Soft and Hard Scale QCD Dynamics in Mesons Peter Tandy Center for - PowerPoint PPT Presentation

Soft and Hard Scale QCD Dynamics in Mesons Peter Tandy Center for Nuclear Research Kent State University Mazatlan Nov09 – p. 1/5

Topics Overview of DSE modeling of meson physics—mainly soft scale Masses, decays, form factors Including a hard scale: DIS: quark distributions in π, K mesons Mesons involving a heavy quark Summary Mazatlan Nov09 – p. 2/5

Lattice-QCD and DSE-based modeling � q, q, G ) e −S [¯ q,q,G ] Lattice: �O� = D ¯ qqG O (¯ Euclidean metric, x-space, Monte-Carlo Issues: lattice spacing and vol, sea and valence m q , fermion Det Large time limit ⇒ nearest hadronic mass pole δq ( x ) e −S [¯ δ q,q,G ]+(¯ η,q )+(¯ q,η )+( J,G ) � EOMs (DSEs): 0 = D ¯ qqG Euclidean metric, p-space, continuum integral eqns Issues: truncation and phenomenology—not full QCD Analtyic contin. ⇒ nearest hadronic mass pole Can be quick to identify systematics, mechanisms, · · · Mazatlan Nov09 – p. 3/5

DSE-based modeling of Hadron Physics Soft physics: truncate DSEs to min: 2-pt, 3-pt fns Should be relativistically covariant—-convenient for decays, Form Factors, etc No boosts needed on wavefns of recoiling bound st. ∞ d.o.f. → few quasi-particle effective d.o.f. Do not make a 3-dimensional reduction Preserve 1-loop QCD renorm group behavior in UV Preserve global symmetries, conserved em currents, etc Preserve PCAC ⇒ Goldstone’s Thm Can’t preserve local color gauge covariance—-just choose Landau gauge [RG fixed pt] Parameterize the deep infrared (large distance) QCD coupling Mazatlan Nov09 – p. 4/5

Constraints on Modeling Preserve vector WTI, and axial vector WTI E.g. τ τ − iP µ Γ 5 µ ( k ; P ) = S − 1 ( k + ) γ 5 2 S − 1 ( k − ) 2 + γ 5 − 2 m q ( µ ) Γ 5 ( k ; P ) ⇒ kernels of DSE q and K BSE are related Ladder-rainbow is the simplest implementation Goldstone Theorem preserved, ps octet masses good, indep of model details π ( p 2 ) = iγ 5 4 tr S − 1 [ 1 Γ 0 0 ( p 2 )] + · · · DCSB ⇒ π : f 0 π Here, 1-body and 2-body systems are the same Mazatlan Nov09 – p. 5/5

Ladder-Rainbow Model K • λ a λ a 4 πα eff ( q 2 ) D free K BSE → − γ µ µν ( q ) γ ν 2 2 qq � µ =1 GeV = − (240 MeV ) 3 , incl vertex dressing → α eff ( q 2 ) IR � ¯ α eff ( q 2 ) → α 1 − loop ( q 2 ) s UV p-k -1 -1 = + p p k P . Maris & P .C. Tandy, PRC60, 055214 (1999) M ρ , M φ , M K ⋆ good to 5%, f ρ , f φ , f K ⋆ good to 10% Mazatlan Nov09 – p. 6/5

Summary of light meson results Vector mesons (PM, Tandy, PRC60, 055214) Range of light meson observables m u = d = 5 . 5 MeV , m s = 125 MeV at µ = 1 GeV m ρ / ω 0.770 GeV 0.742 Pseudoscalar (PM, Roberts, PRC56, 3369) f ρ / ω 0.216 GeV 0.207 expt. calc. m K ⋆ 0.892 GeV 0.936 qq � 0 (0.236 GeV) 3 (0.241 † ) 3 - � ¯ µ f K ⋆ 0.225 GeV 0.241 0.138 † m π 0.1385 GeV m φ 1.020 GeV 1.072 0.093 † f π 0.0924 GeV f φ 0.236 GeV 0.259 0.497 † m K 0.496 GeV Strong decay (Jarecke, PM, Tandy, PRC67, 035202) f K 0.113 GeV 0.109 g ρππ 6.02 5.4 Charge radii (PM, Tandy, PRC62, 055204) g φ KK 4.64 4.3 r 2 0.44 fm 2 0.45 π g K ⋆ K π 4.60 4.1 r 2 0.34 fm 2 0.38 K + Radiative decay (PM, nucl-th/0112022) r 2 -0.054 fm 2 -0.086 K 0 g ρπγ / m ρ 0.74 0.69 γπγ transition (PM, Tandy, PRC65, 045211) g ωπγ / m ω 2.31 2.07 g πγγ 0.50 0.50 ( g K ⋆ K γ / m K ) + 0.83 0.99 r 2 0.42 fm 2 0.41 ( g K ⋆ K γ / m K ) 0 πγγ 1.28 1.19 Weak K l 3 decay (PM, Ji, PRD64, 014032) Scattering length (PM, Cotanch, PRD66, 116010) λ + ( e 3 ) 0.028 0.027 a 0 0.220 0.170 0 7.6 · 10 6 s − 1 Γ ( K e 3 ) 7.38 a 2 0.044 0.045 0 5.2 · 10 6 s − 1 Γ ( K µ 3 ) 4.90 a 1 0.038 0.036 1 bsampl Mazatlan Nov09 – p. 7/5

