Roy-Steiner equations for πn

Transcription

1 Roy-Steiner equations for πn J. Ruiz de Elvira Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universität Bonn The Seventh International Symposium on Chiral Symmetry Beijing, October 2013 In collaboration with: C. Ditsche, M. Hoferichter, B. Kubis, U.-G. Meißner. [JHEP 1206 (2012) 043] and [JHEP 1206 (2012) 063]

2 Outline 1 Motivation: Why Roy-Steiner equations for πn? 2 Introduction: Roy(Steiner)-equations for ππ vs πn 3 Roy-Steiner equations for πn 4 Solving Roy-Steiner equations for πn Solving the t-channel subproblem Solving the s-channel subproblem 5 Summary & Outlook

3 Contents 1 Motivation: Why Roy-Steiner equations for πn? 2 Introduction: Roy(Steiner)-equations for ππ vs πn 3 Roy-Steiner equations for πn 4 Solving Roy-Steiner equations for πn Solving the t-channel subproblem Solving the s-channel subproblem 5 Summary & Outlook

4 Motivation: Why πn scattering? Low energies: test chiral dynamics in the baryon sector low-energy theorems e.g. for the scattering lengths Higher energies: resonances, baryon spectrum Input for NN scattering: LECs c i, πnn coupling Crossed channel ππ NN: nucleon form factors probe the structure of the nucleon σ πn term ππnn

5 Motivation: Why Roy-Steiner equations? Roy(-Steiner) eqs. = Partial-Wave (Hyberbolic) Dispersion Relations coupled by unitarity and crossing symmetry Respect all symmetries: analyticity, unitarity, crossing Model independent the actual parametrization of the data is irrelevant once it is used in the integral. Framework allows for systematic improvements (subtractions, higher partial waves,...) PW(H)DRs help to study processes with high precision: ππ-scattering: [Ananthanarayan et al. (2001), García-Martín et al. (2011)] πk-scattering: [Büttiker et al. (2004)] γγ ππ scattering: [Hoferichter et al. (2011)]

6 Contents 1 Motivation: Why Roy-Steiner equations for πn? 2 Introduction: Roy(Steiner)-equations for ππ vs πn 3 Roy-Steiner equations for πn 4 Solving Roy-Steiner equations for πn Solving the t-channel subproblem Solving the s-channel subproblem 5 Summary & Outlook

7 Warm up: Roy-equations for ππ ππ ππ fully crossing symmetric in Mandelstam variables s, t, and u = 4M π s t Start from twice-subtracted fixed-t DRs of the generic form T I (s, t) = c(t) + 1 π 4m 2 π [ ds s 2 s 2 (s s) + u2 (s u) ] ImT I (s, t) Subtraction functions c(t) are determined via crossing symmetry functions of the I=0,2 scattering lengths: a 0 0 and a 2 0 PW-expansion of these DRs yields the Roy-equations [Roy (1971)] tj(s) I = STJ(s) I + (2J + 1) J =0 I =0,1,2 4m 2 π K II JJ (s, s) kernels analytically known ds K II JJ (s, s)t I J (s )

8 Solving Roy-equations: flow information Roy-equations rigorously valid for a finite energy range introduce a matching point s m only partial waves with J J max are solved assume isospin limit Input Output High-energy region: Imt IJ (s) for s s m and for all J Higher partial waves: Imt IJ (s) for J > J max and for all s Self-consistent solution for δ IJ (s) for J J max and s th s s m Constraints on subtraction constants

9 Roy-Steiner equations for πn: difficulties Key difficulties compared to ππ Roy-equations Crossing: coupling between πn πn (s-channel) and ππ NN (t -channel) hyperbolic dispersion relations [Hite, Steiner 1973], [Büttiker et al. 2004] Unitarity in t-channel, e.g. in single-channel approximation Imf J ±(t) = σ π t f J ±(t)t l J(t) t I J f I J Watson s theorem: phase of f J ±(t) equals δ IJ [Watson 1954] solve with Muskhelishvili-Omnès techniques [Muskhelishvili 1953, Omnès 1958] { } Omnès function: Ω I J (t) = exp t tm π t dt δj I (t) th t (t t) Large pseudo-physical region in t -channel KK intermediate states for s-wave in the region of the f 0(980)

10 Contents 1 Motivation: Why Roy-Steiner equations for πn? 2 Introduction: Roy(Steiner)-equations for ππ vs πn 3 Roy-Steiner equations for πn 4 Solving Roy-Steiner equations for πn Solving the t-channel subproblem Solving the s-channel subproblem 5 Summary & Outlook

