Meson spectroscopy and pion cloud effect on baryon masses Stanislav Kubrak, Christian Fischer, Helios Sanchis-Alepuz, Richard Williams Justus-Liebig-University, Giessen 13.06.2014 SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 1 / 24
Talk Objectives Talk Objectives 1. Pion cloud effect on baryon masses 2. Light meson spectroscopy for J = 0..3 3. Charmonium spectroscopy and gluon shape impact SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 2 / 24
Pion Cloud Effect - why it is crucial? Evidence of pion clouds The excess of ū sea / d sea = 1.14 in nucleon sea. P. Amaudruz at al, Phys. Rev. Lett. 66, 2712(1991) The ratio E2/M1 in Nucleon-Delta transition. K. Joo et al. (CLAS), Phys. Rev. Lett. 88, 122001 (2002), hep-ex/0110007. This effect is expect to impact the nucleon form factors behaviour: a) The pion cloud probed at long wavelengths. b) The nucleon core probed at high Q2 The ratio E2/M1 as a function of Q 2. G. Eichmann and C. Fischer,Eur.Phys.J.A48,9(2012) SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 3 / 24
DSE and BSE with the pion cloud There are familiar rainbow-ladder Dyson-Schwinger and Bethe-Salpeter equations, but with an additional pion exchange term: System of DSEs and BSEs Christian S. Fischer, Richard Williams, Phys.Rev.D78:074006,2008 Internal quark-pion vertex: Γ π (p 2 ) = B(p2 ) chiral f π K, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 4 / 24
Pion cloud effect Quark mass functions Pion dressing functions M(p 2 ) [GeV] 0,6 0,5 0,4 0,3 0,2 RL1, η=2.0 RL1, η=1.6 RL2, η=2.0 RL2, η=1.6 RL2+π, η=2.0 RL2+π, η=1.6 30 E(p) wo pion F(p) wo pion G(p) wo pion H(p) wo pion E(p) with pion F(p) with pion 20 G(p) with pion H(p) with pion 10 0,1 0 10-4 10-3 10-2 10-1 10 0 10 1 10 2 10 3 p 2 [GeV 2 ] 0 0.0001 0.001 0.01 0.1 1 10 100 1000 Properties Provides dynamical quark mass generation Fulfil Gell-Mann-Oakes-Renner relation Conserves Axial WTI SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 5 / 24
Fadeev equation for baryons Fadeev Equation Helios Sanchis-Alepuz, SK, Christian S. Fischer, PLB 733C (2014) Evolution of Nucleon and Delta masses M (GeV) SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 6 / 24
Talk Objectives Talk Objectives 1. Pion cloud effect on baryon masses 2. Light meson spectroscopy for J = 0..3 3. Charmonium spectroscopy and gluon shape impact SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 7 / 24
Rainbow-ladder truncation Quark DSE Kernel as an effective one-gluon exchange S 1 (p) = Z 2 S 1 0 + g 2 Z 1f C F γ µ S(k)Γ ν (k, p)d µν k Meson BSE α eff (p 2 ) = π η7 Λ 2 p2 e p2 η 2 Λ 2 + pqcd [Γ(p; P)] tu = λ K (2) rs;tu (p, k; P) [χ(k; P)] sr. k Λ = 0.72GeV, η = 1.8 ± 0.2 P. Maris, R. Tandy, PRC 78 K, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 8 / 24
Devil is in details Poles in quark propagator 10 Eigenvalue extrapolation 1.0 5 0.5 0-1.0 0.0 ImI p 2 M - 0.5-0.5 0.0 ReI p 2 M 0.5 1.0 Barycentric rational interpolation: -1.0 PN 1 2 C = t + i2tmbs 2 MBS wi i=0 x xi yi wi i=0 x xi R(x) = PN 1, Cauchy Integral Im(p2) wk = f(z0) Re(p2) k X i=k d k ( 1) i+d Y j=i,j6=k 1. xk xj J. Berrut, L. Trefethen, SIAM Rev 46 f (z0 ) = 1 2πi I C dzf (z) z z0 SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 9 / 24
Comparison of extrapolations Meson eigenvalue curves 1.2 2++ (within contour) 2++ (Cauchy) 2++ (Extrapolation) 1.4 1.2 2-- (within contour) 2-- (Cauchy) 2-- (Extrapolation) 1 Eigenvalue 0.8 1 0.8 0.6 0.6 Eigenvalue 0.4 1.6 1.4 1.2 1 0.8 0.6 0.4 0.4 0.6 0.8 1 1.2 M[GeV] 2-+ (within contour) 2-+ (Cauchy) 2-+ (Extrapolation) 0.2 0.4 0.6 0.8 1 1.2 1.4 M[GeV] 0.4 0.4 0.6 0.8 1 1.2 1.4 M[GeV] 1.6 3+- (within contour) 1.4 3+- (Cauchy) 3+- (Extrapolation) 1.2 1 0.8 0.6 0.4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 M[GeV] SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 10 / 24
