On the mass spectrum of glueballs with even charge parity 4 6 1 1 5 − − 1 1 3 * = −2 −2 * * * 4 * * * * * + 1 1 2 −6 −2 3 * * 2 1 BSE 1 lattice[Morningstar, Peardon, 1999] 1 1 1 = 2 + +2 +2 +2 + ... − lattice[Athenodorou, Teper, 2020] 0 0 2 4 6 8 10 Markus Q. Huber MQH, Phys.Rev.D 101, arXiv:2003.13703 MQH, Fischer, Sanchis-Alepuz, Eur.Phys.J.C 80, Institute of Theoretical Physics, Giessen arXiv:2004.00415 University ACHT2021 April 21, 2021 Markus Q. Huber Giessen University April 21, 2021 1/16 Pentaquarks First observations 2015 (LHCb) Tetraquarks Increasing number of confirmed states Hybrids Glueballs States of pure ’radiation’ Introduction QCD bound states Bound states in QCD Mesons Baryons Markus Q. Huber Giessen University April 21, 2021 2/16 Introduction QCD bound states Bound states in QCD Mesons Baryons Pentaquarks First observations 2015 (LHCb) Tetraquarks Increasing number of confirmed states Hybrids Glueballs States of pure ’radiation’ Markus Q. Huber Giessen University April 21, 2021 2/16 Recent analysis of BESIII data [Sarantsev, Denisenko, Thoma, Klempt ’21]: M = 1865 25+10 MeV, ± −30 Γ = 370 50+30 MeV ± −20 Future experiments: e.g., PANDA, GlueX Introduction QCD bound states Glueball observations Experimental candidates, but situation not conclusive. Scalar glueball: 0++, mixing with scalar isoscalar mesons Candidate reaction: J= γ + 2g ! Markus Q. Huber Giessen University April 21, 2021 3/16 Introduction QCD bound states Glueball observations Experimental candidates, but situation not conclusive. Scalar glueball: 0++, mixing with scalar isoscalar mesons Candidate reaction: J= γ + 2g ! Recent analysis of BESIII data [Sarantsev, Denisenko, Thoma, Klempt ’21]: M = 1865 25+10 MeV, ± −30 Γ = 370 50+30 MeV ± −20 Future experiments: e.g., PANDA, GlueX Markus Q. Huber Giessen University April 21, 2021 3/16 Unquenching on the lattice [Gregory et al. ’12]: Much higher statistics required (poor signal-to-noise ratio) Continuum extrapolation and inclusion of fermionic operators still to be done Mixing with qq¯ challenging Tiny (e.g., 0++, 2++) to moderate unquenching effects (e.g., 0−+) found Introduction QCD bound states Glueball calculations Yang-Mills theory “Isolated” problem: only gluons Clean picture: well-established lattice results Markus Q. Huber Giessen University April 21, 2021 4/16 Introduction QCD bound states Glueball calculations Yang-Mills theory “Isolated” problem: only gluons Clean picture: well-established lattice results Unquenching on the lattice [Gregory et al. ’12]: Much higher statistics required (poor signal-to-noise ratio) Continuum extrapolation and inclusion of fermionic operators still to be done Mixing with qq¯ challenging Tiny (e.g., 0++, 2++) to moderate unquenching effects (e.g., 0−+) found Markus Q. Huber Giessen University April 21, 2021 4/16 Introduction QCD bound states Glueball calculations Yang-Mills theory “Isolated” problem: only gluons Clean picture: well-established lattice results Functional methods: High quality input available for bound state equations Unquenching on the lattice [Gregory et al. ’12]: Much higher statistics required (poor signal-to-noise ratio) Continuum extrapolation and inclusion of fermionic operators still to be done Mixing with qq¯ challenging Tiny (e.g., 0++, 2++) to moderate unquenching effects (e.g., 0−+) found Markus Q. Huber Giessen University April 21, 2021 4/16 Bethe-Salpeter