Low-energy aspects of amplitude analysis: chiral perturbation theory and dispersion relations

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1 Low-energy aspects of amplitude analysis: chiral perturbation theory and dispersion relations Bastian Kubis Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) Bethe Center for Theoretical Physics Universität Bonn, Germany Techniques of Amplitude Analysis Jefferson Lab Advanced Study Institute, May 31 to June 13, 2012 Chiral perturbation theory and dispersion relations p. 1

2 Schedule (tentative) Lecture 1: Introduction to chiral perturbation theory chiral symmetry construction of effective Lagrangian power counting Lecture 2: The pion vector form factor dispersion relations, calculation of discontinuities... Omnès solution application(s) Lecture 3: Dispersion relations for 3-body decays quark-mass ratios and η 3π construction of a solution based on Omnès functions Chiral perturbation theory and dispersion relations p. 2

3 Strong and weak QCD 0.5 α s (Q) Data Theory Deep Inelastic Scattering e + e - Annihilation Hadron Collisions Heavy Quarkonia Λ(5) MS NLO NNLO α s(μ Z) { 245 MeV QCD O(α MeV s) 181 MeV Q [GeV] Lattice anti-screening: strong coupling becomes weak at high energies! asymptotic freedom at high energies ("weak QCD") confinement at low energies ("strong QCD"): no quarks + gluons, only (colour-neutral) hadrons baryons (rgb) + mesons (r r) perturbation theory in α s at low energies: impossible! Chiral perturbation theory and dispersion relations p. 3

4 QCD: the spectrum of hadrons Mass [MeV] φ a 0, f 0 η ρ, ω η K Λ Σ p, n π 0 mesons baryons what does this spectrum have to do with the theory of quarks and gluons? Chiral perturbation theory and dispersion relations p. 4

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9 Illustration: spontaneous symmetry breaking figures courtesy of A. Wirzba Chiral perturbation theory and dispersion relations p. 5

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14 Weinberg s power counting argument ν = d V d (d 2)+2L+2 example: ππ scattering ν = 2 only lowest-order tree graphs: V d>2 = 0, L = 0 Chiral perturbation theory and dispersion relations p. 6

15 Weinberg s power counting argument ν = d V d (d 2)+2L+2 example: ππ scattering ν = 4 one-loop graphs with L (2) : V d>2 = 0, L = 1 or one insertion from L (4) : V 4 = 1, V d>4 = 0, L = 0 Chiral perturbation theory and dispersion relations p. 6

16 Weinberg s power counting argument ν = d V d (d 2)+2L+2 example: ππ scattering ν = 6 two-loop graphs with L (2) : V d>2 = 0, L = 2 or one-loop with one vertex from L (4) : V 4 = 1, V d>4 = 0, L = 1 or two insertions from L (4) : V 4 = 2, V d>4 = 0, L = 0 or one insertion from L (6) : V 4 = 0, V 6 = 1, V d>6 = 0, L = 0 Chiral perturbation theory and dispersion relations p. 6

17 The LagrangianL (4) in SU(2) L (4) = l 1 4 D µu D µ U 2 + l 2 4 D µu D ν U D µ U D ν U + l 3 16 χ U +χu 2 + l 4 4 D µud µ χ +D µ χd µ U +l 5 F R,µν U F µν L U + il 6 2 Fµν R D µu D ν U +F µν L D µud ν U l 7 16 χ U χu 2 +L WZW Symbols: D µ U = µ U i[v µ,u] i{a µ,u} χ = 2B(s+ip), s = M+... F µν R = µ r ν ν r µ i[r µ,r ν ], F µν L r µ = v µ +a µ, l µ = v µ a µ covariant derivative (pseudo)scalar sources =... field strength tensors right-/left-handed currents Wess Zumino Witten term / chiral anomaly L WZW : of odd intrinsic parity / odd number of Goldstone bosons describes processes such as π 0 γγ, γπ π 0 π... Chiral perturbation theory and dispersion relations p. 7

18 The LagrangianL (4) in SU(2) L (4) = l 1 4 D µu D µ U 2 + l 2 4 D µu D ν U D µ U D ν U + l 3 16 χ U +χu 2 + l 4 4 D µud µ χ +D µ χd µ U +l 5 F R,µν U F µν L U + il 6 2 Fµν R D µu D ν U +F µν L D µud ν U l 7 16 χ U χu 2 +L WZW Physics: l 1,2 = O( 4 ): needs 4 pions (e.g.) D-wave ππ scattering l 3 = O(m 2 q), l 4 = O( 2 m q ): "symmetry breakers", control m q -dependence of M 2 π, F π l 5 : requires 2 currents: radiative π decay π + l + ν l γ l 6 : t-dependence / radius of π vector (charge) form factor l 7 : isospin-breaking correction (m u m d ) 2 to M 2 π 0 Chiral perturbation theory and dispersion relations p. 7

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20 The pion vector form factor F V (s ππ ) ChPT at one loop data Omnès representation -0,2 0 0,2 0,4 0,6 0,8 1 s ππ [GeV 2 ] Stollenwerk et al Chiral perturbation theory and dispersion relations p. 8

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28 Spectra forη, η π + π γ dγ/de γ [arb. units] E γ [GeV] dγ/de γ [arb. units] E γ [GeV] P(s ππ ) P(s ππ ) s ππ [GeV 2 ] s ππ [GeV 2 ] Stollenwerk et al Chiral perturbation theory and dispersion relations p. 9

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31 Combined result on quark mass ratios (1) Combined information on Q η 3π vs. various corrections to Dashen s theorem: Bijnens, Ghorbani Kambor et al. Lanz et al. (prel.) Q Dashen Lattice Bijnens&Prades Donoghue&Perez Anant.&Moussallam η > 3π (M K + - M K 0) em [MeV] Chiral perturbation theory and dispersion relations p. 10

32 Combined result on quark mass ratios (2) additional constraints needed to find position on the ellipse: m u m d m s m d Q from η decay 15 MILC 09 PACS-CS 08 RBC/UKQCD 08 PDG 08 RBC 07 Bijnens & Ghorbani Namekawa & Kikukawa 06 MILC 04 Nelson, Fleming & Kilcup 03 Gao, Yan & Li 97 5 χpt fails χpt must Kaiser 97 5 Leutwyler 96 be reordered Schechter et al. 93 Donoghue, Holstein & Wyler 92 Gerard 90 Cline 89 0 Gasser and Leutwyler Langacker & Pagels 79 Weinberg 77 Gasser & Leutwyler 75 Leutwyler 2009 Chiral perturbation theory and dispersion relations p. 11

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40 From unitarity to integral equations: solution integral equations including the inhomogeneities ˆM I : M 0 (s) = Ω 0 (s) {α 0 +β 0 s+γ 0 s 2 + s3 ds sinδ 0 (s ) ˆM } 0 (s ) π s 3 Ω 0 (s ) (s s iǫ) 4M 2 π + 2 similar for M 1,2 (s); 4 subtraction constants to be fixed Khuri, Treiman 1960; Aitchison 1977; Anisovich, Leutwyler 1998 solve these equations iteratively by a numerical procedure Re M(s,t=u) tree level 1 iteration 2 iterations final result Im M(s,t=u) 2 1 tree level 1 iteration 2 iterations final result s [M π 2 ] s [M π 2 ] Schneider, Kubis; compare Colangelo et al fast convergence: close to final result after 2 iterations Chiral perturbation theory and dispersion relations p. 12

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