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, 1 (314)
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 A. Krassnigg, Heviz10, 2 (314)
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 A. Krassnigg, Heviz10, 3 (314)
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 A. Krassnigg, Heviz10, 4 (314)
Motivation :: QCD and Hadrons ◮ Study hadrons as composites of quarks and gluons . . . ◮ . . . including: ◮ Chiral symmetry and D χ SB ◮ correct perturbative limit (via α p ( Q 2 )) ◮ quark and gluon confinement ◮ Poincar´ e covariance ◮ Calculate observables! A. Krassnigg, Heviz10, 5 (314)
Motivation :: QCD and Hadrons ◮ Study hadrons as composites of quarks and gluons . . . ◮ . . . including: ◮ Chiral symmetry and D χ SB ◮ correct perturbative limit (via α p ( Q 2 )) ◮ 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 A. Krassnigg, Heviz10, 5 (314)
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 A. Krassnigg, Heviz10, 6 (314)
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 A. Krassnigg, Heviz10, 6 (314)
Even More Motivation A. Krassnigg, Heviz10, 7 (314)
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 A. Krassnigg, Heviz10, 8 (314)
Gap Equation Z ( p 2 ) S ( p ) = i γ · p + M ( p 2 ) D S 0 S [ ] − 1 = [ ] − 1 + γ S Γ D Σ = γ S Γ A. Krassnigg, Heviz10, 9 (314)
Gap Equation Z ( p 2 ) S ( p ) = current quark mass m ζ i γ · p + M ( p 2 ) D S 0 S [ ] − 1 = [ ] − 1 + γ S Γ D Σ = γ S Γ A. Krassnigg, Heviz10, 9 (314)
Gap Equation Z ( p 2 ) S ( p ) = current quark mass m ζ i γ · p + M ( p 2 ) D S 0 S [ ] − 1 = [ ] − 1 + γ S Γ D Σ = γ S Γ ◮ Side note: Weak coupling expansion reproduces every diagram in perturbation theory, but: ◮ Perturbation theory: m ζ = 0 ⇒ M ( p 2 ) ≡ 0 ◮ Here: get nonperturbative solution A. Krassnigg, Heviz10, 9 (314)
Quark Mass Function Z ( p 2 ) Solution of gap equation: S ( p ) = i γ · p + M ( p 2 ) A. Krassnigg, Heviz10, 10 (314)
Quark Mass Function Z ( p 2 ) Solution of gap equation: S ( p ) = i γ · p + M ( p 2 ) 1 10 2 ) [GeV] 0 10 M(p -1 χ limit 10 u/d quark s quark -2 c quark 10 b quark 2 = p 2 M -3 10 -2 -1 0 1 2 10 10 10 10 10 2 [GeV 2 ] p A. Krassnigg, Heviz10, 10 (314)
Quark Mass Function Z ( p 2 ) Solution of gap equation: S ( p ) = i γ · p + M ( p 2 ) M 2 ( p 2 ) = p 2 ⇒ Euclidean constituent quark mass M E 1 10 q M E / m ζ 2 ) [GeV] χ ∞ 0 10 u/d 100 M(p s 7 -1 c 1.7 χ limit 10 u/d quark b 1.2 s quark -2 c quark 10 → D χ SB b quark 2 = p 2 M -3 10 -2 -1 0 1 2 10 10 10 10 10 2 [GeV 2 ] p A. Krassnigg, Heviz10, 10 (314)
Example BSA :: Pseudoscalar Meson ◮ Pseudoscalar meson Bethe-Salpeter amplitude: Γ j π ( k ; P ) = τ j γ 5 [ iE π ( k ; P ) + γ · PF π ( k ; P ) Γ π + γ · k G π ( k ; P ) + σ µν k µ P ν H π ( k ; P )] A. Krassnigg, Heviz10, 11 (314)
Example BSA :: Pseudoscalar Meson ◮ Pseudoscalar meson Bethe-Salpeter amplitude: Γ j π ( k ; P ) = τ j γ 5 [ iE π ( k ; P ) + γ · PF π ( k ; P ) Γ π + γ · k G π ( k ; P ) + σ µν k µ P ν H π ( k ; P )] ◮ P : total momentum, k : relative momentum ◮ Variables: k 2 , P 2 , z := � k · � P → angle variable A. Krassnigg, Heviz10, 11 (314)
