η 3π and quark masses
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1 η 3π and quark masses Stefan Lanz Department of Astronomy and Theoretical Physics, Lund University International Workshop on Chiral Dynamics, August 6 10, 2012, Jefferson Lab Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
2 Outline 1 Introduction 2 Dalitz plot measurements 3 Theoretical work 4 Our dispersive analysis 5 Comparison of results Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
3 Introduction Outline 1 Introduction 2 Dalitz plot measurements 3 Theoretical work 4 Our dispersive analysis 5 Comparison of results Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
4 Introduction Light quark masses Light quark masses not directly accessible to experiment due to confinement m u,d,s scale of QCD small contribution to hadronic quantities Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
5 Introduction Light quark masses Light quark masses not directly accessible to experiment due to confinement m u,d,s scale of QCD small contribution to hadronic quantities Gell-Mann Oakes Renner relations: [ Gell-Mann, Oakes & Renner 68 ] (meson mass) 2 = (spontaneousχsb) (explicitχsb) Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
![Gell-Mann Oakes Renner relations: [ Gell-Mann, Oakes & Renner 68 ] (meson mass) 2 =](/docs-images/46/21703525/images/page_5.jpg "(spontaneousχsb) (explicitχsb) Stefan Lanz (Lund University) η 3π and quark masses Chiral")
6 Introduction Light quark masses Light quark masses not directly accessible to experiment due to confinement m u,d,s scale of QCD small contribution to hadronic quantities Gell-Mann Oakes Renner relations: [ Gell-Mann, Oakes & Renner 68 ] (meson mass) 2 = (spontaneousχsb) (explicitχsb) quark condensate qq Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
![Renner relations: [ Gell-Mann, Oakes & Renner 68 ] (meson mass) 2 = (spontaneousχsb)](/docs-images/46/21703525/images/page_6.jpg "(explicitχsb) quark condensate qq Stefan Lanz (Lund University) η 3π and quark masses Chiral")
7 Introduction Light quark masses Light quark masses not directly accessible to experiment due to confinement m u,d,s scale of QCD small contribution to hadronic quantities Gell-Mann Oakes Renner relations: [ Gell-Mann, Oakes & Renner 68 ] (meson mass) 2 = (spontaneousχsb) (explicitχsb) quark condensate qq quark masses m q Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
![relations: [ Gell-Mann, Oakes & Renner 68 ] (meson mass) 2 = (spontaneousχsb) (explicitχsb) quark](/docs-images/46/21703525/images/page_7.jpg "condensate qq quark masses m q Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics")
8 Introduction Light quark masses Gell-Mann Oakes Renner relations m 2 π + = B 0(m u + m d ) m 2 π 0 = B 0 (m u + m d ) m 2 K + = B 0(m u + m s ) m 2 K 0 = B 0 (m d + m s ) m 2 η = B 0 mu+m d+4m s 3 Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
9 Introduction Light quark masses Gell-Mann Oakes Renner relations m 2 π + = B 0(m u + m d ) m 2 π 0 = B 0 (m u + m d )+ 2ǫ 3 B 0 (m u m d )+... m 2 K + = B 0(m u + m s ) ǫ m 2 K 0 = B 0 (m d + m s ) m 2 η = B 0 mu+m d+4m s 3 2ǫ 3 B 0 (m u m d )+... Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
10 Introduction Light quark masses Gell-Mann Oakes Renner relations m 2 π + = B 0(m u + m d )+ π em +... m 2 π 0 = B 0 (m u + m d )+ 2ǫ 3 B 0 (m u m d )+... m 2 K + = B 0(m u + m s )+ K em +... π/k em (35 MeV) 2 m 2 K 0 = B 0 (m d + m s ) π em = K em [ Dashen 69 ] m 2 η = B 0 mu+m d+4m s 3 2ǫ 3 B 0 (m u m d )+... Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
![.. π/k em (35 MeV) 2 m 2 K 0 = B 0 (m d + m s ) π em = K em [ Dashen 69 ] m 2 η = B 0 mu+m d+4m](/docs-images/46/21703525/images/page_10.jpg "s 3 2ǫ 3 B 0 (m u m d )+.")
