η 3π and light quark masses - PDF Free Download

η 3π and light quark masses Theoretical Division Los Alamos National Laboratory Jefferson Laboratory, Newport News November 5, 013 In collaboration with G. Colangelo, H. Leutwyler and S. Lanz (ITP-Bern)

Outline : 1. Introduction and Motivation. How to measure the light quark masses? 3. η π + π - π 0 decays 4. Dispersive analysis of η π + π - π 0 decays 5. Preliminary results 6. Conclusion and outlook

1. Introduction and Motivation

1.1 Quantum Chromodynamics Description of the strong interactions L QCD =! 1 4 G µ" a G a µ" N F + # q k ( i$ µ D µ! m k )q k k=1 G a µ! (x) = " µ A a! # "! A a µ + g S f abc A b c µ A! # D µ =! µ " ig a s A a (x) µ 7 unknowns in the Lagrangian: Ø strong coupling constant Ø 6 quark masses m k! S (µ) = g (µ) S 4" Not predicted by the theory, should be measured by experiment Problem: no direct access to the quarks due to confinement! 4

1. Why measuring the quark masses? Fundamental unknowns of the Lagrangian In the following, consider the 3 light flavours u,d,s High precision physics at low energy as a key of new physics? m d - m u : small isospin breaking corrections but to be taken into account for high precision physics Ex: V us from K ± 0 l 3 ( K ± π l ± νl ) decays NA6, KLOE- u s g V us W µ,e g ν µ,e + LANL, Los Alamos, 11 January 011

1.3 QCD at low energy At low energy, impossible to describe QCD with perturbation theory since α S becomes large [PDG 1] Need non perturbative methods Model independent methods: Effective field theory Ex: ChPT for light quarks Numerical simulations on the lattice Dispersion relations Confinement 6

1.4 Dispersion Relations Method that relies on analyticity, unitarity and crossing symmetry Model independent Connect different energy regions Summation of all the rescattering processes Very successful for describing hadronic decays at low energy, e.g. Ø ππ scattering, ππ form factors Ananthanarayan et al 01, Descotes-Genon et al 01 Pich & Portoles 01, Gomez-Dumm & Roig 13 Decay of a light Higgs boson Donoghue, Gasser & Leutwyler 98 Probing lepton flavour violating couplings of the Higgs from τ ππν τ Celis, Cirigliano & E.P. 13 7

1.4 Dispersion Relations Method that relies on analyticity, unitarity and crossing symmetry Model independent Connect different energy regions Summation of all the rescattering processes Very successful for describing hadronic decays at low energy, e.g. Ø π scattering, π form factors Buttiker et al 01, Jamin, Oller & Pich 01 Bernard, Oertel, E.P., Stern 06 09, Bernard & E.P. 08 Determination of V us from K l3 decays Flavianet Kaon WG 08, 10 hadronic τ decays Antonelli, Cirigliano, Lusiani & E.P. 13 8

. How to measure the light quark masses?

.1 Meson masses from ChPT m u,d,s!! QCD : masses treated as small perturbations expansion in powers of Gell-Mann-Oakes-Renner relations: (meson mass) = (spontaneous ChSB) x (explicit ChSB) m q qq m q From LO ChPT without e.m effects: Electromagnetic effects: Dashen s theorem ( M M 0) ( M M 0) O ( e m + + = ) K K em π π em Dashen 69 10

.1 Meson masses from ChPT Quark mass ratios Weinberg 77 1

. Lattice QCD Compute the quark masses from first principles L QCD on the lattice Ø QCD Lagrangian as input Ø Calculate the spectrum of the low-lying states for different quark masses Ø Tune the values of the quark masses such that the QCD spectrum is reproduced Ø Set the scale by adding an external input or extract quark mass ratios NB: computation in the isospin limit: m = m = mˆ To get m, needs handle on e.m. effects: u! m d Ø Input from phenomenology (e.g., Kaon mass difference) u d m u + m d Ø Put photons on the lattice See FLAG 10 13 13

.3 η π + π - π 0 Decay forbidden by isospin symmetry A = ( m u! m d ) A 1 + " em A α em effects are small Sutherland 66, Bell & Sutherland 68 Baur, Kambor, Wyler 96, Ditsche, Kubis, Meissner 09 Decay rate measures the size of isospin breaking (m u m d ) in the SM: L QCD L IB ( ) mu md = uu dd Clean access to (m u m d ) 14