Example BSA :: Pseudoscalar Meson ◮ Pseudoscalar meson Bethe-Salpeter amplitude: Γ j π ( k ; P ) = τ j γ 5 [ iE π ( k ; P ) + γ · PF π ( k ; P ) Γ π + γ · k G π ( k ; P ) + σ µν k µ P ν H π ( k ; P )] ◮ P : total momentum, k : relative momentum ◮ Variables: k 2 , P 2 , z := � k · � P → angle variable ◮ pseudoscalar piece A. Krassnigg, Heviz10, 11 (314)
Example BSA :: Pseudoscalar Meson ◮ Pseudoscalar meson Bethe-Salpeter amplitude: Γ j π ( k ; P ) = τ j γ 5 [ iE π ( k ; P ) + γ · PF π ( k ; P ) Γ π + γ · k G π ( k ; P ) + σ µν k µ P ν H π ( k ; P )] ◮ P : total momentum, k : relative momentum ◮ Variables: k 2 , P 2 , 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 A. Krassnigg, Heviz10, 11 (314)
Homogeneous BSE ◮ BSE for q ¯ q or qq bound states ( χ = S Γ h S ) � d 4 q [ χ ( q ; P )] sr K tu Γ h tu ( p ; P ) = rs ( q , p ; P ) . S Γ π Γ π = K S A. Krassnigg, Heviz10, 12 (314)
Homogeneous BSE ◮ BSE for q ¯ q or qq bound states ( χ = S Γ h S ) � Γ h tu ( p ; P ) λ ( P 2 ) = d 4 q [ χ ( q ; P )] sr K tu rs ( q , p ; P ) . S Γ π Γ π = K S ◮ homogeneous → eigenvalue equation A. Krassnigg, Heviz10, 12 (314)
Homogeneous BSE :: Solution Strategy Solution strategy for homogeneous BSE (see also talk by Martina Blank) 1.4 gr 1st exc 1.3 1.2 1.1 λ 1 0.9 0.8 1.25 1.3 1.35 1.4 1.45 M [GeV] A. Krassnigg, Heviz10, 13 (314)
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 A. Krassnigg, Heviz10, 14 (314)
AV WTI ◮ Axial-vector Ward-Takahashi identity τ j τ j P µ Γ j S − 1 ( k + ) i γ 5 2 S − 1 ( k − ) 5 µ ( k ; P ) = 2 + i γ 5 − 2 i m ( ζ ) Γ j 5 ( k ; P ) , A. Krassnigg, Heviz10, 15 (314)
AV WTI ◮ Axial-vector Ward-Takahashi identity τ j τ j P µ Γ j S − 1 ( k + ) i γ 5 2 S − 1 ( k − ) 5 µ ( k ; P ) = 2 + i γ 5 − 2 i m ( ζ ) Γ j 5 ( k ; P ) , ◮ Consequence: Gap and BSE kernels related A. Krassnigg, Heviz10, 15 (314)
AV WTI ◮ Axial-vector Ward-Takahashi identity τ j τ j P µ Γ j S − 1 ( k + ) i γ 5 2 S − 1 ( k − ) 5 µ ( k ; P ) = 2 + i γ 5 − 2 i m ( ζ ) Γ j 5 ( k ; P ) , ◮ Consequence (residues): f π n m 2 π n = 2 m ( ζ ) ρ π n ( ζ ) ; with n = gr , exc1 , . . . A. Krassnigg, Heviz10, 15 (314)
AV WTI ◮ Axial-vector Ward-Takahashi identity τ j τ j P µ Γ j S − 1 ( k + ) i γ 5 2 S − 1 ( k − ) 5 µ ( k ; P ) = 2 + i γ 5 − 2 i m ( ζ ) Γ j 5 ( k ; P ) , ◮ Consequence (residues): f π n m 2 π n = 2 m ( ζ ) ρ π n ( 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¨ oll, A. K., and C. D. Roberts, Phys. Rev. C 70 , 042203 (2004) A. Krassnigg, Heviz10, 15 (314)
Rainbow-Ladder (RL) Truncation ◮ Satisfy the AVWTI! ◮ Simplest truncation to do this: A. Krassnigg, Heviz10, 16 (314)
Rainbow-Ladder (RL) Truncation ◮ Satisfy the AVWTI! ◮ Simplest truncation to do this: D 0 G Σ = ◮ Rainbow approx. for gap eq. γ γ S A. Krassnigg, Heviz10, 16 (314)
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