11 Introduction Light quark masses Gell-Mann Oakes Renner relations m 2 π + = B 0(m u + m d )+ π em +... m 2 π 0 = B 0 (m u + m d )+ 2ǫ 3 B 0 (m u m d )+... m 2 K + = B 0(m u + m s )+ K em +... m 2 K 0 = B 0 (m d + m s ) m 2 η = B 0 mu+m d+4m s 3 2ǫ 3 B 0 (m u m d )+... (m u m d ) well hidden Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
12 Introduction Light quark masses Quark masses from the lattice more on this from others [ talks by Bernard, Lellouch, Sachrajda, Izubuchi,... ] relations between meson masses and quark masses from QCD m u m d needs handle on e.m. effects input from phenomenology (e.g., Kaon mass difference) put photons on the lattice recent review from FLAG [ Colangelo et al. 11 ] Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
![.. ] relations between meson masses and quark masses from QCD m u m d needs handle on e.m. effects input from phenomenology (e.](/docs-images/46/21703525/images/page_12.jpg "g., Kaon mass difference) put photons on the lattice recent review from FLAG [ Colangelo et al.")
13 Introduction Quark masses and η 3π What has η 3π to do with quark masses? η 3π depends on m q in special way: violates isospin generated by L IB = mu m d (ūu 2 dd) I = 1 operator Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
14 Introduction Quark masses and η 3π What has η 3π to do with quark masses? η 3π depends on m q in special way: violates isospin generated by L IB = mu m d (ūu 2 dd) I = 1 operator decay amplitude proportional to (m u m d ) measure for strength of isospin breaking in QCD Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
15 Introduction Quark masses and η 3π Electromagnetic corrections Q u Q d e.m. interactions break isospin can contribute to η 3π Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
16 Introduction Quark masses and η 3π Electromagnetic corrections Q u Q d e.m. interactions break isospin can contribute to η 3π but: contribution very small [ Sutherland 66, Bell & Sutherland 68 ] one-loop contributions known and small [ Baur, Kambor, Wyler 96, Ditsche, Kubis, Meißner 09 ] recent claim that η 3π 0 is mainly e.m. based on incomplete 2-loop calculation [ Nehme, Zein 11 ] Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
17 Introduction Quark masses and η 3π Electromagnetic corrections Q u Q d e.m. interactions break isospin can contribute to η 3π but: contribution very small [ Sutherland 66, Bell & Sutherland 68 ] one-loop contributions known and small [ Baur, Kambor, Wyler 96, Ditsche, Kubis, Meißner 09 ] recent claim that η 3π 0 is mainly e.m. based on incomplete 2-loop calculation [ Nehme, Zein 11 ] clean access to (m u m d ) Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
18 Introduction Quark masses and η 3π The quark mass ratio Q A η 3π B 0 (m u m d ) Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
19 Introduction Quark masses and η 3π A η 3π B 0 (m u m d ) = The quark mass ratio Q 1 mk 2(m2 K m2 π ) Q 2 mπ 2 +O(M 3 ) Q 2 = m2 s ˆm 2 m 2 d m2 u Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
20 Introduction Quark masses and η 3π The quark mass ratio Q 1 mk 2(m2 K m2 π ) Q 2 m 2 +O(M 3 ) π A η 3π B 0 (m u m d ) = 1 R (m2 K m2 π )+O(M2 ) Q 2 = m2 s ˆm 2 m 2 d m2 u R = ms ˆm m d m u Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
21 Introduction Quark masses and η 3π The quark mass ratio Q 1 mk 2(m2 K m2 π ) Q 2 m 2 +O(M 3 ) π A η 3π B 0 (m u m d ) = 1 R (m2 K m2 π )+O(M2 ) Q 2 = m2 s ˆm 2 m 2 d m2 u R = ms ˆm m d m u define normalised amplitude: A(s, t, u) = 1 Q 2 m 2 K(m 2 K m 2 π) 2 3m 2 πf 2 π Γ exp A(s, t, u) 1/Q 4 M(s, t, u) Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
22 Introduction Other aspects What else is interesting? slow convergence of chiral series: experiment [ PDG 12 ] Γ c = 66 ev + 94 ev +... = 296 ev current algebra [ Cronin 67, Osborn & Wallace 70 ] one-loop χpt [ Gasser & Leutwyler 84 ] Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