.4 Quark mass ratios Mass formulae to second chiral order Gasser & Leutwyler 85 with The same O(m) correction appears in both ratios Take the double ratio Q m m mˆ s d mu = M K M! M K " M! M K 0 " M $ 1 + O(m ( K + ) q QCD #,e )% & " $ # m!! m d + m u % ' & Very Interesting quantity to determine since Q does not receive any correction at NLO! 15

.4 Quark mass ratios The same O(m) correction appears in both ratios Take the double ratio Q m m mˆ s d mu = M K M! M K " M! M K 0 " M $ 1 + O(m ( K + ) q QCD #,e )% & Very Interesting quantity to determine since Q does not receive any correction at NLO! Using Dashen s theorem and inserting Weinberg LO values Q D = 4. 16

.4 Quark mass ratios From Q Ellipse in the plane m s /m d, m u /m d Leutwyler s ellipse 17

.5 Quark mass ratios Estimate of Q: B 0 ( m u! m d ) = 1 M K Q M K! M " ( ) M " + O(M 3 ) Ø From corrections to the Dashen s theorem ( ) = ( 0) ( 0) ( + + + ) B m m M M M M O e m 0 d u K K The corrections can be large due to e m s corrections, difficult to estimate due to LECs π π Ø From η π + π - π 0 : A!"3# $ B 0 ( m u % m d )! "#3$ % A % Q &4 18

.4 Quark mass ratios m Use Q to determine and from lattice determinations of and u m d m s ˆm m u = mˆ m mˆ 4mQ ˆ s and m d = mˆ + m mˆ 4mQ ˆ s m ˆm Q From lattice determinations of and + Light quark masses: m u, m d, m s s In the following, extraction of Q from η π + π - π 0 19

3. η π + π - π 0 decays

3.1 Definitions η decay: η π + π - π 0 4 ( ) ( 0 ) + 0 4 π π π η = π δ η + π π π out i p p p p A( s, t, u) Mandelstam variables ( ) = + +, π π t = ( p + p 0 π π ), u= ( p ) 0 + p s p p π π + s+ t+ u= Mη + M 0 + M 3s π π + 0 only two independent variables Current Algebra ( ) 3( ) B0 md mu s s 0 Astu (,, ) = 1 ( ) ( ) + + Om Oem + 3 3F Mη M π π Osborn, Wallace 70 Relate the amplitude to meson masses using Dashen s theorem LO chiral prediction: Γ = ± Problem! Γ η 3π = 66 ev and exp 197 9 ev in 1985 1

3. Solution of the puzzle Discrepancy current algebra vs. experiment discovered and discussed in the 70 s Solution found in the 80 s: Large final state interactions Roiesnel & Truong 81 η

3.3 Solution of the puzzle Chiral Perturbation Theory Systematic method to take into account these effects At one loop, only one LEC related to ππ scattering Γ η 3π = 160 ± 50 ev Gasser & Leutwyler 85 3 ( ) L 3 = 3.5 ± 1.1 10 Important theoretical error: ± 50 ev estimate of the higher order corrections, typical SU(3) error of 5% Seemed to solve the problem ( Γ exp = 197 ± 9 ev in 1985), but now Γ exp = 95 ± 0 ev! 3

3.4 Amplitude beyond one loop Possible sources of discrepancy Ø Electromagnetic effects, control of Ο(e m) corrections ChPT with photons: corrections small of ~1% Baur, Kambor & Wyler 95 Ø Higher order corrections: ChPT at two loops but many LECs to determine at O(p6)! Use of dispersion relations Ø analyticity, unitarity and crossing symmetry Bijnens & Ghorbani 07 Kambor, Wiesendanger & Wyler 96 Anisovich & Leutwyler 96 Ø Take into account all the rescattering effects + +... 4

3.5 Neutral Channel : η π 0 π 0 π 0 Decay amplitude η 3π A 1 α Γ + Z with Z 3 3T i = 1 3 i= 1 Qn Q M 3M n η π 0!! 0.015 α = 0.0317 ± 0.0016 PDG 10 Important discrepancy between ChPT and experiment! Help of a dispersive treatment? 5

