Meson spectra and beyond from the BSE Andreas Krassnigg University - - PowerPoint PPT Presentation

Meson spectra and beyond from the BSE

Andreas Krassnigg

University of Graz, Austria

Non-Perturbative Methods in Quantum Field Theory, Heviz, March 2010

• A. Krassnigg, Heviz10,

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Work with

◮ Univ. Graz :: M. Blank

Special thanks to

◮ Univ. Zagreb :: D. Horvati´

c, D. Klabuˇ car

Work performed at/supported by/in collaboration with

◮ University of Graz ◮ Austrian Research Foundation FWF ◮ FWF Doctoral Prg. ”Hadrons in Vacuum, Nuclei, and Stars” ◮ University of Zagreb ◮ ¨

Osterreichische Forschungsgemeinschaft ¨ OFG

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Outline

Motivation QCD and Hadrons Dyson-Schwinger Equations Equations and Solutions Quark DSE Bethe-Salpeter Equation BSE Solution Strategies Symmetries and Exact Results AV WTI Truncation and Model Building Numerical Results Masses and Leptonic Decay Constants χ-Transition Temperatures Conclusion and Outlook

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Outline

Motivation QCD and Hadrons Dyson-Schwinger Equations Equations and Solutions Quark DSE Bethe-Salpeter Equation BSE Solution Strategies Symmetries and Exact Results AV WTI Truncation and Model Building Numerical Results Masses and Leptonic Decay Constants χ-Transition Temperatures Conclusion and Outlook

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Motivation :: QCD and Hadrons

◮ Study hadrons as composites of

quarks and gluons . . .

◮ . . . including:

◮ Chiral symmetry and DχSB ◮ correct perturbative limit (via αp(Q2)) ◮ quark and gluon confinement ◮ Poincar´

e covariance

◮ Calculate observables!

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Motivation :: QCD and Hadrons

◮ Study hadrons as composites of

quarks and gluons . . .

◮ . . . including:

◮ Chiral symmetry and DχSB ◮ correct perturbative limit (via αp(Q2)) ◮ quark and gluon confinement ◮ Poincar´

e covariance

◮ Calculate observables! ◮ Dyson Schwinger Equations:

a modern method in relativistic QFT

• C. D. Roberts and S. M. Schmidt, Prog. Part. Nucl. Phys. 45 (2000) S1
• R. Alkofer and L. von Smekal, Phys. Rept. 353 (2001) 281
• P. Maris and C. D. Roberts, Int. J. Mod. Phys. E 12 (2003) 297
• C. S. Fischer, J. Phys. G 32 (2006) R253
• C. D. Roberts, M. S. Bhagwat, A. Holl, S. V. Wright, Eur. Phys. J. Special Topics 140 (2007) 53
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Motivation :: Dyson-Schwinger Equations

◮ Euclidean Green functions (also calculated on the lattice)

satisfy the [Dyson, Schwinger] equations

◮ Each function satisfies integral equation involving other

functions ⇒

◮ Infinite set of coupled integral equations ◮ Truncation scheme ⇒ ◮ Generating tool for perturbation theory

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Motivation :: Dyson-Schwinger Equations

◮ Euclidean Green functions (also calculated on the lattice)

satisfy the [Dyson, Schwinger] equations

◮ Each function satisfies integral equation involving other

functions ⇒

◮ Infinite set of coupled integral equations ◮ Truncation scheme ⇒ ◮ Nonperturbative truncation scheme ◮ Respect symmetries ◮ Obtain correct chiral-limit results ◮ Construct (sophisticated) models ◮ Perform reliable calculations of hadron properties ◮ Propagators and Bethe-Salpeter amplitudes (BSAs)

→ can be used to calculate observables

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Even More Motivation

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Coming up next . . .

Motivation QCD and Hadrons Dyson-Schwinger Equations Equations and Solutions Quark DSE Bethe-Salpeter Equation BSE Solution Strategies Symmetries and Exact Results AV WTI Truncation and Model Building Numerical Results Masses and Leptonic Decay Constants χ-Transition Temperatures Conclusion and Outlook

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Gap Equation

S(p) = Z(p2) iγ · p + M(p2)

[ S ]−1 = [ S0 ]−1 + γ Γ S D Σ = γ Γ S D

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Gap Equation

S(p) = Z(p2) iγ · p + M(p2)

[ S ]−1 = [ S0 ]−1 + γ Γ S D Σ = γ Γ S D

current quark mass mζ

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Gap Equation

S(p) = Z(p2) iγ · p + M(p2)