23 Introduction Other aspects What else is interesting? slow convergence of chiral series: experiment [ PDG 12 ] Γ c = 66 ev + 94 ev +... = 296 ev current algebra [ Cronin 67, Osborn & Wallace 70 ] one-loop χpt [ Gasser & Leutwyler 84 ] enhanced by large final state rescattering effects [ Roiesnel & Truong 81 ] Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
24 Introduction Other aspects What else is interesting? possible tension among charged and neutral channel experiments Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
25 Introduction Other aspects What else is interesting? possible tension among charged and neutral channel experiments charged and neutral channel amplitudes are related: A n (s, t, u) = A c (s, t, u)+a c (t, u, s)+a c (u, s, t) allows for consistency check among measurements more on this later... Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
26 Introduction Other aspects What else is interesting? A n (s, t, u) 2 1+2αZ GAMS-2000 (1984) [ Alde et al. 84 ] Crystal Barrel@LEAR (1998) [ Abele et al. 98 ] Crystal Ball@BNL (2001) [ Tippens et al. 01 ] SND (2001) [ Achasov et al. 01 ] WASA@CELSIUS (2007) [ Bashkanov et al. 07 ] WASA@COSY (2008) [ Adolph et al. 09 ] Crystal Ball@MAMI-B (2009) [ Unverzagt et al. 09 ] Crystal Ball@MAMI-C (2009) [ Prakhov et al. 09 ] KLOE (2010) [ Ambrosino et al. 10 ] PDG average [ PDG 12 ] α Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
27 Introduction Other aspects What else is interesting? χpt O(p 4 ) [ GL 85, Bijnens&Gasser 02 ] χpt O(p 6 ) [ Bijnens&Ghorbani 07 ] Kambor et al. [ Kambor et al. 96 ] Kampf et al. [ Kampf et al. 11 ] NREFT [ Schneider et al. 11 ] GAMS-2000 (1984) [ Alde et al. 84 ] Crystal Barrel@LEAR (1998) [ Abele et al. 98 ] Crystal Ball@BNL (2001) [ Tippens et al. 01 ] SND (2001) [ Achasov et al. 01 ] WASA@CELSIUS (2007) [ Bashkanov et al. 07 ] WASA@COSY (2008) [ Adolph et al. 09 ] Crystal Ball@MAMI-B (2009) [ Unverzagt et al. 09 ] Crystal Ball@MAMI-C (2009) [ Prakhov et al. 09 ] KLOE (2010) [ Ambrosino et al. 10 ] PDG average [ PDG 12 ] α Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
28 Introduction Kinematics Kinematics p π + s = (p π + + p π ) 2 t = (p π 0 + p π ) 2 p η p π u = (p π 0 + p π +) 2 s+t + u = m 2 η + 2m2 π + + m2 π 0 3s 0 p π 0 only two independent variables Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
29 Introduction Kinematics Kinematics s = (p π + + p π ) 2 p π + t = (p π 0 + p π ) 2 p η θ s p π u = (p π 0 + p π +) 2 s+t + u = m 2 η + 2m2 π + + m2 π 0 3s 0 p π 0 only two independent variables, e.g., s & t u cosθ S Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
30 Introduction Adler Zero Adler Zero soft pion theorem, i.e., valid in SU(2) chiral limit [ Adler 65 ] decay amplitude has a zero if p π + 0 p π 0 s = u = 0, t = m 2 η s = t = 0, u = m 2 η Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
31 Introduction Adler Zero Adler Zero soft pion theorem, i.e., valid in SU(2) chiral limit [ Adler 65 ] decay amplitude has a zero if p π + 0 p π 0 s = u = 0, t = m 2 η s = t = 0, u = m 2 η for m π 0 Adler zeros at s = u = 4 3 m2 π, t = mη 2 + mπ/3 2 s = t = 4 3 m2 π, u = mη 2 + mπ/3 2 protected by SU(2) chiral symmetry no O(m s ) corrections Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
32 Dalitz plot measurements Outline 1 Introduction 2 Dalitz plot measurements 3 Theoretical work 4 Our dispersive analysis 5 Comparison of results Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
33 Dalitz plot measurements Dalitz plot variables 1 Y 0 1 X = 3 2m ηq c (u t) Y = 3 2m ηq c ( (mη m π 0) 2 s ) 1 Q c = m η 2m π + m π 0 Z = X 2 + Y X Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
