Three-point Green Functions in the resonance region: LEC s

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Tyrone Morgan
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1 Three-point Green unctions in the resonance region: LE s Jorge Portolés Instituto de ísica orpuscular SI-UEG, alencia (Spain)

2 Summary LE s in hiral Perturbation Theory : how do we get them? The role of esonance hiral Theory Three-point Green unctions of QD currents: - esonance hiral Theory approach - Meromorphic unction approach onclusions Task force : incenzo irigliano, Gerhard Ecker, Markus Eidemüller, oland Kaiser, Antonio Pich, J.P.

esonance hiral Theory approach - Meromorphic unction approach onclusions Task

3 LE s in hiral Perturbation Theory ( ) ( 4 ) ( 6 ) L χ PT = L + L + L +... L () µ uu µ χ + = + 4 (N = 3) Even-intrinsic-parity sector only L 10 (4) µ 4 = L1 uu µ +... = L i Oi i = 1 [Gasser & Leutwyler, 1985] L 90 ( 6) α µν 6 = 1 uu α hµνh +... = i Oi i = 1 [earing & Scherer, 1996] [Bijnens, olangelo & Ecker, 1999,000] L,? within χpt : π,s, K l3, i i High-precision predictions

.. = L i Oi i = 1 [Gasser & Leutwyler, 1985] L 90 ( 6) α µν 6 = 1 uu α hµνh +.

4 How do we get them? [Gasser & Leutwyler, 1984] [Donoghue et al., 1989] [Ecker et al., 1989] G [ ] exp LT χ, ; λ j exp ( U; λj) d i ( U ) = i LNo Local q M L ( U; λ ) = L ( U, L, ) Local j Local i i

5 The role of esonance hiral Theory 4 Op ( ) [Ecker et al, 1989] ( ) 1 L T L + Lki n Lin s L χ = + t( = 0,1) 1 int( s 1 G = ) = f + i u u + A f µν µ ν A µν µν + µν µν 1 ( 0) µ Lint s = = cd Suµ u + cm Sχ+ + idm P χ L1 L L3 L 4 L 5.. G c 8M 6 d MS G 4M 3G c + 4M d MS cc d 3M m S cc d m MS

6 The role of esonance hiral Theory 4 Op ( ) [Ecker et al, 1989] ( ) 1 L T L + Lki n Lin s L χ = + t( = 0,1) 1 int( s 1 G = ) = f + i u u + L int χ( p ) A f µν µ ν A µν µν + µν µν 1 ( 0) µ Lint s = = cd Suµ u + cm Sχ+ + idm P χ L1 L L3 L 4 L 5.. G c 8M 6 d MS G 4M 3G c + 4M d MS cc d 3M m S cc d m MS

7 6 Op ( ) [irigliano et al, 005] L = L + L =,A,S,P 1 int int int χ 4 ( p ) 70 operators λ i L int χ 1 ( p ) 38 operators 1 λ i p χ( ) 7 operators 1 3 λ i

8 e.g. = cc d 1 M m 4 S 1 c ( c c ) d d c cc m m d m m P m S d m SP 34 = λ λ + λ MS MP MP MS MM S P = G M = d λ + M λ m P P MM P

9 e.g. = cc d 1 M m 4 S 1 c ( c c ) d d c cc m m d m m P m S d m SP 34 = λ λ + λ MS MP MP MS MM S P = G M = d λ + M λ m P P MM P L χ T is NOT QD for arbitrary values of the couplings

MM S P = G + 88 4 90 4M = d λ + M λ m P P 90 1 9 MM

10 QD E M ρ hiral Symmetry SU L (N ) SU (N ) E M ρ Perturbative QD hiral Perturbation Theory Large N Asymptotic behaviour of spectral functions E ~ M ρ esonance hiral Theory µ (1 -- ) A µ (1 ++ ) QD L (,, ) eff = λ Oi µ Aµ Π i i ector meson dominance

spectral functions E ~ M ρ esonance hiral Theory µ (1 -- ) A

11 What QD tells us on Green unctions of QD currents? 1) Low energy expansion of Green unctions (Ward Id.) ) Large N limit of QD: - Infinite tower of non-decaying hadronic states - Description through a meromorphic function 3) Operator Product Expansion (OPE) at high energies 4) Brodsky-Lepage behaviour of form factors of hadron currents : 1 H, Q Q [Ecker, Gasser et al, 1989]

) ) Large N limit of QD: - Infinite tower of non-decaying hadronic states - Description

12 What QD tells us on Green unctions of QD currents? 1) Low energy expansion of Green unctions (Ward Id.) ) Large N limit of QD: - Infinite tower of non-decaying hadronic states - Description through a meromorphic function 3) Operator Product Expansion (OPE) at high energies 4) Brodsky-Lepage behaviour of form factors of hadron currents : 1 H, Q Q [Ecker, Gasser et al, 1989]

13 What QD tells us on Green unctions of QD currents? 1) Low energy expansion of Green unctions (Ward Id.) ) Large N limit of QD: (modelization) - Infinite tower of non-decaying hadronic states - Description through a meromorphic function 3) Operator Product Expansion (OPE) at high energies (1 st m) 4) Brodsky-Lepage behaviour of form factors of hadron currents : 1 H, Q Q [Ecker, Gasser et al, 1989]

) ) Large N limit of QD: (modelization) - Infinite tower of non-decaying hadronic states -

