Dyson Schwinger 101 Research in the DSE approach Esther Weil Justus-Liebig-Universität Gießen Institut für Theoretische Physik γ µ December, 12th 2017 Cold Quantum Coffee-Heidelberg
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen Goal : Compute Hadron Properties (such as spectra, form factors, scattering amplitudes … ) using QCD’s Greens functions on a quark-gluon level. QCD effective derive Dyson-Schwinger equations (DSE) action “The quantum equation of motion“ + Bethe Salpeter equations describing bound states non-perturbative, at all momentum scales, light and heavy quarks. But for numerical evaluation a truncation is needed. Intro 2 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen Christian Fischers group, JLU Gießen Group topics include: • Elastic & transition form factors Eichmann, Fischer, Weil, Williams • Tetraquarks Wallbott, Eichmann, Fischer eg. PLB, 753, 1508.07178 (2016) • Exotic & Hybrid mesons Söhngen, Eshraim, Williams, Huber • QCD phase diagram Gunkel, Isserstedt, Haque • Meson and Bayron spectroscopy Eichmann, Sanchis-Alepuz, Williams, Fischer - recent Review (2016) • Nucleon Compton scattering Eichmann, Fischer, PRD 87 (2014) Eichmann, G. Few-Body Syst (2016) 57: 965 • Muons g-2 Phys. Rev. D 86 , 099901 (2012) Bonnet, Eichmann, Fischer, Williams eg. Intro 3 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen A − M ) ψ ( x ) − 1 L = ¯ QCD Lagrangian ψ ( x )( i/ ∂ + g / 4 F a µ ν F µ ν a degrees of quarks, gluons freedom in QCD but those can not be seen in nature. We observe only hadrons mesons tetraquarks? baryons hybrids? glueballs? asymptotic freedom quark and gluons are confined calls for non-perturbative methods such as the Dyson-Schwinger approach ! Intro webific.ific.uv.es/web/en 4 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen quark propagator The so called self-energy includes all possible quark self interactions. Are these included? Origin of the equation Deriving the quark Dyson-Schwinger equation diagrammatically (i) QCD’s action 5 A − M ) ψ ( x ) − 1 6 ⁄ ¯ d 4 x ψ ( x )( i/ ∂ + g / S = 4 F a µ ν F µ ν Starting from a we obtain the following action diagrammatically (ii) Take the functional derivative with respect to the quark fields Theory stuff 5 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen quark propagator The so called self-energy includes all possible quark self interactions. Are these included? Origin of the equation Deriving the quark Dyson-Schwinger equation diagrammatically (i) QCD’s action 5 A − M ) ψ ( x ) − 1 6 ⁄ ¯ d 4 x ψ ( x )( i/ ∂ + g / S = 4 F a µ ν F µ ν Starting from a we obtain the following action diagrammatically (ii) Take the functional derivative with respect to the quark fields Theory stuff 5 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen quark propagator The so called self-energy includes all possible quark self interactions. Are these included? Origin of the equation Deriving the quark Dyson-Schwinger equation diagrammatically (i) QCD’s action 5 A − M ) ψ ( x ) − 1 6 ⁄ ¯ d 4 x ψ ( x )( i/ ∂ + g / S = 4 F a µ ν F µ ν Starting from a we obtain the following action diagrammatically (ii) Take the functional derivative with respect to the quark fields Theory stuff 5 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen derivation inspired by Gernot Eichmann, Marcus Huber, Reinhard Alkhofer and Kai Schwenzer. (ii) Take the functional derivative with respect to the quark fields and are related thru S Γ a Legendre-Trafo. connecting the dots in all (iii) Replace the action by the effective action possible ways lead to the quantum equation of motion (iv) Take another derivative and set the field to zero quark Dyson-Schwinger equation and there is more! - equations for the different Greens functions Theory stuff 6 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen quark propagator + = Dyson-Schwinger equation (DSE) gluon propagator ghost propagator equation for bound + = Bethe-Salpeter equation (BSE) states and vertices vertices P + = Γ µ ( p, P ) γ µ + K eg. quark-gluon vertex, quark-photon vertex bound states q + P K Γ p Γ = q quark propagator q − eg. for a meson Theory stuff 7 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen includes higher order Greens functions quark propagator + gluon propagator quark-gluon vertex ghost propagator gluon propagator + all these equations are rigorous and non-perturbative vertices but but we have an infinite set of coupled integral equations! + Need to