34 Dalitz plot measurements KLOE measurement KLOE measurement of the charged channel only modern high-statistics Dalitz plot measurement [ KLOE 08 ] η π + π π 0 events from e + e φ ηγ 2 d N dxdy Y X [ Figure from Ambrosino et al. 08 ] Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
35 Dalitz plot measurements KLOE measurement KLOE result for Dalitz plot parameters Dalitz plot parametrisation: A c (s, t, u) 2 1+aY + by 2 + cx + dx 2 + exy + fy 3 + gx 3 + hx 2 Y + lxy 2 Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
36 Dalitz plot measurements KLOE measurement KLOE result for Dalitz plot parameters Dalitz plot parametrisation: A c (s, t, u) 2 1+aY + by 2 + cx + dx 2 + exy + fy 3 + gx 3 + hx 2 Y + lxy 2 charge conjugation s ymmetry under X X Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
37 Dalitz plot measurements KLOE measurement KLOE result for Dalitz plot parameters Dalitz plot parametrisation: A c (s, t, u) 2 1+aY + by 2 + cx + dx 2 + exy + fy 3 + gx 3 + hx 2 Y + lxy 2 charge conjugation s ymmetry under X X h consistent with zero Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
38 Dalitz plot measurements KLOE measurement KLOE result for Dalitz plot parameters Dalitz plot parametrisation: A c (s, t, u) 2 1+aY + by 2 + cx + dx 2 + exy + fy 3 + gx 3 + hx 2 Y + lxy 2 charge conjugation s ymmetry under X X h consistent with zero result: [ KLOE 08 ] a = b = 0.124±0.012 d = f = 0.14±0.02 Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
39 Dalitz plot measurements KLOE measurement KLOE result for Dalitz plot parameters Dalitz plot parametrisation: A c (s, t, u) 2 1+aY + by 2 + cx + dx 2 + exy + fy 3 + gx 3 + hx 2 Y + lxy 2 charge conjugation s ymmetry under X X h consistent with zero result: [ KLOE 08 ] a = b = 0.124±0.012 d = f = 0.14±0.02 Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
40 Dalitz plot measurements KLOE measurement KLOE result for Dalitz plot parameters Dalitz plot parametrisation: older experiments: A c (s, t, u) 2 1+aY AGS@BNL + by 2 + cx + dx 2 + exy + fy 3 + gx 3 + hx 2 Y + lxy 2 [ Gormley et al. 70 ] charge conjugation Princeton-Pennsylvania s ymmetry under Accelerator [ Layter et al. 73 ] X X Crystal Barrel@LEAR [ Abele et al. 98 ] h consistent with zero upcoming analyses: result: [ KLOE 08 ] WASA@COSY [ Adlarson, tbp ] a = b = 0.124±0.012 KLOE [ Caldeira Balkeståhl, tbp ] d = f = 0.14±0.02 Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
41 Dalitz plot measurements MAMI-C measurement MAMI-C measurement of the neutral channel η 3π 0 events from γp ηp smallest uncertainties on α Data / MC Slope Fit (d) χ 2 / ndf 31.5 / 18 p0 1± 0.0 p ± similar but independent 0.9 measurement from MAMI-B A n (s, t, u) 2 1+2αZ + 6βY (X 2 Y z η 3π 0 ) + 2γZ 2 [ figure from Prakhov et al. 09 ] Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
42 Dalitz plot measurements MAMI-C measurement MAMI-C measurement of the neutral channel η 3π 0 events from γp ηp smallest uncertainties on α Data / MC Slope Fit (d) χ 2 / ndf 31.5 / 18 p0 1± 0.0 p ± similar but independent 0.9 measurement from MAMI-B A n (s, t, u) 2 1+2αZ + 6βY (X 2 Y z η 3π 0 ) + 2γZ 2 [ figure from Prakhov et al. 09 ] Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
43 Theoretical work Outline 1 Introduction 2 Dalitz plot measurements 3 Theoretical work 4 Our dispersive analysis 5 Comparison of results Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
44 Theoretical work Overview What has been done? Q α tension e.m. contributions in χpt [ Baur et al., 96, Ditsche et al. 09 ] two-loop χpt [ Bijnens & Ghorbani 07 ] non-relativistic EFT [ Schneider, Kubis & Ditsche 11 ] analytical dispersive ( ) [ Kampf, Knecht, Novotný & Zdráhal 11 ] resummed χpt ( ) ( ) [ Kolesar et al. 11 ] numerical dispersive [ Colangelo, SL, Leutwyler, Passemar tbp ] Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
