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/ψ 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/ψ 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
8 2 6
4 1 2
0 0 -3 -2 -1 0 1 2 4 6 8 10 10 10 10 10 10
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 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: DSE vs. FRG:
2 2
1
1 0 DSE scaling
DSE decoupling - 0 1 3-loop 3PI EOM FRG scaling
3-loop 3PI EOM (SP) FRG decoupling -2 Lattice 2-loop DSE (SP) -1 0 1 2 3 4 5 0 1 2 3 4 5
[Cucchieri, Maas, Mendes ’08; Sternbeck et al. ’17; [Cucchieri, Maas, Mendes ’08; Sternbeck et al. ’17; Cyrol et al. ’16; MQH ’20] MQH ’20]
Markus Q. Huber Giessen University April 21, 2021 10/16 Correlation functions Results Concurrence of functional methods
Exemplified with three-gluon vertex.
3PI vs. 2-loop DSE: DSE vs. FRG:
2 2
1
1 0 DSE scaling
DSE decoupling - 0 1 3-loop 3PI EOM FRG scaling
3-loop 3PI EOM (SP) FRG decoupling -2 Lattice 2-loop DSE (SP) -1 0 1 2 3 4 5 0 1 2 3 4 5
[Cucchieri, Maas, Mendes ’08; Sternbeck et al. ’17; [Cucchieri, Maas, Mendes ’08; Sternbeck et al. ’17; Cyrol et al. ’16; MQH ’20] 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]
Markus Q. Huber Giessen University April 21, 2021 10/16 Consider the eigenvalue problem (Γ is the BSE amplitude)
Γ(P) = λ(P)Γ( P). K· λ(P2) = 1 is a solution to the BSE Glueball mass P2 = M2 ⇒ − Calculation requires quantities for
k 2 = P2 + k 2 2√P2 k 2 cos θ = M2 + k 2 2 i M √k 2 cos θ. ± ± − ± Complex momentum arguments. ⇒
Glueballs BSE Solving a BSE
S(k ) −
= K(k,p,P ) Γ(k, P ) Γ(p, P ) S(k+)
Markus Q. Huber Giessen University April 21, 2021 11/16 Calculation requires quantities for
k 2 = P2 + k 2 2√P2 k 2 cos θ = M2 + k 2 2 i M √k 2 cos θ. ± ± − ± Complex momentum arguments. ⇒
Glueballs BSE Solving a BSE
S(k ) −
= K(k,p,P ) Γ(k, P ) Γ(p, P ) S(k+)
Consider the eigenvalue problem (Γ is the BSE amplitude)
Γ(P) = λ(P)Γ( P). K· λ(P2) = 1 is a solution to the BSE Glueball mass P2 = M2 ⇒ −
Markus Q. Huber Giessen University April 21, 2021 11/16 Glueballs BSE Solving a BSE
S(k ) −
= K(k,p,P ) Γ(k, P ) Γ(p, P ) S(k+)
Consider the eigenvalue problem (Γ is the BSE amplitude)
Γ(P) = λ(P)Γ( P). K· λ(P2) = 1 is a solution to the BSE Glueball mass P2 = M2 ⇒ − Calculation requires quantities for
k 2 = P2 + k 2 2√P2 k 2 cos θ = M2 + k 2 2 i M √k 2 cos θ. ± ± − ± Complex momentum arguments. ⇒
Markus Q. Huber Giessen University April 21, 2021 11/16 Test extrapolation for solvable system: Heavy meson
1.2
1.0
0.8
0.6
0.4
0.2
0.0 0 2 4 6
[MQH, Sanchis-Alepuz, Fischer ’20]
Glueballs Method Extrapolation of λ(P2)
Extrapolation method Extrapolation to time-like P2 using Schlessinger’s continued fraction method (proven superior to default Padé approximants) [Schlessinger ’68] Average over extrapolations using subsets of points for error estimate
Markus Q. Huber Giessen University April 21, 2021 12/16 Glueballs Method Extrapolation of λ(P2)
Extrapolation method Extrapolation to time-like P2 using Schlessinger’s continued fraction method (proven superior to default Padé approximants) [Schlessinger ’68] Average over extrapolations using subsets of points for error estimate
Test extrapolation for solvable system: Heavy meson
1.2
1.0
0.8
0.6
0.4
0.2
0.0 0 2 4 6
[MQH, Sanchis-Alepuz, Fischer ’20]
Markus Q. Huber Giessen University April 21, 2021 12/16 1.2 1.2
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0 0 5 10 0 5 10 15
Physical solutions for λ(P2) = 1.
Glueballs Method Extrapolation of λ(P2) for glueballs
Higher eigenvalues: Excited states.