introduce a truncation for numerical evaluations ! bound states Theory stuff 8 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen Truncate, but how? To preserve chiral symmetry we use a consistent truncation in each equation. We choose an Ansatz for higher order Greens function: Rainbow-ladder In the case of the rainbow-ladder approximation we combine the quark-gluon vertex and gluon propagator to a truncation tree-level vertex + effective coupling DSE self-energy term Ansatz = tree-level + BSE kernel effective coupling α ( k 2 ) K = ∂ Σ Σ = ∂ Γ α ( k 2 ) ∂ S = ∂ S = self-energy and kernel are related through a symmetry relation, Axial-Vector Ward-Takehashi identity Numerical evaluations 9 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen Combine gluon with quark-gluon vertex to an effective coupling α ( k 2 ) + tree-level vertex Maris, Roberts,Tandy, PRC 56 (1997), PRC 60 (1999) Ansatz effective coupling: α ( k 2 ) = α IR ( k 2 / Λ 2 , η ) + α UV ( k 2 ) α ( k 2 ) scale is adjusted to observables, like . f π Λ masses m u =m d , m s from m , m K π from perturbation theory - UV : rigorous field theory part α UV parameter : band of results - IR : modeled part η Numerical evaluations 10 /27
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen DSE results for the quark propagator , Using the rainbow-ladder approximation: for different quark masses: Fischer J. Phys G. 32 (2006) works well for not as well for pseudo-scalar and vector excited scalar, axial- mesons, also form- vector mesons factors & decays - too light. similar for the baryons . 350 MeV excited bayrons More complicated, two different approaches. Eichmann, Fischer, Williams, Alkhofer PPN 91 ( 2016 ) Quark mass 3 MeV function [GeV]: excited state are problematic! Bottom Charm Calculations beyond Strange Up/down rainbow-ladder Chiral limit Phys. Rev. D 93 , 034026 quark mass generation through Using RL, we are ready dynamical chiral symmetry breaking . to calculate various QCD observables. the kinematic regime where we can limitations in kinematic regim due to poles calculate is limited due to poles in the in the calculations, e.g.. quark poles. quark propagators. Numerical evaluations 11 /27
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen Next : π 0 → γγ PLB V 774 , 425-429 Pion transition form factor arXiv:1704.05774v3 [hep-ph] π 0 + Λ µ ν ( Q, Q 0 ) = e 2 F ( Q 2 , Q 0 2 ) ε µ ναβ Q 0 α Q β 4 π 2 f π SU(3) flavor From the anomaly we know anomaly F (0 , 0) = 1 A DLER , B ELL , In the chiral limit ( ) m π = 0 J ACKIW , 1969 Recent Research 12 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen Next : π 0 → γγ PLB V 774 , 425-429 Pion transition form factor arXiv:1704.05774v3 [hep-ph] We need these building blocks: γ µ quark propagator π 0 Pion amplitude Quark-photon vertex Λ µ ν ( Q, Q 0 ) = e 2 F ( Q 2 , Q 0 2 ) ε µ ναβ Q 0 α Q β 4 π 2 f π SU(3) flavor From the anomaly we know anomaly F (0 , 0) = 1 A DLER , B ELL , In the chiral limit ( ) m π = 0 J ACKIW , 1969 Recent Research 12 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen F ( Q 2 , Q Õ 2 ) = η + F ( Q 2 , Q Õ 2 )  Result for the form factor in the asymmetric 4 π 2 f 2 and symmetric ( ) limit Q 2 = Q Õ 2 π 1.0 1.0 kinematic restriction due to the limitations 2.0 � ( �� , � ´ � ) of the building blocks, e.g quark Belle � 2 propagator in the complex plane 0.8 0.8 2 ( ′ ) F Q , Q BaBar CLEO 1.5 asym CELLO Q 2 symmetric limit for all sym 0.6 0.6 Q 2 < 4GeV 2 asymmetric cone 1.0 0.4 0.4 In comparison 0.5 Dispersion relation DSE (asym) DSE (sym) 0.2 0.2 M. Hoferichter et al.(2014), arXiv:1410.4691 [hep-ph] 0.0 Lattice 0 5 10 15 20 �� ���� Q 2 �� [ ���� ] �� ���� estimate of the continuum limit �� ���� 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 A. Geradin et al. (2016), arXiv:1508.07178 [hep-lat] 0.0 0.2 0.4 0.6 0.8 1.0 combined effort of different non- �� [ ���� ] �� [ ���� ] perturbative methods. singly-virtual up to 2 ∼ 2GeV 2 Non-perturbative input to further electro-magnetic pion decays Recent Research 13 /18
Dyson-Schwinger 101 Tues , 12.12.17 Esther Weil JLU Giessen Next : Phys. Rev. D 96 , 014021 π 0 → e + e − Rare decay of the pion arXiv:1704.06046v2 [hep-ph] Discrepancy between theory and experiment of 2 σ BR exp ( π 0 e + e − ) ≈ 6 . 87(36) × 10 − 8 theory calculations A ( t ) ∼ ∆ ( π 0 ) The branching ratio is given by with t 0 = − m 2 π / 4 and the scalar amplitude Recent Research 14 /18
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