Partitioning optimization Heavy and light quark s contours 20 10 Heavy quark (Maximum) Light quark (Maximum) Heavy quark Light quark 0 Im(p 2 )[GeV 2 ] 10 20 30 40 50 20 0 20 40 60 80 100 Re(p 2 )[GeV 2 ] C Heavyquark = k + ζp BS C Lightquark = k + (1 ζ)p BS 1 Optimal partitioning: ζ = Mheavy max /Mmax light + 1 SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 11 / 24
n n spectrum n n spectrum 2 π0(1800) ρ(1700) π2(1670) ρ3(1690) PDG η = 1.8 η = 2.0 1.5 π0(1300) ρ(1450) a0(1450) a2(1320) b1(1235) a1(1260) π1(1400) M[GeV] 1 ρ(770) 0.5 π0 0 0 + 1 0++ 1+ 1++ 2++ 2 2 + 3 3+ 3++ 0 0+ 1 + 2+ J PC SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 12 / 24
s s spectrum s s spectrum M[Gev] 2.4 2.2 2 1.8 1.6 1.4 η(1405) φ(2175) φ(1680) f ' 2(1525) φ ' 3(1850) PDG η = 1.8 η = 2.0 1.2 1 φ(1020) 0.8 0.6 0 + 1 0++ 1+ 1++ 2++ 2 2 + 3 3+ 3++ 0 0+ 1 + 2+ 3 + J PC SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 13 / 24
9 n s spectrum 14 PDG n s spectrum 2 1.8 K * (1680) K2(1820) K2(1770) K3(1780) PDG η = 1.8 η = 2.0 1.6 M[GeV] 1.4 1.2 K0 * (1430) K * (1410) K1(1400) K2 * (1430) K1(1270) 1 K * (892) 0.8 0.6 0.4 K0(493) 0 0+ 1 1+ 2 2+ 3 3+ J P SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 14 / 24
Regge trajectories Natural parity 4.0 φ 3 (1850) Unnatural parity 4.0 3.0 isoscalar ss f 2 (1525) 3.0 a 3 (1875) M 2 [GeV 2 ] 2.0 φ(1020) ρ(1690) M 2 [GeV 2 ] 2.0 a 1 (1260) ρ 2 (???) 1.0 0.0 a 2 (1320) isovector nn ρ(770) 1 -- 2 ++ 3 -- J PC 1.0 0.0 isovector nn 1 ++ 2 -- 3 ++ J PC K, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 15 / 24
Radial excitations Leading amplitudes for ground and excited states 0.06 Leading vector meson amplitude ρ(770) ρ(1465) 0.04 0.02 0 0.02 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 p 2 [GeV 2 ] SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 16 / 24
Talk Objectives Talk Objectives 1. Pion cloud effect on barion masses 2. Light meson spectroscopy for J = 0..3 3. Charmonium spectroscopy and gluon shape impact SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 17 / 24
Lattice charmonium Charmonium spectrum on the lattice Liuming Liu et al., JHEP 07 (2012) 126 SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 18 / 24
c c spectrum c c spectrum M[Gev] 5 4.5 4 3.5 ψ(4.66) ψ(4.42) X(4.36) X(4.26) ψ(4.15) ψ(4.04) X(3.918) X(3.927) X(3.872) ψ(3.77) ψ(2s)(3.68) ηc(2s)(3.63) hc(3.52) χc2(3.55) χc1(3.51) χc0(3.41) PDG η = 1.25 3 J/ψ(3.09) ηc(2.98) 2.5 0 + 1 0++ 1+ 1++ 2++ 2 2 + 3 3+ 3++ 0 0+ 1 + 2+ 3 + J PC SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 19 / 24
Gluon shape Gluon shape α eff (x 2 ) = πη 7 P n (x 2 )e x2 η 2 + pqcd, x 2 = p2 Λ 2, P n(x 2 ) = n b i x 2i i=0 Particularly: P 2 (x 2 ) = x 2 + bx 4 1 Maris-Tandy model 0.1 0.01 b = 0 b = 0.2..1.0 0.001 1 p 2 [GeV 2 ] SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 20 / 24
Shape impact ψ(4.42) X(4.36) X(4.26) ψ(4.15) Shape impact on the charmonium ψ(4.04) 4 X(3.918) X(3.872) 3.8 ψ(3.77) 3.6 ηc(2s)(3.63) ψ(2s)(3.68) hc(3.52) χc1(3.51) M[Gev] 3.4 χc0(3.41) 3.2 J/ψ(3.09) 3 ηc(2.98) PDG b = 0.2..1 2.8 0 + 1 0++ 1+ 1++ J PC SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 21 / 24
Gradient descent Cost function to re-gauge quark mass J (m q ) = As example ( 1 λ J/Ψ (M exp J/Ψ ) ) 2 P 2 (x 2 ) = x 2 + bx 4 and j = Ψ(2S), Ψ(3S) Cost function to fit gluon shape parameters J (b 0,.., b n ) = j ( ) 2 1 λ j (M exp j ) Use gradient descent to obtain b i : b i = b i J (b 0,.., b n ) b i the fit gives: b = 0.288 SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 22 / 24
Further Development Further Development Increase a variability of the effective interaction (Scaling vs Decoupling) Charmonium weak decay constants Charm form factors and transition form factors Pion form factor with pion cloud effect SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 23 / 24
Conclusion Conclusion M N M splitting and indication of missing self-gluon interaction J = 3 light meson spectrum Regge behaviour Similar pattern for J = 2 and J = 3 with the lattice in charmonium spectrum SK, C. Fischer, H. Sanchis-Alepuz, R. Williams Meson () spectroscopy and pion cloud effect 13.06.2014 24 / 24