45 Theoretical work NREFT analysis NREFT analysis Q α tension non-relativistic EFT [ Schneider, Kubis & Ditsche 11 ] expansion in small π three momenta in η rest frame explicitly includes two pion rescattering processes inputs: O(p 4 ) η 3π amplitude from χpt empirical ππ scattering phases results only for shape, but not normalisation Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
46 Theoretical work NREFT analysis NREFT analysis Q α tension non-relativistic EFT [ Schneider, Kubis & Ditsche 11 ] α = ± correct sign, marginal agreement with experiment tension between charged and neutral channel experiments: α 1 4 (b + d 1 4 a2 ) [ Bijnens & Ghorbani 07 ] Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
47 Theoretical work NREFT analysis NREFT analysis Q α tension non-relativistic EFT [ Schneider, Kubis & Ditsche 11 ] α = ± correct sign, marginal agreement with experiment tension between charged and neutral channel experiments: α = 1 4 (b + d 1 4 a2 )+ can be calculated in NREFT (no η 3π input from χpt needed!) Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
48 Theoretical work NREFT analysis NREFT analysis Q α tension non-relativistic EFT [ Schneider, Kubis & Ditsche 11 ] α = ± correct sign, marginal agreement with experiment tension between charged and neutral channel experiments: α = 1 4 (b + d 1 4 a2 ) can be calculated in NREFT (no η 3π input from χpt needed!) from KLOE Dalitz plot parameters: α = 0.059±0.007 main reason for disagreement: b NREFT = > b KLOE = Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
49 Theoretical work Dispersive analysis by Kampf et al. Dispersive analysis by Kampf et al. Q α tension analytical dispersive ( ) [ Kampf, Knecht, Novotný & Zdráhal 11 ] analytical dispersive analysis relying on two-loop χpt and KLOE data 6 subtraction constants two rescattering processes reproduces two-loop result Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
50 Theoretical work Dispersive analysis by Kampf et al. Dispersive analysis by Kampf et al. Q α tension analytical dispersive ( ) [ Kampf, Knecht, Novotný & Zdráhal 11 ] analytical dispersive analysis relying on two-loop χpt and KLOE data 6 subtraction constants two rescattering processes reproduces two-loop result main result: subtraction constants from fit to KLOE data (normalisation fixed by imaginary part of two-loop result along t = u) Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
51 Theoretical work Dispersive analysis by Kampf et al. Dispersive analysis by Kampf et al. Q α tension analytical dispersive ( ) [ Kampf, Knecht, Novotný & Zdráhal 11 ] Adler zero strongly violated incompatible with SU(2) chiral symmetry M(s, 3s0 2s, s) one-loop χpt KKNZ dispersive s = u in m 2 π Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
52 Our dispersive analysis Outline 1 Introduction 2 Dalitz plot measurements 3 Theoretical work 4 Our dispersive analysis 5 Comparison of results Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
53 Our dispersive analysis Dispersion relations Method Q α tension numerical dispersive [ Colangelo, SL, Leutwyler, Passemar tbp ] includes arbitrary number of rescattering processes Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
54 Our dispersive analysis Dispersion relations Method Q α tension numerical dispersive [ Colangelo, SL, Leutwyler, Passemar tbp ] includes arbitrary number of rescattering processes two main steps: derive & solve dispersion relations [ Anisovich & Leutwyler 96 ] fix subtraction constants Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
55 Our dispersive analysis Dispersion relations Dispersion relations relies on decomposition [ Fuchs, Sazdijan & Stern 93, Anisovich & Leutwyler 96 ] M(s, t, u) = M 0 (s)+(s u)m 1 (t)+(s t)m 1 (u)+m 2 (t)+m 2 (u) 2 3 M 2(s) Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