Markus Q. Huber Giessen University April 21, 2021 13/16 Glueballs Method Extrapolation of λ(P2) for glueballs
Higher eigenvalues: Excited states.
1.2 1.2
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0 0 5 10 0 5 10 15
Physical solutions for λ(P2) = 1.
Markus Q. Huber Giessen University April 21, 2021 13/16 Under conjecture that choice of solution is a gauge choice: Explicit test of gauge independence! Tested that results are independent of family of solutions.
Glueballs Results + Glueballs masses for 0±
Spin-0 glueballs
4
∗∗ 3 Lattice 0 ++: Conjectured based on 2 irred. rep. of octahedral BSE group lattice[Morningstar, Peardon, 1999] 1 lattice[Athenodorou, Teper, 2020]
0
All results for r0 = 1/418(5) MeV. [MQH, Fischer, Sanchis-Alepuz ’20]
Markus Q. Huber Giessen University April 21, 2021 14/16 Glueballs Results + Glueballs masses for 0±
Spin-0 glueballs
4
∗∗ 3 Lattice 0 ++: Conjectured based on 2 irred. rep. of octahedral BSE group lattice[Morningstar, Peardon, 1999] 1 lattice[Athenodorou, Teper, 2020]
0
All results for r0 = 1/418(5) MeV. [MQH, Fischer, Sanchis-Alepuz ’20]
Under conjecture that choice of solution is a gauge choice: Explicit test of gauge independence! Tested that results are independent of family of solutions.
Markus Q. Huber Giessen University April 21, 2021 14/16 Glueballs Results + Glueball masses for J±
6
5 *
* * *
4 * * * * Lattice: * *: identification with some 3 uncertainty * * †: conjecture based on 2 BSE irred. rep of octahedral
1 lattice[Morningstar, Peardon, 1999] group preliminarylattice[Athenodorou, Teper, 2020]
0
[MQH, Fischer, Sanchis-Alepuz, in preparation]
Markus Q. Huber Giessen University April 21, 2021 15/16 I Comparison with lattice results I Concurrence of different functional methods
Quantitatively reliable correlation functions (Euclidean) from functional equations
Connection to observables: Glueballs
Systematic improvements (now) possible
Direct access to analytic structure [Fischer, MQH ’20]
Summary Summary
Parameter-free determination of glueball masses from functional methods.
Markus Q. Huber Giessen University April 21, 2021 16/16 I Comparison with lattice results I Concurrence of different functional methods Connection to observables: Glueballs
Systematic improvements (now) possible
Direct access to analytic structure [Fischer, MQH ’20]
Summary Summary
Parameter-free determination of glueball masses from functional methods.
Quantitatively reliable correlation functions (Euclidean) from functional equations
Markus Q. Huber Giessen University April 21, 2021 16/16 I Concurrence of different functional methods Connection to observables: Glueballs
Systematic improvements (now) possible
Direct access to analytic structure [Fischer, MQH ’20]
Summary Summary
Parameter-free determination of glueball masses from functional methods.
Quantitatively reliable correlation functions (Euclidean) from functional equations I Comparison with lattice results
Markus Q. Huber Giessen University April 21, 2021 16/16 Connection to observables: Glueballs
Systematic improvements (now) possible
Direct access to analytic structure [Fischer, MQH ’20]
Summary Summary
Parameter-free determination of glueball masses from functional methods.
Quantitatively reliable correlation functions (Euclidean) from functional equations I Comparison with lattice results I Concurrence of different functional methods
Markus Q. Huber Giessen University April 21, 2021 16/16 Systematic improvements (now) possible
Direct access to analytic structure [Fischer, MQH ’20]
Summary Summary
Parameter-free determination of glueball masses from functional methods.
Quantitatively reliable correlation functions (Euclidean) from functional equations I Comparison with lattice results I Concurrence of different functional methods Connection to observables: Glueballs
Markus Q. Huber Giessen University April 21, 2021 16/16 Direct access to analytic structure [Fischer, MQH ’20]
Summary Summary
Parameter-free determination of glueball masses from functional methods.
Quantitatively reliable correlation functions (Euclidean) from functional equations I Comparison with lattice results I Concurrence of different functional methods Connection to observables: Glueballs
Systematic improvements (now) possible
Markus Q. Huber Giessen University April 21, 2021 16/16 Summary Summary
Parameter-free determination of glueball masses from functional methods.
Quantitatively reliable correlation functions (Euclidean) from functional equations I Comparison with lattice results I Concurrence of different functional methods Connection to observables: Glueballs
Systematic improvements (now) possible
Direct access to analytic structure [Fischer, MQH ’20]
Markus Q. Huber Giessen University April 21, 2021 16/16 Summary Summary
Parameter-free determination of glueball masses from functional methods.