11 πn-scattering basics π a (q) + N(p) π b (q ) + N(p ) 200 Isospin Structure: T ba = δ ba T + + ɛ ab T Lorentz Structure: I {+, } ) T I = ū(p ) (A I q+ q + B 2 I u(p) D I = A I + νb I, ν = s u 4m Isospin basis: I s {1/2, 3/2} {T +, T } {T 1/2, T 3/2 } PW projection: s-channel pw: fl± I t-channel pw: f± J Bose symmetry even/odd J I = +/ u M Π zs 1 zu 1 t u t Ν s M Π 2 s

12 πn-scattering basics: Unitarity relations s-channel unitarity relations (I s {1/2, 3/2}): Im f Is l± (W) = q f Is l± (W) 2 θ ( ( ) 1 η I s l± W W (W)) θ ( ) W W inel 4q f I l± f I l± t-channel unitarity relations: 2-body intermediate states: ππ + KK + Im f± J (t) = ( σπ t t I t J (t)) f J ± (t) θ ( ) t t π + 2cJ 2 k 2J t σt K ( g I t J (t)) h J ± (t) θ ( t t K ) N π π N K π N f J ± π t I J π + N h J ± K g I J π Only linear in f J ±(t) less restrictive

13 Roy-Steiner equations for πn: HDR s Hyperbolic DRs:(s a)(u a) = b = (s a)(u a) with a, b R A + (s, t; a) = 1 π s + ds B + (s, t; a) = N + (s, t) + 1 π [ 1 s s + 1 s u 1 ] s Im A + (s, t ) + 1 a π s + ds N + (s, t) = g 2 ( 1 m 2 s 1 m 2 u Why HDR? t π dt Im A+(s, t ) t t [ 1 s s 1 ] s Im B + (s, t ) + 1 dt ν Im B + (s, t ) u π ν t t t π ) similar for A, B and N [Hite/Steiner (1973)] Combine all physical regions crucial for t-channel projection Evade double-spectral regions the PW decompositions converge Range of convergence can be maximized by tuning the free hyperbola parameter a No kinematical cuts, manageable kernel functions

14 Roy-Steiner equations for πn: derivation Recipe to derive Roy-Steiner equations: Expand imaginary parts in terms of s- and t-channel partial waves Project onto s- and t-channel partial waves Combine the resulting equations using s- and t-channel PW unitarity relations Similar structure to ππ Roy equations Validity: assuming Mandelstam analyticity s-channel optimal for a = 23.2Mπ 2 s [ s + = (m + M π) 2, Mπ 2 ] W [W + = 1.08 GeV, 1.38 GeV] t-channel optimal for a = 2.71M 2 π t [t π = 4M 2 π, M 2 π] t [ t π = 0.28 GeV, 2.00 GeV].

15 Roy-Steiner equations for πn: subtractions Subtractions are necessary to ensure the convergence of DR integrals asymptotic behavior Can be introduced to lessen the dependence of the low-energy solution on the high-energy behavior Parametrize high-energy information in (a priori unknown) subtraction constants matching to ChPT Subthreshold expansion around ν = t = 0 where Ā + (ν, t) = a + mn ν2m t n Ā (ν, t) = m,n=0 a mn ν2m+1 t n m,n=0 Ā + (s, t) = A + (s, t) g2 m Ā (s, t) = A (s, t), similar expansion for B + (s, t) and B (s, t)

16 Solving t-channel Solving s-channel Contents 1 Motivation: Why Roy-Steiner equations for πn? 2 Introduction: Roy(Steiner)-equations for ππ vs πn 3 Roy-Steiner equations for πn 4 Solving Roy-Steiner equations for πn Solving the t-channel subproblem Solving the s-channel subproblem 5 Summary & Outlook

17 Solving t-channel Solving s-channel Solving Roy-Steiner equations for πn: Recoupling schemes s-channel subproblem: Kernels are diagonal for I {+, }, but unitarity relations are diagonal for I s {1/2, 3/2} all partial-waves are interrelated Once the t-channel PWs are known Structure similar to ππ Roy-equations t-channel subproblem: Only higher PWs couple to lower ones Only PWs with even or odd J are coupled No contribution from f J + to f J+1 Leads to Muskhelishvili-Omnès problem