DSE kernel constrained from Lattice QCD Mazatlan Nov09 – p. 8/5

Lattice-assisted DSE Results 2.0 1.8 Evident vertex enhancement 1.6 2 ) v ( p Curvature in low m q depn 1.4 M IR ( p 2 ) 40% below linear 1.2 Chiral Extrapolation 1.0 0 2 4 6 8 10 2 (GeV 2 ) p qq � qu − lat µ =1 GeV = − (190 MeV) 3 � ¯ 0.400 qq � qu − lat ≈ � ¯ qq � expt / 2 � ¯ 2 ) (GeV) f π 30% low 0.300 2 = 0.38 GeV 0.200 M ( p 0.100 0.000 0.025 0.050 0.075 0.100 0.125 m ( ζ =19 GeV) (GeV) Mazatlan Nov09 – p. 9/5

Qu-lattice S ( p ) , D ( q ) mapped to a DSE kernel S ( p ) = Z ( p ) [ i � p + M ( p )] − 1 Old data New ’improved action’ data m q = 0.168GeV m q = 0.030GeV 0.5 m q = 0.225GeV m q = 0.055GeV 0.4 m q = 0.110GeV M (p) [GeV] m q = 0.0GeV 0.3 0.2 0.1 0 0 1 2 3 4 p [GeV] Mazatlan Nov09 – p. 10/5

Quenched lattice ⇒ m q Depn of DSE Kernel 4 10 DSE-LR (MT) 2 ,m=0)*D(q 2 ) V(q 3 10 2 10 2 2 )/q 1 10 4 π α eff (q 0 10 -1 10 chiral quark -2 10 -3 10 -2 -1 0 1 2 3 10 10 10 10 10 10 2 [GeV 2 ] q Bhagwat,Pichowsky,Roberts,Tandy, PRC68, 015203 (2003) Mazatlan Nov09 – p. 11/5

Quenched lattice ⇒ m q Depn of DSE Kernel 4 10 DSE-LR (MT) 2 ,m=0)*D(q 2 ) V(q 3 10 2 , m u )*D(q 2 ) V(q 2 10 2 2 )/q 1 10 4 π α eff (q 0 10 -1 10 u-quark -2 10 -3 10 -2 -1 0 1 2 3 10 10 10 10 10 10 2 [GeV 2 ] q Bhagwat,Pichowsky,Roberts,Tandy, PRC68, 015203 (2003) Mazatlan Nov09 – p. 11/5

Quenched lattice ⇒ m q Depn of DSE Kernel 4 10 DSE-LR (MT) 2 ,m=0)*D(q 2 ) V(q 3 10 2 , m u )*D(q 2 ) V(q 2 , m s )*D(q 2 ) V(q 2 10 2 2 )/q 1 10 4 π α eff (q 0 10 -1 10 s-quark -2 10 -3 10 -2 -1 0 1 2 3 10 10 10 10 10 10 2 [GeV 2 ] q Bhagwat,Pichowsky,Roberts,Tandy, PRC68, 015203 (2003) Mazatlan Nov09 – p. 11/5

Quenched lattice ⇒ m q Depn of DSE Kernel 4 10 DSE-LR (MT) 2 ,m=0)*D(q 2 ) V(q 3 10 2 , m u )*D(q 2 ) V(q 2 , m s )*D(q 2 ) V(q 2 10 2 , m c )*D(q 2 ) V(q 2 2 )/q 1 10 4 π α eff (q 0 10 -1 10 c-quark -2 10 -3 10 -2 -1 0 1 2 3 10 10 10 10 10 10 2 [GeV 2 ] q Bhagwat,Pichowsky,Roberts,Tandy, PRC68, 015203 (2003) Mazatlan Nov09 – p. 11/5