amplitude Ingredients: Quark propagator S Interaction kernel K Constrained by symmetries Nonperturbative diagram: full momentum dependent dressings numerical solution ! Introduction Bound state equations Hadrons from bound state equations Example: Meson Z Integral equation: Γ(q; P) = dk Γ(k; P) S(k+) S(k−) K (k; q; P) Markus Q. Huber Giessen University April 21, 2021 5/16 Introduction Bound state equations Hadrons from bound state equations Bethe-Salpeter amplitude Example: Meson Z Integral equation: Γ(q; P) = dk Γ(k; P) S(k+) S(k−) K (k; q; P) Ingredients: Quark propagator S Interaction kernel K Constrained by symmetries Nonperturbative diagram: full momentum dependent dressings numerical solution ! Markus Q. Huber Giessen University April 21, 2021 5/16 Not quite. Gluons couple to ghosts Include ’ghostball’-part. (First step: no quarks ! ! Yang-Mills theory) Need , and 4 , solve for and . Mass × ! Construction of kernel Consistency with input: Apply same construction principle. Previous BSE calculations for glueballs: Input is important for I [Meyers, Swanson ’13] quantitative) predictive I [Sanchis-Alepuz, Fischer, Kellermann, von Smekal ’15] power! I [Souza et al. ’20] [MQH, Fischer, Sanchis-Alepuz ’20] I [Kaptari, Kämpfer ’20] Introduction Bound state equations Glueball BSE Need and , solve for . Mass ! Markus Q. Huber Giessen University April 21, 2021 6/16 Gluons couple to ghosts Include ’ghostball’-part. (First step: no quarks ! ! Yang-Mills theory) Need , and 4 , solve for and . Mass × ! Construction of kernel Consistency with input: Apply same construction principle. Previous BSE calculations for glueballs: Input is important for I [Meyers, Swanson ’13] quantitative) predictive I [Sanchis-Alepuz, Fischer, Kellermann, von Smekal ’15] power! I [Souza et al. ’20] [MQH, Fischer, Sanchis-Alepuz ’20] I [Kaptari, Kämpfer ’20] Introduction Bound state equations Glueball BSE Need and , solve for . Mass ! Not quite. Markus Q. Huber Giessen University April 21, 2021 6/16 Need , and 4 , solve for and . Mass × ! Construction of kernel Consistency with input: Apply same construction principle. Previous BSE calculations for glueballs: Input is important for I [Meyers, Swanson ’13] quantitative) predictive I [Sanchis-Alepuz, Fischer, Kellermann, von Smekal ’15] power! I [Souza et al. ’20] [MQH, Fischer, Sanchis-Alepuz ’20] I [Kaptari, Kämpfer ’20] Introduction Bound state equations Glueball BSE = + = + Gluons couple to ghosts Include ’ghostball’-part. (First step: no quarks ! ! Yang-Mills theory) Markus Q. Huber Giessen University April 21, 2021 6/16 Previous BSE calculations for glueballs: Input is important for I [Meyers, Swanson ’13] quantitative) predictive I [Sanchis-Alepuz, Fischer, Kellermann, von Smekal ’15] power! I [Souza et al. ’20] [MQH, Fischer, Sanchis-Alepuz ’20] I [Kaptari, Kämpfer ’20] Introduction Bound state equations Glueball BSE = + = + Gluons couple to ghosts Include ’ghostball’-part. (First step: no quarks ! ! Yang-Mills theory) Need , and 4 , solve for and . Mass × ! Construction of kernel Consistency with input: Apply same construction principle. Markus Q. Huber Giessen University April 21, 2021 6/16 Introduction Bound state equations Glueball BSE = + = + Gluons couple to ghosts Include ’ghostball’-part. (First step: no quarks ! ! Yang-Mills theory) Need , and 4 , solve for and . Mass × ! Construction of kernel Consistency with input: Apply