Quantitatively reliable correlation functions (Euclidean) from functional equations I Comparison with lattice results I Concurrence of different functional methods Connection to observables: Glueballs
Systematic improvements (now) possible
Direct access to analytic structure [Fischer, MQH ’20]
Thank your for your attention.
Markus Q. Huber Giessen University April 21, 2021 16/16 PC ++ µν Example: For J = 0 glueball take O(x) = Fµν (x)F (x): D(x y) = O(x)O(y) − h i Lattice: Mass from this correlator by exponential Euclidean time decay.→ Complicated object in a diagrammatic language: 2-, 3- and 4-gluon contributions Put total momentum on-shell and consider individual 2-, 3- and 4-gluon contributions. Each can have a pole at the glueball mass. →
A4-part of D(x y), total momentum on-shell: −
P 2 M 2 →−
More details. . . Glueballs as bound states
Hadron masses from correlation functions of color singlet operators.
Markus Q. Huber Giessen University April 21, 2021 17/16 Put total momentum on-shell and consider individual 2-, 3- and 4-gluon contributions. Each can have a pole at the glueball mass. →
A4-part of D(x y), total momentum on-shell: −
P 2 M 2 →−
More details. . . Glueballs as bound states
Hadron masses from correlation functions of color singlet operators. PC ++ µν Example: For J = 0 glueball take O(x) = Fµν (x)F (x): D(x y) = O(x)O(y) − h i Lattice: Mass from this correlator by exponential Euclidean time decay.→ Complicated object in a diagrammatic language: 2-, 3- and 4-gluon contributions
Markus Q. Huber Giessen University April 21, 2021 17/16 More details. . . Glueballs as bound states
Hadron masses from correlation functions of color singlet operators. PC ++ µν Example: For J = 0 glueball take O(x) = Fµν (x)F (x): D(x y) = O(x)O(y) − h i Lattice: Mass from this correlator by exponential Euclidean time decay.→ Complicated object in a diagrammatic language: 2-, 3- and 4-gluon contributions Put total momentum on-shell and consider individual 2-, 3- and 4-gluon contributions. Each can have a pole at the glueball mass. →
A4-part of D(x y), total momentum on-shell: −
P 2 M 2 →−
Markus Q. Huber Giessen University April 21, 2021 17/16 More details. . . Charge parity
Transformation of gluon field under charge conjugation: Aa η(a)Aa µ → − µ where +1 a = 1, 3, 4, 6, 8 η(a) = 1 a = 2, 5, 7 − Color neutral operator with two gluon fields: Aa Aa η(a)2Aa Aa = Aa Aa . µ ν → µ ν µ ν C = +1 ⇒ Negative charge parity, e.g.: d abcAa Ab Ac d abcη(a)η(b)η(c)Aa Ab Ac = µ ν ρ → − µ ν ρ d abcAa Ab Ac . − µ ν ρ Only nonvanishing elements of the symmetric structure constant d abc : zero or two indices equal to 2, 5 or 7.
Markus Q. Huber Giessen University April 21, 2021 18/16 More details. . . Landau gauge vertices
Ghost-gluon vertex: Three-gluon vertex:
1.5 2 DSE 1.4 Sternbeck et al., 2017
1.3 1 Cucchieri et al., 2008
1.2 0 1.1
1.0 -1
0.9
0.8 -2 10-2 10-1 100 101 0 1 2 3 4 5
[Maas ’19; MQH ’20] [Cucchieri, Maas, Mendes ’08; Sternbeck et al. ’17; MQH ’20]
Nontrivial kinematic dependence Four-gluon vertex: 5.0 of ghost-gluon vertex Simple kinematic dependence of three-gluon vertex 2.5 Four-gluon vertex from solution
0.0 10-2 10-1 100 101
[MQH ’20]
Markus Q. Huber Giessen University April 21, 2021 19/16 Renormalization: First 14 parameter-free subtraction of 12 quadratic divergences 10 One unique free parameter 8 6 (family⇒ of solutions) 4
2
0 10-3 10-2 10-1 100 101
More details. . . Some properties of the Landau gauge solution
[MQH ’20]
Slavnov-Taylor identities (gauge 101 invariance): Vertex couplings 100
- agree down to GeV regime 10 1
- 10 2 α - ghg 10 3
α3g - 10 4 α4g
-5 10 - - 10 2 10 1 100 101 102
Markus Q. Huber Giessen University April 21, 2021 20/16 More details. . . Some properties of the Landau gauge solution
[MQH ’20]
Slavnov-Taylor identities (gauge 101 invariance): Vertex couplings 100
- agree down to GeV regime 10 1