Quenched lattice ⇒ m q Depn of DSE Kernel 4 10 DSE-LR (MT) 2 ,m=0)*D(q 2 ) V(q 3 10 2 , m u )*D(q 2 ) V(q 2 , m s )*D(q 2 ) V(q 2 10 2 , m c )*D(q 2 ) V(q 2 2 , m b )*D(q 2 ) 2 )/q V(q 1 10 4 π α eff (q 0 10 -1 10 b-quark -2 10 -3 10 -2 -1 0 1 2 3 10 10 10 10 10 10 2 [GeV 2 ] q Bhagwat,Pichowsky,Roberts,Tandy, PRC68, 015203 (2003) Mazatlan Nov09 – p. 11/5

Quark Confinement—positivity violation Confinement/positivity analysis (Osterwalder-Schrader axiom No. 3) Fourier transf σ S ( p 4 , � p = 0) to Eucl time T 0 10 -1 10 -2 10 | ∆ S ( T )| -3 10 -4 10 -5 10 -6 10 0 10 20 30 5 15 25 -1 ) T (GeV solid = lattice prop, dashed = MT DSE, dotted = cc pole eg Mazatlan Nov09 – p. 12/5

DSE and Lattice results for M V and M ps Mazatlan Nov09 – p. 13/5

Pion electromagnetic form factor d 4 q � Λ µ = ( P ′ + P ) µ F π ( Q 2 ) Γ π S i Γ µ S Γ π S � ¯ � = N c (2 π ) 4 Tr π γ π Mazatlan Nov09 – p. 14/5

Pion F ( Q 2 ) : Low Q 2 Mazatlan Nov09 – p. 15/5

Kaon F ( Q 2 ) : Low Q 2 Mazatlan Nov09 – p. 16/5

Pion electromagnetic form factor 0.5 0.4 2 ] 2 ) [GeV 0.3 2 F π (Q Our prediction 0.2 VMD ρ pole Q CERN ’80s 0.1 Cornell ’70s 0 0 1 2 3 4 2 [GeV 2 ] Q PM and Tandy, PRC62,055204 (2000) [nucl-th/0005015] Mazatlan Nov09 – p. 17/5

Pion electromagnetic form factor 0.5 0.4 2 ] 2 ) [GeV 0.3 2 F π (Q Our prediction 0.2 VMD ρ pole Q CERN ’80s 0.1 JLab, 2001 0 0 1 2 3 4 2 [GeV 2 ] Q JLab data from Volmer et al , PRL86, 1713 (2001) [nucl-ex/0010009] PM and Tandy, PRC62,055204 (2000) [nucl-th/0005015] Mazatlan Nov09 – p. 17/5

Pion electromagnetic form factor 0.5 Our prediction VMD ρ pole 0.4 2 ] CERN ’80s 2 ) [GeV JLab, 2001 JLab at 12 GeV 0.3 pert. QCD 2 F π (Q JLab, 2006b 0.2 JLab, 2006a Q 0.1 0 0 2 4 6 2 [GeV 2 ] Q PM and Tandy, PRC62,055204 (2000) [nucl-th/0005015] 2006a: V. Tadevosyan et al , [nucl-ex/0607007], 2006b: T. Horn et al , [nucl-ex/0607005] Mazatlan Nov09 – p. 17/5

1-loop chiral correction to r π vs m π 0.5 Ladder-rainbow DSE Expt 0.4 2 C / f π 1--loop Ch PT r /f π 2 2 ] Ch PT contact/core term 12 L 9 0.3 2 [ fm r π 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 m π [GeV] P . Maris and PCT, in preparation Mazatlan Nov09 – p. 18/5

1-loop chiral correction to r π vs m π 0.5 Ladder-rainbow DSE Expt 0.4 2 C / f π 1--loop Ch PT r /f π 2 2 ] Ch PT contact/core term 12 L 9 0.3 2 [ fm r π 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 m π [GeV] P . Maris and PCT, in preparation Mazatlan Nov09 – p. 18/5

γ ⋆ π 0 → γ Transition Form Factor Q 1.0 1.0 0.9 CELLO 0.8 CLEO P-Q/2 P+Q/2 0.0 0.1 all 8 covariants 5 covariants BL monopole 0.6 2 )/g expt f(Q Abelian axial anomaly + π pole 0.4 in Γ 5 µ ⇒ G (0 , 0) Chiral limit G (0 , 0) = 1 0.2 2 ⇒ Γ πγγ to 2% 0.0 0.0 1.0 2.0 3.0 2 [GeV 2 ] Q Mazatlan Nov09 – p. 19/5

γ ⋆ πγ ⋆ Asymptotic Limit Lepage and Brodsky, PRD22, 2157 (1980): LC-QCD/OPE ⇒ 0 10 DSE results VMD dipole bare vertices 2 f π 2 / Q 2 (4/3) π -1 10 2 ) 2 ,Q -2 10 F(Q -3 10 -4 10 -2 -1 0 1 2 3 10 10 10 10 10 10 2 [GeV 2 ] Q Mazatlan Nov09 – p. 20/5