same construction principle. Previous BSE calculations for glueballs: Input is important for I [Meyers, Swanson ’13] quantitative) predictive I [Sanchis-Alepuz, Fischer, Kellermann, von Smekal ’15] power! I [Souza et al. ’20] [MQH, Fischer, Sanchis-Alepuz ’20] I [Kaptari, Kämpfer ’20] Markus Q. Huber Giessen University April 21, 2021 6/16 Introduction Bound state equations Kernel construction From 3PI effective action truncated to three-loops: [Fukuda ’87; McKay, Munczek ’89; Sanchis-Alepuz, Williams ’15; MQH, Fischer, Sanchis-Alepuz ’20] = + Need , , , . ! Some diagrams vanish for certain quantum numbers. Full QCD: Same for quarks Mixing with mesons. ! Markus Q. Huber Giessen University April 21, 2021 7/16 4 coupled integral equations with full kinematic dependence. Sufficient numerical accuracy required for renormalization. One- and two-loop diagrams [Meyers, Swanson ’14; MQH ’17]. Correlation functions Equations Equations of motion from 3-loop 3PI effective action 1 1 − = − − 1 1 − − 1 1 + 1 1 = −2 −2 −6 −2 1 1 1 = 2 + +2 +2 +2 + ... − = + + + ... Gluon and ghost fields: Elementary fields of Yang-Mills theory in the Landau gauge Self-contained system of equations with the scale as the only input. Truncation? Markus Q. Huber Giessen University April 21, 2021 8/16 Correlation functions Equations Equations of motion from 3-loop 3PI effective action 1 1 − = − − 1 1 − − 1 1 + 1 1 = −2 −2 −6 −2 1 1 1 = 2 + +2 +2 +2 + ... − = + + + ... Gluon and ghost fields: Elementary fields of Yang-Mills theory in the Landau gauge Self-contained system of equations with the scale as the only input. Truncation 3-loop expansion of 3PI effective action [Berges ’04] ! 4 coupled integral equations with full kinematic dependence. Sufficient numerical accuracy required for renormalization. One- and two-loop diagrams [Meyers, Swanson ’14; MQH ’17]. Markus Q. Huber Giessen University April 21, 2021 8/16 Correlation functions Results Landau gauge propagators Gluon dressing function: Gluon propagator: 14 4 12 3 10 2 8 6 1 4 0 2 4 6 2 8 10 0 10-3 10-2 10-1 100 101 Family of solutions: Ghost dressing function: Nonperturbative completions of 6 5 Landau gauge [Maas ’10]? 4 Realized by condition on G(0) 3 [Fischer, Maas, Pawlowski ’08; Alkofer, MQH, 2 Schwenzer ’08] 1 0 2 4 6 8 10 [Sternbeck ’06; MQH ’20] Markus Q. Huber Giessen University April 21, 2021 9/16 DSE vs. FRG: 2 1 0 DSE scaling DSE decoupling -1 FRG scaling FRG decoupling -2 Lattice 0 1 2 3 4 5 [Cucchieri, Maas, Mendes ’08; Sternbeck et al. ’17; Cyrol et al. ’16; MQH ’20] Beyond this truncation Further dressings of three-gluon vertex [Eichmann, Williams, Alkofer, Vujinovic ’14] Effects of four-point functions [MQH ’16, MQH ’17, Corell et al. ’18, MQH ’18] Correlation functions Results Concurrence of functional methods Exemplified with three-gluon vertex. 3PI vs. 2-loop DSE: 2 1 0 3-loop 3PI EOM 3-loop 3PI EOM (SP) 2-loop DSE (SP) -1 0 1 2 3 4 5 [Cucchieri, Maas, Mendes ’08; Sternbeck et al. ’17; MQH ’20] Markus Q. Huber Giessen University April 21, 2021 10/16 2 1 0 DSE scaling DSE decoupling -1 FRG scaling FRG decoupling -2 Lattice 0 1 2 3 4 5 Beyond this truncation Further dressings of three-gluon vertex [Eichmann, Williams, Alkofer, Vujinovic ’14] Effects of four-point functions [MQH ’16, MQH ’17, Corell et al. ’18, MQH ’18] Correlation functions Results Concurrence of functional methods Exemplified with three-gluon vertex.