4. Dispersive Analysis of η π + π - π 0 decays

4.1 A new dispersive analysis Dispersive analysis in the 90 s Why a new analysis? Kambor, Wiesendanger & Wyler 96 Anisovich & Leutwyler 96 Walker 97 Ø New inputs available: extraction ππ phase shifts has improved Ananthanarayan et al 01, Colangelo et al 01 Descotes-Genon et al 01 Kaminsky et al 01, Garcia-Martin et al 09 Ø New experimental programs, precise Dalitz plot measurements CBall-Brookhaven, CLAS (Jlab), KLOE (Frascati) TAPS/CBall-MAMI (Mainz), WASA-Celsius (Uppsala), WASA-Cosy (Juelich) Ø Possible improvements: Inelasticity Electromagnetic effects, complete analysis of O(e m) effects Ditche, Kubis, Meissner 09 Isospin breaking effects: new techniques NREFT Ditche, Kubis, Meissner, Rusetsky, Schneider 09

4.1 A new dispersive analysis Compare to other approaches Ø η 3π computed at NNLO in ChPT Bijnens & Ghorbani 07 Ø η 3π with analytical dispersive method Kampf, Knecht, Novotný, Zhadral 11 Strategy: New dispersive analysis with new inputs following Anisovich & Leutwyler approach for charged decays but Ø Electromagnetic effects neglected except for the phase space Ø Isospin limit Future improvements possible and planned Aim: determine Q with the best precision 8

4.1 A new dispersive analysis Determination of Q: Γ η 3π A Ø Γ η 3π experimentally measured Ø 1 (,, ) M M K Astu = M M(,, stu) Q M F K π π 3 3 π Mstu (,, ) computed from dispersive treatment Extraction of Q Neutral channel: η π 0 π 0 π 0 Astu (,, ) = Astu (,, ) + Atus (,,) + Aust (,,) 9

4. Method: Representation of the amplitude Dispersion relations From the discontinuity, reconstruct the amplitude everywhere in the complex plane need the discontinuity 30

4. Method: Representation of the amplitude Dispersion relations From the discontinuity, reconstruct the amplitude everywhere in the complex plane need the discontinuity 31

4. Method: Representation of the amplitude Dispersion relations From the discontinuity, reconstruct the amplitude everywhere in the complex plane need the discontinuity 3

4. Method: Representation of the amplitude Dispersion relations From the discontinuity, reconstruct the amplitude everywhere in the complex plane need the discontinuity 33

4. Method: Representation of the amplitude Decomposition of the amplitude as a function of isospin states M(,, s t u) = M0() s + ( s u) M1() t + ( s t) M1( u) + M() t + M( u) M() s 3 Ø M I Fuchs, Sazdjian & Stern 93 Anisovich & Leutwyler 96 isospin I rescattering in two particles Ø Amplitude in terms of S and P waves exact up to NNLO (O(p 6 )) Ø Main two body rescattering corrections inside M I Dispersion relation for the M I s ( M I (s) =! I (s) P I (s) + sn ) * " # $ 4 M " ds' s' n sin% I (s') ˆM I (s') ( )! I (s') s'& s & i' +, -. ( 0! I (s) = exp s 0 ) * " / & ' 4 M " # ds' I (s') + 1 3 s'(s'$ s $ i% ), - 3 Omnès function 34

4. Method: Representation of the amplitude Decomposition of the amplitude as a function of isospin states M(,, s t u) = M0() s + ( s u) M1() t + ( s t) M1( u) + M() t + M( u) M() s 3 Ø M I Fuchs, Sazdjian & Stern 93 Anisovich & Leutwyler 96 isospin I rescattering in two particles Ø Amplitude in terms of S and P waves exact up to NNLO (O(p 6 )) Ø Main two body rescattering corrections inside M I Dispersion relation for the M I s ( M I (s) =! I (s) P I (s) + sn ) * " # $ 4 M " ds' sin% I (s') ˆM I (s') + s' n! I (s') ( s'& s & i' ), -. ( 0! I (s) = exp s 0 ) * " / Omnès function ˆM I (s) : singularities in the t and u channels, depend on the other M I (s) subtract M I (s) from the partial wave projection of Mstu (,, ) Angular averages of the other functions Coupled equations & ' 4 M " # ds' I (s') + 1 3 s'(s'$ s $ i% ), - 3 36

4.3 Iterative Procedure 38

4.4 Subtraction constants Extension of the numbers of parameters compared to Anisovich & Leutwyler 96 P 0 (s) =! 0 + " 0 s + # 0 s + $ 0 s 3 P 1 (s) =! 1 + " 1 s + # 1 s P (s) =! + " s + # s In the work of Anisovich & Leutwyler 96 matching to one loop ChPT Use of the SU() x SU() chiral theorem The amplitude has an Adler zero along the line s=u Now data on the Dalitz plot exist from KLOE, WASA and MAMI Use the data to directly fit the subtraction constants Solution linear in the subtraction constants Anisovich & Leutwyler 96 M(s, t,u) =! 0 M! (s, t,u) + " 0 0 M "0 (s, t,u) +... makes the fit much easier 39

