convergence of a given observable, difficult to assign a global chiral error to LECs M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

36 The ruler plot vs. ChPT Lattice QCD simulations can be performed at different quark/pion masses Pion mass dependence of m N up to NNNLO in ChPT, using Input from Roy Steiner solution 1.3 mn [GeV] Ruler approximation LHCP 28 χqcd 212 ETMC Mπ [GeV] thanks to A. Walker-Loud for providing the lattice data range of convergence of the chiral expansion is very limited huge cancellation amongst terms to produce a linear behavior M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

37 Summary & Outlook Summary Derived a closed system of Roy Steiner equations for πn Numerical solution and error analysis of the full system of RS eqs. Precise determination of the σ πn Roy Steiner formalism reproduces KH8 result with KH8 input With modern input for scattering lengths and coupling constant σ πn increases Precise determination of threshold parameters Extraction of the ChPT LECs Study of the chiral convergence Outlook Dispersive determination of pole parameters Matching to ChPT with explicit s LECs extraction and study of the chiral convergence Large-N c constraints on LECs M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

38 Outlook: ππ-continuum contribution to the nucleon form factors Proton Radius Puzzle strong constraints from analyticity and unitarity π J µ em N N π first inelastic correction ππ continuum, rigorous constraint fixed from: RS t-channel partial waves pion form factor update of Höhler spectral functions, including also isospin breaking criticism by Lee et al. 215 M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th, 215 3

39 Spare slides M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

40 Roy-Steiner equations for πn: flow of information Higher partial waves Im fl± I, l 2, s sm s-channel partial waves solve Roy Steiner equations for s sm Inelasticities ηl± I, l 1, s sm f + + f + 1+ f + 1 High-energy region Im f I l±, s sm f + f 1+ f 1 Subtraction constants πn coupling constant ππ scattering phases δ It J t-channel partial waves solve Roy Steiner equations for t tm f + f 1 ± f 2 ± f 3 ± f 4 ± High-energy region Im f J ±, t tm M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,

41 Uncertainties Statistical errors (at intermediate energies) important correlations between subthreshold parameters shallow fit minima Sum rules for subthreshold parameters become essential to reduce the errors Input variation (small) small effect for considering s-channel KH8 input very small effects from L > 5 s-channel PWs small effect from the different S-wave extrapolation for t > 1.3 GeV negligible effect of ρ and ρ very significant effects of the D-waves (f 2 (1275)) F-waves shown to be negligible matching conditions (close to Wm) scattering length (SL) errors (on S-waves and subthreshold parameters) very important for the σ πn M. Hoferichter & J. Ruiz de Elvira (INT & HISKP) Roy Steiner-equation analysis of πn scattering Bonn, December 8th,