56 Our dispersive analysis Dispersion relations Dispersion relations relies on decomposition [ Fuchs, Sazdijan & Stern 93, Anisovich & Leutwyler 96 ] M(s, t, u) = M 0 (s)+(s u)m 1 (t)+(s t)m 1 (u)+m 2 (t)+m 2 (u) 2 3 M 2(s) dispersion relation for each M I (s): [ Anisovich & Leutwyler 96 ] { } M I (s) = Ω I (s) P I (s)+ sn I ds sinδ I (s )ˆM I (s ) π s n I Ω I (s ) (s s iǫ) 4m 2 π s Omnès function: Ω I (s) = exp π 4m 2 π ds δ I (s ) s (s s iǫ) [ Omnès 58 ] Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
57 Our dispersive analysis Dispersion relations Dispersion relations relies on decomposition [ Fuchs, Sazdijan & Stern 93, Anisovich & Leutwyler 96 ] M(s, t, u) = M 0 (s)+(s u)m 1 (t)+(s t)m 1 (u)+m 2 (t)+m 2 (u) 2 3 M 2(s) dispersion relation for each M I (s): [ Anisovich & Leutwyler 96 ] { } M I (s) = Ω I (s) P I (s)+ sn I ds sinδ I (s )ˆM I (s ) π s n I Ω I (s ) (s s iǫ) 4m 2 π s Omnès function: Ω I (s) = exp π input needed for 4m 2 π ds δ I (s ) s (s s iǫ) [ Omnès 58 ] ππ phase shifts [ Ananthanarayan, Colangelo, Gasser & Leutwyler 01 ] subtraction constants Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
58 Our dispersive analysis Dispersion relations Dispersion relations relies on decomposition [ Fuchs, Sazdijan & Stern 93, Anisovich & Leutwyler 96 ] M(s, t, u) = M 0 (s)+(s u)m 1 (t)+(s t)m 1 (u)+m 2 (t)+m 2 (u) 2 3 M 2(s) dispersion relation for each M I (s): [ Anisovich & Leutwyler 96 ] { } M I (s) = Ω I (s) P I (s)+ sn I ds sinδ I (s )ˆM I (s ) π s n I Ω I (s ) (s s iǫ) 4m 2 π s Omnès function: Ω I (s) = exp π input needed for 4m 2 π ds δ I (s ) s (s s iǫ) [ Omnès 58 ] ππ phase shifts [ Ananthanarayan, Colangelo, Gasser & Leutwyler 01 ] subtraction constants Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
59 Our dispersive analysis Subtraction constants Taylor coefficients M I (s) = a I + b I s+c I s 2 + d I s Taylor coefficients subtraction constants Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
60 Our dispersive analysis Subtraction constants Taylor coefficients M I (s) = a I + b I s+c I s 2 + d I s Taylor coefficients subtraction constants a I, b I,... R, but α I,β I,... C imaginary parts of subtraction constants suppressed splitting into M I (s) not unique because of s + t + u = m 2 η + 2m 2 π + + m2 π 0 gauge freedom to fix some Taylor coefficients arbitrarily Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
61 Our dispersive analysis Subtraction constants Matching to one-loop χpt Subtraction constants from theory alone: M 0 (s) = a 0 + b 0 s + c 0 s 2 + d 0 s M 1 (s) = a 1 + b 1 s + c 1 s M 2 (s) = a 2 + b 2 s + c 2 s 2 + d 2 s use only TC up to s 2 Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
62 Our dispersive analysis Subtraction constants Matching to one-loop χpt Subtraction constants from theory alone: M 0 (s) = a 0 + b 0 s + c 0 s 2 + d 0 s M 1 (s) = a 1 + b 1 s + c 1 s M 2 (s) = a 2 + b 2 s + c 2 s 2 + d 2 s use only TC up to s 2 gauge 4 TC to tree level value Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
63 Our dispersive analysis Subtraction constants Matching to one-loop χpt Subtraction constants from theory alone: M 0 (s) = a 0 + b 0 s + c 0 s 2 + d 0 s M 1 (s) = a 1 + b 1 s + c 1 s M 2 (s) = a 2 + b 2 s + c 2 s 2 + d 2 s use only TC up to s 2 gauge 4 TC to tree level value set 4 TC to one-loop value Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
64 Our dispersive analysis Subtraction constants Matching to one-loop χpt Subtraction constants from theory alone: M 0 (s) = a 0 + b 0 s + c 0 s 2 + d 0 s M 1 (s) = a 1 + b 1 s + c 1 s M 2 (s) = a 2 + b 2 s + c 2 s 2 + d 2 s use only TC up to s 2 gauge 4 TC to tree level value set 4 TC to one-loop value dispersive, one loop Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