18 Solving t-channel Solving s-channel Solving t-channel equations Elastic-channel approximation: generic form of the integral equation f (t) = (t) + (a + bt)(t 4m 2 ) + t2 (t 4m 2 ) π dt Im f (t ) t t (t 2 4m 2 )(t t) (t): Born terms, s-channel integrals, higher t -channel partial waves left-hand cut Introduce subtractions at ν = t = 0 subthreshold parameters a, b Solution in terms of Omnès function: f (t) = (t) + (t 4m 2 )Ω(t)(1 t Ω(0))a + t(t 4m 2 )Ω(t)b { Ω(t) t2 (t 4m 2 tm ) dt (t )Im Ω(t ) 1 π t 2 (t 4m 2 )(t t) + { t Ω(t) = exp π t π t m 4M 2 π } dt δ(t ) t t t 4M 2 π } dt Ω(t ) 1 Im f (t ) t ( t 4m 2 )(t t)

19 Solving t-channel Solving s-channel Solving t-channel: P-wave results Important for electromagnetic nucleon form factors elastic channel approximation t m = 0.98 GeV and Im f J ±(t) = 0 above First step: check consistency with KH80 [Höhler 1983] Input needed: ππ phase shifts: [Caprini, Colangelo, Leutwyler, (in preparation)] πn phase shifts: SAID [Arndt et al. 2008], KH80 πn at high energies: Regge model [Huang et al. 2010] πn parameters: KH80 f 1 + n-sub / GeV 1 2Mπ Mπ tm 0-sub( a ) 0-sub 1-sub( a ) 1-sub 2-sub( a ) 2-sub KH80 f 1 n-sub / GeV 2 2Mπ Mπ tm 0-sub( a ) 0-sub 1-sub( a ) 1-sub 2-sub( a ) 2-sub KH t / GeV t / GeV MO solutions in general consistent with KH80 results

20 Solving t-channel Solving s-channel Solving t-channel: S-wave results Important for the σ πn term KK channel important two-channel Muskhelishvili-Omnès problem also needed: K K s-wave partial waves: [Büttiker. (2004)] KN s-wave pw: SAID [Arndt et al. 2008], KH80 Hyperon couplings from [Jülich model 1989] KN subthreshold parameters neglected MO solutions in general consistent with KH80 results

21 Solving t-channel Solving s-channel Solving s-channel: General form General form of the s-channel integral equation fl+ I (W) = I l+ (W) + 1 { } dw Kll I π (W, W ) Im fl I + (W ) + Kll I (W, W ) Im f(l I +1) (W ) W l =0 + form of ππ Roy-Equations I l+(w) t-channel contribution and pole term valid up to W m = 1.38 GeV Input: Output: RS t-channel solutions for S and P waves s-channel partial waves for J > 1 [SAID analysis] s-channel partial waves for W m < W < 2.5 GeV [SAID analysis] high energy contribution for W > 2.5 GeV: Regge model [Huang et al. 2010] Self-consistent solution for S and P waves for s th s s m Constraints on subtraction constants subthreshold parameters

22 Solving t-channel Solving s-channel Solving s-channel: consistency with KH80 Consistency with KH80 parametrize SAID S and P waves up to W<Wm Imposing a continuous and differentiable matching point Compare between the input (LHS) and the output (RHS) important discrepancies

23 Solving t-channel Solving s-channel Solving s-channel: fitting subthreshold parameters Next step fit only subthreshold parameters, but keep phase shifts fixed Minimize the χ-like function: χ 2 = ( N l,i s,± j=1 Re f Is l± (W j) F[f Is l± ](W j) F[f Is l± ](Wj) right hand side of RS-equations subthreshold parameters change in less than 10% ) 2 Still important differences, especially for the S-waves the pw parametrizations have to be included in the fit In progress

24 Contents 1 Motivation: Why Roy-Steiner equations for πn? 2 Introduction: Roy(Steiner)-equations for ππ vs πn 3 Roy-Steiner equations for πn 4 Solving Roy-Steiner equations for πn Solving the t-channel subproblem Solving the s-channel subproblem 5 Summary & Outlook

25 Summary & Outlook What has been done: Derived a closed system of Roy-Steiner equations (PWHDRs) for πn scattering Constructed unitarity relations including KK intermediate states for the t-channel PWs Optimized the range of convergence by tuning a for s- and t-channel each Implemented subtractions at several orders Solved the t-channel MO problem for a single- and two-channel approximation t-channel RS/MO machinery works Numerical solution of the s-channel subproblem fitting the subthreshold parameters What needs to be done: Self-consistent, iterative solution of the full RS system lowest PWs & low-energy parameters Possible improvements: higher PWs, more inelastic input,...