[ S ]−1 = [ S0 ]−1 + γ Γ S D Σ = γ Γ S D

current quark mass mζ

◮ Side note: Weak coupling expansion reproduces every diagram

in perturbation theory, but:

◮ Perturbation theory: mζ = 0 ⇒ M(p2) ≡ 0 ◮ Here: get nonperturbative solution

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Quark Mass Function

Solution of gap equation: S(p) = Z(p2) iγ · p + M(p2)

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Quark Mass Function

Solution of gap equation: S(p) = Z(p2) iγ · p + M(p2)

10

• 2

10

• 1

10 10

1

10

2

p

2 [GeV 2]

10

• 3

10

• 2

10

• 1

10 10

1

M(p

2) [GeV]

χ limit u/d quark s quark c quark b quark M

2 = p 2

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Quark Mass Function

Solution of gap equation: S(p) = Z(p2) iγ · p + M(p2)

10

• 2

10

• 1

10 10

1

10

2

p

2 [GeV 2]

10

• 3

10

• 2

10

• 1

10 10

1

M(p

2) [GeV]

χ limit u/d quark s quark c quark b quark M

2 = p 2

q ME/mζ χ ∞ u/d 100 s 7 c 1.7 b 1.2 → DχSB M2(p2) = p2 ⇒ Euclidean constituent quark mass ME

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Example BSA :: Pseudoscalar Meson

◮ Pseudoscalar meson Bethe-Salpeter amplitude:

Γj

π(k; P) = τ jγ5 [iEπ(k; P) + γ · PFπ(k; P)

+ γ · k Gπ(k; P) + σµν kµPν Hπ(k; P)]

Γπ

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Example BSA :: Pseudoscalar Meson

◮ Pseudoscalar meson Bethe-Salpeter amplitude:

Γj

π(k; P) = τ jγ5 [iEπ(k; P) + γ · PFπ(k; P)

+ γ · k Gπ(k; P) + σµν kµPν Hπ(k; P)]

Γπ ◮ P: total momentum, k: relative momentum ◮ Variables: k2, P2, z :=

k · P → angle variable

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Example BSA :: Pseudoscalar Meson

◮ Pseudoscalar meson Bethe-Salpeter amplitude:

Γj

π(k; P) = τ jγ5 [iEπ(k; P) + γ · PFπ(k; P)

+ γ · k Gπ(k; P) + σµν kµPν Hπ(k; P)]

Γπ ◮ P: total momentum, k: relative momentum ◮ Variables: k2, P2, z :=

k · P → angle variable

◮ pseudoscalar piece

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Example BSA :: Pseudoscalar Meson

◮ Pseudoscalar meson Bethe-Salpeter amplitude:

Γj

π(k; P) = τ jγ5 [iEπ(k; P) + γ · PFπ(k; P)

+ γ · k Gπ(k; P) + σµν kµPν Hπ(k; P)]

Γπ ◮ P: total momentum, k: relative momentum ◮ Variables: k2, P2, z :=

k · P → angle variable

◮ pseudoscalar piece ◮ pseudovector pieces:

◮ intrinsic orbital angular momentum ◮ crucial for Lorentz invariance ◮ preserving symmetries (AV WTI) ◮ asymptotic behavior of pion em form factor

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Homogeneous BSE

◮ BSE for q¯

q or qq bound states (χ = S Γh S) Γh tu(p; P) =

• d4q [χ(q; P)]sr K tu

rs (q, p; P) .

Γπ = K S S Γπ

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Homogeneous BSE

◮ BSE for q¯

q or qq bound states (χ = S Γh S) Γh tu(p; P) λ(P2) =

• d4q [χ(q; P)]sr K tu

rs (q, p; P) .

Γπ = K S S Γπ

◮ homogeneous → eigenvalue equation

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Homogeneous BSE :: Solution Strategy

Solution strategy for homogeneous BSE (see also talk by Martina Blank)

1.25 1.3 1.35 1.4 1.45 M [GeV] 0.8 0.9 1 1.1 1.2 1.3 1.4 λ gr 1st exc

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Coming up next . . .