4.4 Subtraction constants Adler zero: the real part of the amplitude along the line s=u has a zero Anisovich & Leutwyler 96 40

Experimental measurements Dalitz plot measurement : Amplitude expanded in X and Y around X=Y=0 A( s, t,u) =!(X,Y ) = N ( 1 + ay + by + dx + fy 3 ) with Q M M M c η + 0 π π 3 X = u t MQ η c ( ) ( ( ) ) η 0 π 3 Y = M M s 1 MQ η c Z = X + Y 41

Experimental measurements : Charged channel Charged channel measurements with high statistics from KLOE and WASA e.g. KLOE: ~1.3 x 10 6 η π + π - π 0 events from + ϕ η γ A c ( s, t,u) = N ( 1 + ay + by + dx + fy 3 ) KLOE 08 ( ( ) ) η 0 π 3 Y = M M s 1 Emilie MQ Passemar η c 3 X = u t MQ η c ( ) 4

Experimental measurements : Neutral channel Neutral channel measurements with high statistics from MAMI-B, MAMI-C and WASA e.g. MAMI-C: ~3 x 10 6 η 3π 0 events from γ p ηp A n ( s, t,u) $ $ = N 1 +! Z + 6"Y X # Y ' % & 3 ( ) + * Z ' & ) % ( Extraction of the slope : 3 X = u t MQ η c ( ) ( ( ) ) η 0 π 3 Y = M M s 1 MQ η c Z = X + Y MAMI-C 09

4.4 Subtraction constants As we have seen, only Dalitz plots are measured, unknown normalization! 1 (,, ) M M K Astu = M M(,, stu) Q M F K π π 3 3 π To determine Q, one needs to know the normalization For the normalization one needs to use ChPT The subtraction constants are P 0 (s) =! 0 + " 0 s + # 0 s + $ 0 s 3 P 1 (s) =! 1 + " 1 s + # 1 s P (s) =! + " s + # s Only 6 coefficients are of physical relevance 44

4.4 Subtraction constants The subtraction constants are P 0 (s) =! 0 + " 0 s + # 0 s + $ 0 s 3 P 1 (s) =! 1 + " 1 s + # 1 s P (s) =! + " s + # s Only 6 coefficients are of physical relevance They are determined from Matching to one loop ChPT! 0 = " 1 = 0 Combine ChPT with fit to the data! and! 0 1 are determined from the data Matching to one loop Ch : Taylor expand the dispersive M I Subtraction constants Taylor coefficients 46

4.4 Subtraction constants Matching to one loop Ch : Taylor expand the dispersive M I Subtraction constants Taylor coefficients M 0 (s) = a 0 + b 0 s + c 0 s + d 0 s 3 +... M 1 (s) = a 1 + b 1 s + c 1 s +... M (s) = a + b s + c s Ø gauge freedom a 0, b 0, a 1, a tree level ChPT values Ø fix the remaining ones with one-loop ChPT c 0, b 1, b, c Ø matching to one loop : d 0 = c 1 = 0 or fit : d 0 and c 1 from the data Problem : this identification assumes there is not significant contributions from higher orders of the chiral expansion not well-justified for the s 3 terms! Solution: Match the SU() x SU() expansion of the dispersive representation with the one of the one loop representation In progress Important : Adler zero should be reproduced! Can be used to constrain the fit 47

5. Preliminary Results

5.1 Dalitz plot distribution of η π + π - π 0 decays The amplitude squared along the line t = u : Good agreement between theory and experiment "M! # The theoretical error bars are large fit the subtraction constants to the data to reduce the uncertainties $ %

5. Z distribution for η π 0 π 0 π 0 decays The amplitude squared in the neutral channel is Here also the agreement looks very good but

5. Z distribution for η π 0 π 0 π 0 decays NRFT in η decays Gullstrom, Kupsc, Rusetsky 08 Schneider, Kubis, Ditsche 08 The uncertainties coming from the matching with ChPT are very large there is room for improvement using the data

5. Z distribution for η π 0 π 0 π 0 decays If one wants to fit the data, at this level of precision the e.m. corrections matter use the one loop e.m. calculations from Ditsche, Kubis and Meissner 08