65 Our dispersive analysis Subtraction constants Matching to one-loop χpt Subtraction constants from theory alone: M 0 (s) = a 0 + b 0 s + c 0 s 2 + d 0 s M 1 (s) = a 1 + b 1 s + c 1 s M 2 (s) = a 2 + b 2 s + c 2 s 2 + d 2 s use only TC up to s 2 use χpt at low energy gauge 4 TC to tree level value extrapolate to physical region using dispersion relations set 4 TC to one-loop value dispersive, one loop Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
66 Our dispersive analysis Subtraction constants Fit to data use data to further constrain subtraction constants: M 0 (s) = a 0 + b 0 s + c 0 s 2 + d 0 s M 1 (s) = a 1 + b 1 s + c 1 s M 2 (s) = a 2 + b 2 s + c 2 s 2 + d 2 s gauge 4 TC to tree level value set 4 TC to one-loop value Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
67 Our dispersive analysis Subtraction constants Fit to data use data to further constrain subtraction constants: M 0 (s) = a 0 + b 0 s + c 0 s 2 + d 0 s M 1 (s) = a 1 + b 1 s + c 1 s M 2 (s) = a 2 + b 2 s + c 2 s 2 + d 2 s gauge 4 TC to tree level value set 4 TC to one-loop value fix 2 TC from fit to data Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
68 Our dispersive analysis Subtraction constants Fit to data use data to further constrain subtraction constants: M 0 (s) = a 0 + b 0 s + c 0 s 2 + d 0 s M 1 (s) = a 1 + b 1 s + c 1 s M 2 (s) = a 2 + b 2 s + c 2 s 2 + d 2 s gauge 4 TC to tree level value set 4 TC to one-loop value fix 2 TC from fit to data gauge 1 TC such that δ 2 = 0 Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
69 Our dispersive analysis Subtraction constants Fit to data use data to further constrain subtraction constants: M 0 (s) = a 0 + b 0 s + c 0 s 2 + d 0 s M 1 (s) = a 1 + b 1 s + c 1 s M 2 (s) = a 2 + b 2 s + c 2 s 2 + d 2 s gauge 4 TC to tree level value set 4 TC to one-loop value fix 2 TC from fit to data gauge 1 TC such that δ 2 = 0 dispersive, fit to KLOE Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
70 Our dispersive analysis Results Dalitz distribution for η π + π π preliminary one-loop χpt KLOE 2.0 Γ(0, Y) Y Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
71 Our dispersive analysis Results Dalitz distribution for η π + π π preliminary one-loop χpt KLOE dispersive, one loop Γ(0, Y) Y Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
72 Our dispersive analysis Results Dalitz distribution for η π + π π preliminary one-loop χpt KLOE dispersive, one loop dispersive, fit to KLOE Γ(0, Y) Y Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
73 Our dispersive analysis Results Dalitz distribution for η 3π preliminary Γ(Z) 0.98 one-loop χpt MAMI-C Z Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
74 Our dispersive analysis Results Dalitz distribution for η 3π preliminary Γ(Z) 0.98 one-loop χpt MAMI-C 0.94 dispersive, one loop Z Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
75 Our dispersive analysis Results Dalitz distribution for η 3π preliminary Γ(Z) 0.98 one-loop χpt MAMI-C 0.94 dispersive, one loop dispersive, fit to KLOE Z Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
76 Our dispersive analysis Results But: electromagnetic corrections dispersive amplitude in isospin limit needs to be accounted for in fits: Dalitz plot distribution: kinematic effects most important (position of cusps) decay rate: kinematic effects not enough (size of phase space) Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
77 Our dispersive analysis Results But: electromagnetic corrections dispersive amplitude in isospin limit needs to be accounted for in fits: Dalitz plot distribution: kinematic effects most important (position of cusps) decay rate: kinematic effects not enough (size of phase space) small effect on Q, but branching ratio is off roughly estimate e.m. effects on Γ [ Gullström et al. 09, Ditsche et al. 09 ] e.m. corrections can amend BR Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