26 Spare slides

27 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 + 0+ f + 1+ f + 1 High-energy region Im f I l±, s sm f 0+ 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 0 + f 1 ± f 2 ± f 3 ± f 4 ± High-energy region Im f J ±, t tm

28 πn-scattering basics: partial waves s-channel projection: fl± I (W) = 1 {(E + m) [ A I l 16πW (s) + (W m)bi l (s)] + (E m) [ A I l±1 (s) + (W + m)bi l±1 (s)]} 1 Xl I (s) = dz s P l (z s)x I (s, t) for X {A, B} and W = s 1 t=t(s,zs)= 2q 2 (1 z s) McDowell symmetry: f I l+ (W) = f I (l+1) ( W) l 0 t-channel projection: f+ J (t) = 1 1 { p 2 dz t P J (z t t) 4π (p tq t) J AI (s, t) m } s=s(t,zt) (p tq t) J 1 ztbi (s, t) s=s(t,zt) 0 f J (t) = 1 J(J + 1) 1 1 ] 4π 2J + 1 (p tq t) J 1 dz t [P J 1 (z t) P J+1 (z t) B I (s, t) s=s(t,zt) 0 J 0 J 1 Bose symmetry even/odd J I = +/

29 Roy-Steiner equations for πn: s-channel s-channel RS equations f I l+ (W) = NI l+ (W) + 1 { dw K I π W + l ll (W, W ) Im f I l + (W ) + K I ll (W, W ) Im f I } (l +1) (W ) =0 + 1 dt π tπ J { G lj (W, t ) Im f J + (t ) + H lj (W, t ) Im f J } (t ) = f I (l+1) ( W) l 0, [Hite/Steiner (1973)] K I ll (W, W ), G lj (W, t ) and H lj (W, t )-Kernels: analytically known, e.g. K I ll (W, W ) = δ ll W W +... l, l 0, Validity: assuming Mandelstam analyticity optimal for a = 23.2M 2 π s [ s + = (m + M π) 2, Mπ 2 ] W [W + = 1.08 GeV, 1.38 GeV]

30 Roy-Steiner equations for πn: t-channel t-channel RS equations f J + (t) = ÑJ + (t) + 1 dw { G Jl (t, W ) Im f I l+ π (W ) + G Jl (t, W ) Im f I } (l+1) (W ) W l= π tπ f J (t) = ÑJ (t) + 1 dw π W l= π tπ dt { K 1 J JJ (t, t ) Im f J + (t ) + K 2 JJ (t, t ) Im f J } (t ) J 0, { H Jl (t, W ) Im f I l+ (W ) + H Jl (t, W ) Im f I } (l+1) (W ) dt J K 3 JJ (t, t ) Im f J (t ) J 1, Validity: assuming Mandelstam analyticity optimal for a = 2.71M 2 π t [t π = 4M 2 π, M 2 π] t [ t π = 0.28 GeV, 2.00 GeV].

31 RS-eqs for πn: Range of convergence Subthreshold expansion around ν = t = 0 A + (ν, t) = g2 m + d d+ 01 t + a+ 10 ν2 + O ( ν 2 t, t 2) A (ν, t) = νa 00 + a 01 νt + a 10 ν3 + O ( ν 5, νt 2, ν 3 t ) B + (ν, t) = g 2 4mν (m 2 s 0 ) 2 + νb O( ν 3, νt ), [ ] B (ν, t) = g 2 2 t m 2 s 0 (m 2 s 0 ) 2 g2 2m 2 + b 00 + b 01 t + +b 10 ν2 + O ( ν 2, ν 2 t, t 2) pseudovector Born terms: D I = A I + νb I Sum rules for subthreshold parameters: D + = d d+ 01 t + d+ 10 ν2 d + mn = a + mn + b + m 1,n, d mn = a mn + b mn. d + 00 = g2 m + 1 ds h 0 (s [ ) Im A + (s, z π s) ] (0,0) + 1 dt [ Im A + π t (t, z t ) ] (0,0) s + t π h 0 (s ) = 2 s s 0 1 s a