14 What QD tells us on Green unctions of QD currents? 1) Low energy expansion of Green unctions (Ward Id.) ) Large N limit of QD: (modelization) - Infinite tower of non-decaying hadronic states - Description through a meromorphic function (1 st m) 3) Operator Product Expansion (OPE) at high energies 4) Brodsky-Lepage behaviour of form factors of hadron currents : 1 H, Q Q [Ecker, Gasser et al, 1989] [Bijnens et al, 003]

through a meromorphic function (1 st m) 3) Operator Product Expansion (OPE) at high energies 4)

15 Three-point Green unctions of QD currents [Bijnens & Prades, 1994] [Moussallam, 1997] [Peris et al, 1998] [Knecht & Nyffeler, 001] [uiz-emenía et al, 003] [Bijnens et al, 003] i j k i( p x p y) ijk + λ λ λ Π 13 1 = ψ 1 ψ ψ ψ ψ 3 ψ Γ Γ Γ ( p, p ) i d xd y e 0 T (0) ( x) ( y) 0 p p 1 i k p 1 + p j ( λp λp ) l im Π, λ ijk 13 1 ( λp p ) l im Π, λ ijk 13 1 l im Π, λ ( p λp ) ijk 13 1 ( λp λ p p ) lim Π,(1 ) + λ ijk OPE + 1/λ expansion

1 ψ ψ ψ ψ 3 ψ Γ Γ Γ 4 4 1 ( p, p ) i d xd y e 0 T (0) ( x) ( y) 0 p p 1 i k p 1 + p j ( λp λp ) l im Π, λ ijk

16 1) esonance hiral Theory approach Π p ( ) ( ) 1, p, M ijk p p Π p, p ijk 13 1 χt 13 1 Direct information on the couplings of L χ T OPE Hadronic decays of the tau lepton, decays of resonances, Hadronic cross-section, etc. ) Meromorphic unction approach (<SPP>) Π P + P + P + P + P = d N Π ijk ijk stu,, M ijk SPP χ SPP [ MS s][ t][ u][ MP t][ MP u] OPE Large N n k n k k l l Pn = cn k, k l, l s t u k= 0 l= 0 s = p, t = p, u = ( p + p ) 1 1

) Meromorphic unction approach (<SPP>) Π P + P + P + P + P = d N Π ijk ijk 0 1 3 4 stu,, M ijk SPP χ SPP [ MS

17 On the constraints from orm actors Brodsky-Lepage behaviour of two-body orm actors of Hadron urrents H 1 Q, Q 3-point Green unctions + π γ, JQD π π J QD 3-point Green unctions + Not always! J QD [Bijnens et al, 003]

18 On the constraints from orm actors Brodsky-Lepage behaviour of two-body orm actors of Hadron urrents H 1 Q, Q 3-point Green unctions + π γ, JQD π π J QD 3-point Green unctions + Not always! J QD [Bijnens et al, 003]

19 Does phenomenology support this procedure? E.g. 1) [uiz-emenía et al, 003] <P> 3 π N ω ω mπ ω π ω α m M M Γ ( ω πγ ) = Mω M 4 M M ( ω πγ) ( ) Me [ = Me ] Γ = th ( ω πγ ) exp ( ) Me Γ = ±

20 ) [irigliano et al, 004] <AP> 3 3 A mπ 1 1 MA α M Γ ( a1(160) πγ ) = MA 4 M 3 α M A M A m π 0 MA 1 4 M M M A ( a πγ ) Γ 1(16) = 1 1 [Moussallam, 1997] [irigliano et al, 004] [Moussallam, 1997] Experiment Γ(a 1 πγ) (Me) ± 0.46 M = 0.77 Ge, M = 1. Ge A

21 ) [irigliano et al, 004] <AP> 3 3 A mπ 1 1 MA α M Γ ( a1(160) πγ ) = MA 4 M One multiplet of pseudoescalar resonances 3 α M A M A m π 0 MA 1 4 M M M A ( a πγ ) Γ 1(16) = 1 1 [Moussallam, 1997] [irigliano et al, 004] [Moussallam, 1997] Experiment Γ(a 1 πγ) (Me) ± 0.46 M = 0.77 Ge, M = 1. Ge A

22 esults <AP> [irigliano et al, 004] 78 = 4 ( 3MA + 4M ) 8MM A 16MM P 8 = ( 4MA + 5M ) 3MM 3MM 4 A A P π νγ 88 = + π 4 4M 8MM P q ( ) 90 = 8M MP K 3 also, 87 89

23 esults <AP> [irigliano et al, 004] 78 = 4 ( 3MA + 4M ) 8MM A 16MM P 8 = ( 4MA + 5M ) 3MM 3MM 4 A A P π νγ 88 = + 4 4M 8MMP 90 = 8M MP OBUST! π ( q ) K 3 also, 87 89

24 <SPP> [irigliano et al, 005] 1 = 4 8M S = MS 16 MS MP f K + K 3 0 π (0) 6 Op ( ) MS 16 MS MP = All of these are OBUST!!

25 onclusions LE s in χpt ouplings λ i in the esonance hiral Theory High-precision χpt predictions (O(p 6 )) Applications of esonance hiral Theory - Hadronic cross-section - esonance decays - Hadronic decays of the tau lepton - (g-) µ...

26 onclusions LE s in χpt ouplings λ i in the esonance hiral Theory High-precision χpt predictions (O(p 6 )) Applications of esonance hiral Theory - Hadronic cross-section - esonance decays - Hadronic decays of the tau lepton - (g-) µ... Procedure Green unctions of QD currents Asymptotic constraints of QD <AP>,<SPP>,.(still a lot to do!) and phenomenologically supported

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