5.3 Qualitative results of our analysis Determination of Q from the dispersive approach : 4 ( ) Γ = 1 M MK M max K π + 0 ds η π π π Q M 691π F M 4 4 3 4 3 smin u ( s) π π η s u+ ( s) du M( s, t, u) Γ η 3π = 95 ± 0 ev PDG 1 Q m m mˆ s d mu Determination of α A n ( s, t,u) = N ( 1 +! Z ) 53

5.3 Qualitative results of our analysis Plot of Q versus α : NB: Isospin breaking has not been accounted for From kaon mass spliting : Q = 0.7 ± 1. Kastner & Neufeld 08 All the data look consistent and push the value of Q towards its lower value in agreement with the kaon mass splitting determination 54

5.3 Qualitative results of our analysis Plot of Q versus α : NB: Isospin breaking has not been accounted for In our first results we manage to have the correct sign for α in agreement with experimental measurements. Improvement compared to previous analyses! 55

5.4 Comparison with KKNZ Amplitude along the line s=u Adler zero not reproduced! 56

5.4 Comparison with KKNZ Adler zero not reproduced! 57

5.5 Light quark masses Courtesy of H.Leutwyler Smaller values for Q smaller values for ms/md and mu/md than LO ChPT 58

5.5 Light quark masses 59

5.5 Light quark masses Courtesy of H.Leutwyler 60

6. Conclusion and outlook

6.1 Conclusion η 3π decays represent a very clean source of information on the quark mass ratio Q A reliable extraction of Q requires having the strong rescattering effects in the final state under control This is possible thanks to dispersion relations need to determine unknown subtraction constants This was done up to now relying exclusively on ChPT but precise measurements have become available Ø In the charged channel: KLOE and WASA Ø In the neutral channel: MAMI-B, MAMI-C, WASA Ø More results are expected: KLOE, CLAS will allow to reduce the uncertainties in the significant way seems to push the value for Q towards low results 6

6. Outlook Analysis still in progress : Ø Determination of the subtraction constants : combine ChPT and the data in the optimal way Ø Take into account the e.m. corrections implementation of the one loop e.m. corrections from Ditsche, Kubis and Meissner 08 to be able to fit to the data charged and neutral channel Ø Matching to NNLO ChPT Constraints from experiment: possible insights on Ø Careful estimate of all uncertainties Ø Inelasticities C i values Our preliminary results give a consistent picture between Ø all experimental measurements: Dalitz plot measurements Ø theoretical requirements: e.g. Adler zero 63

7. Back-up

Comparison with original analysis 65

M I 66

Comparison for Q 67

Comparison for α IFIC, Valencia 19 May 011 68

1.5 Quark masses But in the real world quarks are massive G also explicitly broken by quark masses 0 L = L + L with L m = qm q QCD QCD m and mu 0 0 M = 0 md 0 0 0 m s L m The mass term gives the masses to the Goldstone bosons 69

1.6 Construction of an effective theory: ChPT Effective Field Theory approach: At a given energy scale Ø Degrees of freedom Ø Symmetries Decoupling : Ex : To play pool you don t need to know the movement of earth around the sun Chiral Perturbation Theory (ChPT) CPT, Marseille, June 1st, 01 70

Method: Representation of the amplitude Consider the s channel Partial wave expansion of M(s,t,u): Mstu (,, ) = f() s+ f()cos s θ +... Elastic unitarity with 0 1 Watson s theorem disc f l() s tl() s fl() s t l () s partial wave of elastic ππ scattering M(s,t,u) right-hand branch cut in the complex s-plane starting at the ππ threshold Left-hand cut present due to crossing Same situation in the t- and u-channel 71

Discontinuities of the M I (s) 0 Mˆ 0 ( s ) = M0 + s s0 M1 + M + κ ( s ) zm1 3 9 3 1 n 1 n z M I s = (, ), dz z M 1 I t s z z = cosθ scattering angle Ex: ( ) where ( ) ( ) Non trivial angular averages path to avoid crossing cuts need to deform the integration Anisovich & Anselm 66 University of Lund, 6 May 011 7

3.7 Comparison of values of Q Q =.3 ± 0.9 Q =.1± 0.9 Q = 4. Q =.8 Q = Q = 1.5 Q = 0.7 Q = 0.7 ± 1. Fair agreement with the determination from meson masses 74

Comparison with Q from meson mass splitting M K MK Mπ Q = 1 + O( m ) q is only valid for e=0 M M M π 0 + K K Including the electromagnetic corrections, one has Q D = 4. Corrections to the Dashen s theorem The corrections can be large due to e m s corrections: ( M -M ) ( M -M ) = e M ( A ) ( 1+ A A O e M + 3 + ) K π K + 0 + 0 K em π π em Urech 98, Ananthanarayan & Moussallam 04 75