- 10 2 α - ghg 10 3
α3g - 10 4 α4g
-5 10 - - 10 2 10 1 100 101 102
Renormalization: First 14 parameter-free subtraction of 12 quadratic divergences 10 One unique free parameter 8 6 (family⇒ of solutions) 4
2
0 10-3 10-2 10-1 100 101
Markus Q. Huber Giessen University April 21, 2021 20/16 Contour deformation: Special technique to respect analyticity (avoid branch cuts in the integrand) I QED3 [Maris ’95 (QED)] I Quark propagator [Alkofer, Fischer, Detmold, Maris ’04] I Self-consistent solution: Ray technique, YM propagators [Strauss, Fischer, Kellermann ’12; Fischer, MQH ’20] I Glueball correlators [Windisch, Alkofer, Haase, Liebmann ’13; Windisch, MQH, Alkofer ’13] I Meson decays [Weil, Eichmann, Fischer, Williams ’17; Williams ’18] I Spectral functions at T > 0 [Pawlowski, Strodthoff, Wink ’18] I Quark-photon vertex [Miramontes, Sanchis-Alepuz ’19] I Scalar scattering amplitude [Eichmann, Duarte, Pena, Stadler ’19]
More details. . . Landau gauge propagators in the complex plane
Propagators for complex momenta Reconstruction from Euclidean results: mathematically ill-defined, bias in solution Direct calculation from functional methods possible, e.g., contour deformation or spectral DSEs [Horak, Pawlowski, Wink ’20]
Markus Q. Huber Giessen University April 21, 2021 21/16 More details. . . Landau gauge propagators in the complex plane
Propagators for complex momenta Reconstruction from Euclidean results: mathematically ill-defined, bias in solution Direct calculation from functional methods possible, e.g., contour deformation or spectral DSEs [Horak, Pawlowski, Wink ’20]
Contour deformation: Special technique to respect analyticity (avoid branch cuts in the integrand) I QED3 [Maris ’95 (QED)] I Quark propagator [Alkofer, Fischer, Detmold, Maris ’04] I Self-consistent solution: Ray technique, YM propagators [Strauss, Fischer, Kellermann ’12; Fischer, MQH ’20] I Glueball correlators [Windisch, Alkofer, Haase, Liebmann ’13; Windisch, MQH, Alkofer ’13] I Meson decays [Weil, Eichmann, Fischer, Williams ’17; Williams ’18] I Spectral functions at T > 0 [Pawlowski, Strodthoff, Wink ’18] I Quark-photon vertex [Miramontes, Sanchis-Alepuz ’19] I Scalar scattering amplitude [Eichmann, Duarte, Pena, Stadler ’19]
Markus Q. Huber Giessen University April 21, 2021 21/16 More details. . . Landau gauge propagators in the complex plane
Appearance of branch cuts for complex momenta forbids integration directly to cutoff.
Deformation of integration contour necessary [Maris ’95]. Recent resurgence: [Alkofer et al. ’04; Windisch, MQH, Alkofer, ’13; Williams ’19; Miramontes, Sanchis-Alepuz ’19; Eichmann et al. ’19],...
Ray technique for self-consistent solution of a DSE: [Strauss, Fischer, Kellermann; Fischer, MQH ’20].
Markus Q. Huber Giessen University April 21, 2021 22/16 2 2 i θ Polar coordinates: p = ep e
Current truncation leads to a pole-like structure in the gluon propagator. Analyticity up to ’pole’ confirmed by various tests (Cauchy-Riemann, Schlessinger, reconstruction) No proof of existence of complex conjugate poles due to simple truncation. [Fischer, MQH ’20]
More details. . . Landau gauge propagators in the complex plane
Technique to resp. analyticity (avoid branch cuts in integrand): Contour deformation
1 1 − − 1 + Simpler truncation: = −2
Markus Q. Huber Giessen University April 21, 2021 23/16 More details. . . Landau gauge propagators in the complex plane
Technique to resp. analyticity (avoid branch cuts in integrand): Contour deformation
1 1 − − 1 + Simpler truncation: = −2
2 2 i θ Polar coordinates: p = ep e
Current truncation leads to a pole-like structure in the gluon propagator. Analyticity up to ’pole’ confirmed by various tests (Cauchy-Riemann, Schlessinger, reconstruction) No proof of existence of complex conjugate poles due to simple truncation. [Fischer, MQH ’20]
Markus Q. Huber Giessen University April 21, 2021 23/16