78 Our dispersive analysis Results But: electromagnetic corrections dispersive amplitude in isospin limit needs to be accounted for in fits: Dalitz plot distribution: kinematic effects most important (position of cusps) decay rate: kinematic effects not enough (size of phase space) small effect on Q, but branching ratio is off roughly estimate e.m. effects on Γ [ Gullström et al. 09, Ditsche et al. 09 ] e.m. corrections can amend BR needs to be improved Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
79 Comparison of results Outline 1 Introduction 2 Dalitz plot measurements 3 Theoretical work 4 Our dispersive analysis 5 Comparison of results Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
80 Comparison of results Comparison of Q dispersive (Walker) [ Anisovich & Leutwyler 96, Walker 98 ] dispersive (Kambor et al.) [ Kambor et al. 96 ] dispersive (Kampf et al.) [ Kampf et al. 11 ] χpt O(p 4 ) [ Gasser & Leutwyler 85, Bijnens & Ghorbani 07 ] χpt O(p 6 ) [ Bijnens & Ghorbani 07 ] no Dashen violation [ Weinberg 77 ] with Dashen violation [ Anant et al. 04, Kastner & Neufeld 08 ] Q Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
81 Comparison of results Comparison of Q dispersive (Walker) [ Anisovich & Leutwyler 96, Walker 98 ] dispersive (Kambor et al.) [ Kambor et al. 96 ] dispersive (Kampf et al.) [ Kampf et al. 11 ] χpt O(p 4 ) [ Gasser & Leutwyler 85, Bijnens & Ghorbani 07 ] χpt O(p 6 ) [ Bijnens & Ghorbani 07 ] no Dashen violation [ Weinberg 77 ] with Dashen violation [ Anant et al. 04, Kastner & Neufeld 08 ] dispersive, one loop dispersive, fit to KLOE preliminary Q Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
82 Comparison of results Comparison of α χpt O(p 4 ) [ GL 85, Bijnens&Gasser 02 ] χpt O(p 6 ) [ Bijnens&Ghorbani 07 ] Kambor et al. [ Kambor et al. 96 ] Kampf et al. [ Kampf et al. 11 ] NREFT [ Schneider et al. 11 ] GAMS-2000 (1984) [ Alde et al. 84 ] Crystal Barrel@LEAR (1998) [ Abele et al. 98 ] Crystal Ball@BNL (2001) [ Tippens et al. 01 ] SND (2001) [ Achasov et al. 01 ] WASA@CELSIUS (2007) [ Bashkanov et al. 07 ] WASA@COSY (2008) [ Adolph et al. 09 ] Crystal Ball@MAMI-B (2009) [ Unverzagt et al. 09 ] Crystal Ball@MAMI-C (2009) [ Prakhov et al. 09 ] KLOE (2010) [ Ambrosino et al. 10 ] PDG average [ PDG 10 ] α Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
83 Comparison of results Comparison of α χpt O(p 4 ) [ GL 85, Bijnens&Gasser 02 ] χpt O(p 6 ) [ Bijnens&Ghorbani 07 ] Kambor et al. [ Kambor et al. 96 ] Kampf et al. [ Kampf et al. 11 ] NREFT [ Schneider et al. 11 ] GAMS-2000 (1984) [ Alde et al. 84 ] Crystal Barrel@LEAR (1998) [ Abele et al. 98 ] Crystal Ball@BNL (2001) [ Tippens et al. 01 ] SND (2001) [ Achasov et al. 01 ] WASA@CELSIUS (2007) [ Bashkanov et al. 07 ] WASA@COSY (2008) [ Adolph et al. 09 ] Crystal Ball@MAMI-B (2009) [ Unverzagt et al. 09 ] Crystal Ball@MAMI-C (2009) [ Prakhov et al. 09 ] KLOE (2010) [ Ambrosino et al. 10 ] PDG average [ PDG 10 ] dispersive, one loop dispersive, fit to KLOE preliminary α Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
84 Conclusion & Outlook Conclusion & Outlook η 3π very well suited to gain information on isospin breaking in QCD dispersion relations allow to treat rescattering effects properly dispersive treatment significantly improves one-loop result neutral channel slope parameter can be understood based on charged channel data no clear sign of a tension among experiments more careful treatment of electromagnetic effects needed Stefan Lanz (Lund University) η 3π and quark masses Chiral Dynamics
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