3.6 Corrections to Dashen s theorem Dashen s Theorem ( M - M ) ( M - M ) K + 0 + 0 K em = ( M M 0 ) π π em K - 1.3 MeV + = K em With higher order corrections Lattice : ENJL model: VMD: ( 0 ) M - M 1.9 MeV, Q.8 + = = Ducan et al. 96 K K ( 0 K K ) ( 0 ) em M - M.3 MeV, Q + = = Bijnens & Prades 97 em M - M.6 MeV, Q 1.5 + = = Donoghue & Perez 97 K K ( ) em 0 em Sum Rules: M - M 3. MeV, Q 0.7 + = = Anant & Moussallam 04 K K Update Q = 0.7 ± 1. Kastner & Neufeld 07 76

4. Method: Representation of the amplitude Decomposition of the amplitude as a function of isospin states M(,, s t u) = M0() s + ( s u) M1() t + ( s t) M1( u) + M() t + M( u) M() s 3 Ø M I isospin I rescattering in two particles Ø Amplitude in terms of S and P waves exact up to NNLO (O(p 6 )) Ø Main two body rescattering corrections inside M I Functions of only one variable with only right-hand cut of the partial wave I disc M I() s disc f () s l Fuchs, Sazdjian & Stern 93 Anisovich & Leutwyler 96 Elastic unitarity Watson s theorem I * I l () l() l () t () s disc f s t s f s with l scattering partial wave of elastic ππ 77

4. Method: Representation of the amplitude Knowing the discontinuity of write a dispersion relation for it Cauchy Theorem and Schwarz reflection principle 1 disc M I ( s') MI () s = ds' π s' s iε 4M π M I M I can be reconstructed everywhere from the knowledge of If M I doesn t converge fast enought for s subtract the dispersion relation M () s = P () s + n s π disc M I n 1 ' n 4M π I () s ds' disc M I ( s') s s s i ( ' ε) P n-1 (s) polynomial 78

4.3 Hat functions Discontinuity of : by definition I f () s = M () s + Mˆ () s l M I I I I disc M I() s disc f () s l with Mˆ () s I real on the right-hand cut The left-hand cut is contained in Mˆ () s I Determination of Mˆ : I () s subtract M from the partial wave projection of Mstu (,, ) I ( ) Mstu (,, ) = M() s+ s um() t+... 0 1 Mˆ singularities in the t and u channels, depend on the other I () s Angular averages of the other functions Coupled equations M I 79

4.3 Hat functions Ex: Mˆ ( ) 0 ( s ) = M + s s M + M + κ ( s ) zm 3 9 3 0 0 0 1 1 1 = 1 n n where z M ( s) dz z M ( t s z ) I 1 I (, ), z = cosθ scattering angle Non trivial angular averages path to avoid crossing cuts need to deform the integration Anisovich & Anselm 66 80

4.4 Dispersion Relations for the M I (s) Elastic Unitarity [ l = 1 for I = 1, l = 0 otherwise ] I I ( ) ( ) ˆ I i s I ( ) θ 4 ( ) ( ) l = l = π I + I sin δl ( ) I I δ ππ phase shift l phase of the partial wave fl ( s) disc M disc f s s M M s M s s e δ Watson theorem: elastic ππ scattering phase shifts Solution: Inhommogeneous Omnès problem M s s s s 3 0 s ds' δ 0 0 0( ) =Ω 0( ) α0+ β0 + γ 0 + 3 π ' 4 0( ') ' M Omnès function Similarly for M 1 and M ( ε) 81 π sin ( s') Mˆ ( s') s Ω s s s i s Ω I () = exp ' I δ ( s') '( ' ε) l s ds π 4M s s s i π

4.4 Dispersion Relations for the M I (s) M ( s) =Ω ( s) α + β s+ γ s + 3 0 s ds' δ 0 0 0 0 0 0 0 3 π ' 4 0( ') ' M Omnès function π sin ( s') Mˆ ( s') s Ω s s s i ( ε) Similarly for M 1 and M Four subtraction constants to be determined: α 0, β 0, γ 0 and one more in M 1 (β 1 ) Inputs needed for these and for the ππ phase shifts M 0 : ππ scattering, l=0, I=0 M 1 : ππ scattering, l=1, I=1 M : ππ scattering, l=0, I= I δ l Solve dispersion relations numerically by an iterative procedure 8