Motivation QCD and Hadrons Dyson-Schwinger Equations Equations and Solutions Quark DSE Bethe-Salpeter Equation BSE Solution Strategies Symmetries and Exact Results AV WTI Truncation and Model Building Numerical Results Masses and Leptonic Decay Constants χ-Transition Temperatures Conclusion and Outlook

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AV WTI

◮ Axial-vector Ward-Takahashi identity

PµΓj

5µ(k; P)

= S−1(k+)iγ5 τ j 2 + iγ5 τ j 2 S−1(k−) − 2i m(ζ) Γj

5(k; P),

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AV WTI

◮ Axial-vector Ward-Takahashi identity

PµΓj

5µ(k; P)

= S−1(k+)iγ5 τ j 2 + iγ5 τ j 2 S−1(k−) − 2i m(ζ) Γj

5(k; P), ◮ Consequence: Gap and BSE kernels related

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AV WTI

◮ Axial-vector Ward-Takahashi identity

PµΓj

5µ(k; P)

= S−1(k+)iγ5 τ j 2 + iγ5 τ j 2 S−1(k−) − 2i m(ζ) Γj

5(k; P), ◮ Consequence (residues): fπnm2 πn = 2 m(ζ) ρπn(ζ) ;

with n = gr, exc1, . . .

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AV WTI

◮ Axial-vector Ward-Takahashi identity

PµΓj

5µ(k; P)

= S−1(k+)iγ5 τ j 2 + iγ5 τ j 2 S−1(k−) − 2i m(ζ) Γj

5(k; P), ◮ Consequence (residues): fπnm2 πn = 2 m(ζ) ρπn(ζ) ;

with n = gr, exc1, . . .

◮ valid for every pseudoscalar meson ◮ valid for every current quark mass ◮ ⇒ GMOR, PCAC

• P. Maris, C. D. Roberts, Phys. Rev. C56, 3369 (1997)
• A. H¨
• ll, A. K., and C. D. Roberts, Phys. Rev. C 70, 042203 (2004)
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Rainbow-Ladder (RL) Truncation

◮ Satisfy the AVWTI! ◮ Simplest truncation to do this:

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Rainbow-Ladder (RL) Truncation

◮ Satisfy the AVWTI! ◮ Simplest truncation to do this: ◮ Rainbow approx. for gap eq.

Σ = γ γ S D0 G

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Rainbow-Ladder (RL) Truncation

◮ Satisfy the AVWTI! ◮ Simplest truncation to do this: ◮ Rainbow approx. for gap eq.

Σ = γ γ S D0 G

◮ Ladder approximation for BSE

K = γ γ D0 G

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Rainbow-Ladder (RL) Truncation

◮ Satisfy the AVWTI! ◮ Simplest truncation to do this: ◮ Rainbow approx. for gap eq.

Σ = γ γ S D0 G

◮ Ladder approximation for BSE

K = γ γ D0 G

◮ Bare quark-gluon vertex γν

D Γ → D0 G γ

◮ Bare gluon prop. Dfree µν (p − q) ◮ Effective coupling G multiplies product of qgv and gp ◮ Input needed for G: modeling

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Effective Coupling

◮ What do we know? ◮ Perturbative QCD determines UV regime

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Effective Coupling

◮ What do we know? ◮ Perturbative QCD determines UV regime ◮ IR enhancement necessary for dynamical breaking of chiral

symmetry

◮ Fit e. g. pion mass and decay constant ◮ For that, effective strength is essential ◮ Precise form at low/intermediate Q2 → Ansatz

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Effective Coupling

◮ What do we know? ◮ Perturbative QCD determines UV regime ◮ IR enhancement necessary for dynamical breaking of chiral

symmetry

◮ Fit e. g. pion mass and decay constant ◮ For that, effective strength is essential ◮ Precise form at low/intermediate Q2 → Ansatz ◮ IR: two-parameters via Gaussian:

strength D and width ω

◮ perturbative α in the UV region ◮ P. Maris, P. C. Tandy: series of papers following

• P. Maris and P. C. Tandy, Phys. Rev. C 60, 055214 (1999).

◮ Successful description of light pseudoscalar and vector mesons

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Model Details

◮ Effective coupling α(Q2): ω D =const. (effective strength)

0.5 1 1.5 2

q

2 [GeV 2]

10 20

α(q

2)

ω = 0.30 ω = 0.40 ω = 0.50

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Model Details

◮ Effective coupling α(Q2): ω D =const. (effective strength)

0.5 1 1.5 2

q

2 [GeV 2]

10 20

α(q

2)

ω = 0.30 ω = 0.40 ω = 0.50

ω D = (.72 GeV)3 ⇒ meson ground-state properties unchanged ⇒ one-parameter model

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Model Parameter Dependence

This is a good thing!!

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Coming up next . . .