32 RS-eqs for πn: Range of convergence Assumption: Mandelstam analyticity [Mandelstam (1958,1959)] T(s,t) can be written in terms double spectral densities: ρ st, ρ su, ρ ut T(s, t) = 1 π 2 ds du ρ su(s,u ) (s s)(u u) + 1 π 2 dt du ρ tu(t,u ) (t t)(u u) + 1 π 2 ds dt ρ st(s,t ) (s s)(t t) integration ranges defined by the support of the double spectral densities ρ Boundaries of ρ are given lowest lying intermediate states 200 (I) (II) (III) (IV) Ρut Ρsu u M Π 2 50 They limit the range of validity of the HDRS: 0 t Ν Pw expansion converge 50 Ρst z = cos θ Lehman ellipses [Lehmann (1958)] the hyperbolae (s a)(u a) = b does not enter any double spectral region for a value of a, constraints on b yield ranges in s & t s M Π 2

33 Solving t-channel: P-wave results f 1 + n-sub / GeV 1 2Mπ Mπ tm 0-sub( a ) 0-sub 1-sub( a ) 1-sub 2-sub( a ) 2-sub KH80 f 1 n-sub / GeV 2 2Mπ Mπ tm 0-sub( a ) 0-sub 1-sub( a ) 1-sub 2-sub( a ) 2-sub KH t / GeV t / GeV f 1 + n-sub / GeV 1 2Mπ Mπ sub( tm = 1.1 GeV) 0-sub 1-sub( tm = 1.1 GeV) 1-sub 2-sub( tm = 1.1 GeV) 2-sub KH80 f 1 n-sub / GeV 2 2Mπ Mπ sub( tm = 1.1 GeV) 0-sub 1-sub( tm = 1.1 GeV) 1-sub 2-sub( tm = 1.1 GeV) 2-sub KH t / GeV t / GeV MO solutions in general consistent with KH80 results

34 Solving t-channel equations: S-waves Generic coupled-channel integral equation t m f(t) = (t) + 1 π t π dt T (t )Σ(t )f(t ) t t + 1 π t m dt Im f(t ) t t Formal solution as in the single-channel case (now with Omnès matrix Ω(t)) Two-channel ( Muskhelishvili-Omnès ) problem f 0 f(t) = + (t) h 0 Im Ω(t) = (T(t)) Σ(t)Ω(t) +(t) Two linearly independent solutions Ω 1, Ω 2 [Muskhelishvili 1953] In general no analytical solution for the Omnès matrix but for its determinant [Moussallam 2000] { t det Ω(t) = exp π t m dt ψ(t ) t t (t t) }.

35 Solving t-channel S-wave equations: input Input needed: ππ s-wave partial waves: [Caprini, Colangelo, Leutwyler, (in preparation)] K K s-wave partial waves: [Büttiker. (2004)] πn and KN s-wave pw: SAID [Arndt et al. 2008], KH80 πn at high energies: Regge model [Huang et al. 2010] πn parameters: KH80 Hyperon couplings from [Jülich model 1989] KN subthreshold parameters neglected Two-channel approximation beaks down at t 0 = 1.3 GeV 4π channel From t 0 to t = 2 GeV, different approximations considered

36 Solving t-channel: S-wave results MO solutions in general consistent with KH80 results

37 Solving s-channel: threshold parameters Precise data for pionic atoms [Gotta et al. 2005, 2010] Impose as a constraint scattering lengths from a combined analysis of pionic hydrogen and deuterium [Baru et al. 2011] a 1/2 0+ = (170.5 ± 2.0)10 3 Mπ 1 a 3/2 0+ = ( 86.5 ± 1.8)10 3 Mπ 1 ( ) Re fl± I (s) = q2l a I l± + bi l± q2 + s-channel scattering lengths from RS sum rules 1-sub 2-sub 3-sub KH80 S11 a 1/2 0+ [10 3 Mπ 1 ] ±3 S31 a 3/2 0+ [10 3 M 1 π ] ±4 three subtractions needed

38 Solving s-channel: consistency with KH80 Consistency with KH80 parametrize SAID S and P waves up to W<Wm Imposing a continuous and differentiable matching point Compare between the input (LHS) and the output (RHS) S-WAVES important discrepancies

39 Solving s-channel: consistency with KH80. P-waves

40 Solving s-channel: fitting subthreshold parameters Next step fit only subthreshold parameters, but keep phase shifts fixed Minimize the χ-like function: χ 2 = N l,i s,± j=1 ( Re f I s Is l± (Wj) F[fl± ](Wj)) 2 F[f Is l± ](W j) right hand side of RS-equations small change in the subthreshold parameters

41 Solving s-channel: fitting subthreshold parameters. P-waves

42 Solving s-channel: fitting subthreshold parameters. S-waves Still important differences, especially for the S-waves the pw parametrizations have to be included in the fit In progress