Motivation QCD and Hadrons Dyson-Schwinger Equations Equations and Solutions Quark DSE Bethe-Salpeter Equation BSE Solution Strategies Symmetries and Exact Results AV WTI Truncation and Model Building Numerical Results Masses and Leptonic Decay Constants χ-Transition Temperatures Conclusion and Outlook

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First Things First :: Spectra

Meson masses for J = 0, 1 as functions of current quark mass. Three vertical dotted lines: u/d, s, c quark masses

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 mq(1 GeV) [GeV] 0.5 1 1.5 2 2.5 3 3.5 4 4.5 M [GeV] 0-+ 0++ 1-- 1++ 1+-

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Next :: Even More Spectra

Same for 0−+ and 1−− ground and excited-state meson masses

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 mq(1 GeV) [GeV] 0.5 1 1.5 2 2.5 3 3.5 4 4.5 M [GeV] PS ground VE ground PS excited VE excited

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Beyond Spectra :: Decay Constants

fπ and fρ as functions of current quark mass. Three vertical dotted lines: u/d, s, c quark masses

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 mq(1 GeV) [GeV] 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 fJPC [GeV] 0-+ 1--

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Spectra :: Model Parameter Dep. and Experiment

Learn about the Force, Luke.

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Spectra :: Model Parameter Dep. and Experiment

mX as functions of ω

• A. K., Phys. Rev. D80, 114010 (2009)

0.3 0.4 0.5 ω 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 M [GeV]

• +

1

• u/d

0.3 0.4 0.5 ω 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 s 0.3 0.4 0.5 ω 2.9 3 3.1 3.2 3.3 3.4 3.5 c 0.3 0.4 0.5 ω 9.3 9.4 9.5 9.6 9.7 9.8 9.9 b

Vector states fixed to experiment via fit of current quark masses

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Spectra :: Model Parameter Dep. and Experiment

mX as functions of ω

• A. K., Phys. Rev. D80, 114010 (2009)

0.3 0.4 0.5 ω 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 M [GeV]

• +

++

1

• 1

++

1

+-

u/d 0.3 0.4 0.5 ω 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 s 0.3 0.4 0.5 ω 2.9 3 3.1 3.2 3.3 3.4 3.5 c 0.3 0.4 0.5 ω 9.3 9.4 9.5 9.6 9.7 9.8 9.9 b

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Model Parameter Dependence

This is – again – a good thing!!

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Spectra :: Effective Masses of Diquarks

mX as functions of ω

0.3 0.4 0.5 ω 0.6 0.7 0.8 0.9 1 1.1 1.2 M [GeV] scalar DQ axialvector DQ u/d 0.3 0.4 0.5 ω 1 1.1 1.2 1.3 1.4 1.5 1.6 s 0.3 0.4 0.5 ω 3.3 3.4 3.5 3.6 3.7 3.8 3.9 c 0.3 0.4 0.5 ω 9.7 9.8 9.9 10 10.1 10.2 10.3 b

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Even Better :: More Model Dependence

Don’t underestimate the power of the Force.

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QCD Gap Equation at Finite Temperature

◮ Reminder: ◮ Work in Rainbow-truncation:

(

S(p; ωk)

)−1 =

• S0(p; ωk)

−1 +

Dfree

µν

D (p − q; ωk − ωl) γν S(q; ωl) γµ

◮ Full vertex times full gluon propagator

→ bare vertex times bare propagator dressed with some function D(p − q, ωk − ωl)

◮ Model for interaction: Choice of D(k, Ω)

(k = p − q and Ω = ωk − ωl are gluon momentum and energy)

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Finite Temperature :: Interaction Models

MN: H. J. Munczek and A. M. Nemirovsky, Phys. Rev. D28: 181, 1983 AWW: R. Alkofer, P. Watson, H. Weigel, Phys. Rev. D65: 094026, 2002 MT: P. Maris, P. C. Tandy, Phys. Rev. C60: 055214, 1999 MR: P. Maris, C. D. Roberts, Phys. Rev. C56: 3369, 1997

G( k, Ω; mg) = MN : = D 4π3 T δ3( k)δk−l,0 AWW : = D 4π2 ω6 se−s/ω2 MT : = D 4π2 ω6 se−s/ω2 + FUV (s) MR : = D 4π3 T δ3( k)δk−l,0 + 4π2 ω6 se−s/ω2 + FUV (s)

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Finite Temperature :: Interaction Characteristics

Shape of the differ- ent interactions at T = 0:

(here s = pµpµ)

• 0.001

0.1 10 s 104 0.001 0.01 0.1 1 10 Αs

• MT3
• MT2
• MT1
• MR2
• MR1
• AWW2
• AWW1

◮ All models are used to describe physical states ◮ Parameters fitted to observables at T = 0

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Finite Temperature :: Interaction Models

Summary of model parameters: Model ω D −¯ qq0 mπ m̺ MN 0.5618 (.115 GeV)3 0.14 0.77 AWW1 0.3 1.47 (.245 GeV)3 0.135 0.745 AWW2 0.4 1.152 (.246 GeV)3 0.135 0.748 AWW3 0.5 1.0 (.251 GeV)3 0.137 0.758 MT1 0.3 1.24 (.243 GeV)3 0.139 0.747 MT2 0.4 0.93 (.242 GeV)3 0.139 0.743 MT3 0.5 0.744 (.239 GeV)3 0.141 0.724 MR1 0.3 0.78 (.241 GeV)3 0.139 MR2 0.4 0.78 (.250 GeV)3 0.139 MR3 0.5 0.78 (.255 GeV)3 0.139

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Finite Temperature :: Interaction Characteristics

How can one characterize the different interactions?

◮ Amount of chiral symmetry breaking at T = 0:

chiral condensate

◮ Integrated Strength at zero temperature:

I.S. =

• d4p

(2π)4 GIR (leave out FUV (p2)) ◮ ω D: constant in MT ◮ ω: range of the strongest contribution

Attempt correlation to Tc

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Finite Temperature :: Order Parameter

◮ Order parameter: B0 := B(0, ω0) (or chiral condensate) ◮ Example: MT2

0.06 0.07 0.08 0.09 0.10 T 0.1 0.2 0.3 0.4 0.5 0.6 B0T

◮ Indicates a second order phase transition with Tc ≃ 0.094 GeV

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Finite Temperature :: Tc and Exponents

Mean Field universality class for all Ans¨ atze Results for critical temperature: ω N/A 0.3 0.4 0.5 Model MN Tc 169 Model AWW1 AWW2 AWW3 Tc 82 94 101 Model MT1 MT2 MT3 Tc 82 94 96 Model MR1 MR2 MR3 Tc 120 133 144 Lets visualize this.

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Finite Temperature :: Results

Tc as a function of ω and ω D

0.3 0.4 0.5 ω [GeV] 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 Tc [GeV] AWW MT MR MN 0.2 0.3 0.4 0.5 ω D [GeV

3]

0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 Tc [GeV]

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Finite Temperature :: Results

Tc as a function of D and the chiral condensate

0.6 0.8 1 1.2 1.4 D [GeV

2]

0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 Tc [GeV] 0.1 0.15 0.2 0.25

• <qq>

1/3 [GeV]

0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 Tc [GeV] 0.24 0.25 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15

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WIP - Wish List

Other work in progress

◮ Higher J (tensor mesons) ◮ Higher radial excitations ◮ Heavy-light mesons and radial excitations ◮ Hadronic decays, e. g. π(1300) → ̺ π

(see also talk by Valentin Mader)

◮ Transition form factors

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WIP - Wish List

Other work in progress

◮ Higher J (tensor mesons) ◮ Higher radial excitations ◮ Heavy-light mesons and radial excitations ◮ Hadronic decays, e. g. π(1300) → ̺ π

(see also talk by Valentin Mader)

◮ Transition form factors

Wish list

◮ Good description of excited mesons ◮ Comprehensive general description of hadron properties

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Conclusions

Summary

◮ Dyson-Schwinger equations provide a nonperturbative

continuum approach to QCD

◮ BSE describes bound states in a manifestly covariant way ◮ Present approach: symmetry-preserving calculation of meson

properties applicable to whole range of data in mq

◮ No simple correlation of Tc to model parameters

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Conclusions

Summary

◮ Dyson-Schwinger equations provide a nonperturbative

continuum approach to QCD

◮ BSE describes bound states in a manifestly covariant way ◮ Present approach: symmetry-preserving calculation of meson

properties applicable to whole range of data in mq

◮ No simple correlation of Tc to model parameters

Conclusions

◮ Excellent description of PS and VE meson ground states in

Rainbow-Ladder truncation, including em properties

◮ Excitations are hard to describe well at present → improve the

model (truncation, etc.)

◮ These and Tc provide means to study the long-range behavior

• f the strong interaction between light quarks
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The End

Thank you!

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