Mass dependence of the heavy quark potential and its effects on quarkonium states Alexander Laschka Norbert Kaiser Wolfram Weise Physik Department Technische Universit¨ at M¨ unchen XIV International Conference on Hadron Spectroscopy June 14, 2011 Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 1
Heavy quark-antiquark potential History: phenomenological potential models Fitted to low lying charmonium and bottomonium states Typical shape: “Coulomb-plus-linear” Today: heavy quark-antiquark potential from QCD Characteristic scales of non-relativistic bound states m heavy quark mass hard scale mv heavy quark momentum soft scale mv 2 heavy quark energy ultrasoft scale Effective field theory (EFT) methods QCD ⇒ non-relativistic QCD (NRQCD, pNRQCD, vNRQCD) Topics: Extended range of validity of perturbative potential Spectroscopy at order 1 /m Detailed analysis of the role of quark masses Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 2
Outline Static quark-antiquark potential Heavy quark potential at order 1 /m Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 3
The static potential Non-perturbative sector : lattice studies of quenched and full QCD Static QCD potential (from static Wilson loop) G.S. Bali et al., Phys.Rev.D62 (2000) Y. & M. Koma, Nucl.Phys.B769 (2007) 1.0 full -1 quenched 0.8 QCD -1.5 0.6 V 0 (r) [GeV] [V(r)-V(r 0 )]r 0 β = 6.0 0.4 -2 β = 6.2 fit to r > 0.4 r 0 0.2 κ = 0.1560 -2.5 κ = 0.1565 0.0 κ = 0.1570 κ = 0.1575 -0.2 -3 fit to r > 0.4 r 0 β = 6.0 κ = 0.1580 β = 6.3 -0.4 -3.5 -0.6 0.2 0.3 0.4 0.5 0.6 0.0 0.2 0.4 0.6 0.8 1.0 r/r 0 r [fm] 0.10 0.15 0.20 0.25 0.30 r [fm] Sea quark effects important at small distances Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 4
Fourier transform of the static potential Perturbative sector: static potential is known at three-loop order M. Peter, Phys.Rev.Lett.78 (1997), Y. Schr¨ oder, Phys.Lett.B447 (1999) Three-loop: C. Anzai, Y. Kiyo, Y. Sumino, Phys.Rev.Lett.104 (2010), A. & V. Smirnov, M. Steinhauser, Phys.Rev.Lett.104 (2010) Momentum space � α s ( | � � � 2 q | ) = − 4 πC F α s ( | � q | ) 1 + α s ( | � q | ) q | ) V (0) ( | � ˜ a 1 + a 2 q 2 � 4 π 4 π � α s ( | � � 3 � � � A ln µ 2 q | ) a 3 + 8 π 2 C 3 IR + + . . . 4 π � q 2 where C F = 4 / 3 , C A = 3 , a 1 = 7 , a 2 ≈ 268 . 8 , a 3 ≈ 5199 . 8 ( n f =3) At N 3 LO (three-loop order): infrared divergences ( µ 2 IR ) from ultrasoft gluons Avoid expansion of α s ( | � q | ) about a fixed scale µ Reliable potential from extremely small distances up to r ≈ 0.15 fm needed Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 5
Potential subtracted (PS) scheme PS scheme with numerical Fourier transform Evaluate numerically (with a low-momentum cutoff µ f ) � � α s ( | � � 2 � � d 3 � q r α s ( | � q | ) + α s ( | � q | ) q | ) V (0) ( � (2 π ) 3 e i� q · � r, µ f ) = − 4 πC F 1 a 1 + a 2 + . . . q 2 � 4 π 4 π | � q | >µf ⇓ ⇓ ⇓ LO NLO NNLO 0 0 � 1 � 1 ( V (0) − const ) [GeV] ( V (0) − const ) [GeV] � 2 � 2 LO � 3 � 3 µ f = 0.7 GeV NLO � 4 � 4 µ f = 1.0 GeV NNLO µ f = 1.5 GeV µ f = 1.0 GeV NNNLO � 5 � 5 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 r [fm] r [fm] No free scale parameter µ Unknown constant is moved into the definition of m PS : 2 m pole + V (0) ( r ) = 2 m PS ( µ f ) + V (0) ( r, µ f ) Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 6
Matching and uncertainty estimate Perturbative potential (here NNLO) and lattice potential matched 0 � � � � � � � � � � pert. QCD � � � V (0) ( r ) − V (0) ( 0.5 fm ) [GeV] � � � � � lattice QCD � 1 � � 2 � 3 matching position � 4 0.0 0.1 0.2 0.3 0.4 0.5 r [fm] Differentiable quark-antiquark potential for distances up to ∼ 1 fm Matching at 0.14 fm gives µ f = 0.9 +0 . 3 − 0 . 2 GeV (for charmonium and bottomonium) Grey band: uncertainty of lattice calculation and uncertainty of α s Dot-dashed curve: continuation of the “Coulomb-plus-linear” fit Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 7
Bottomonium spectrum Solve the Schr¨ odinger equation with this matched potential Mass [GeV] 10.6 B¯ B -threshold Υ(4S) static potential 10.4 Υ(3S) 10.2 χ bj(2P) 10.0 Υ(2S) 9.8 χ bj(1P) 9.6 9.4 Υ(1S) Model η b(1S) 9.2 1 S 0 3 S 1 1 P 1 3 P j Experiment Single parameter m PS ( 0.908 GeV ) = 4.78 GeV Can be converted to the MS scheme MS masses [GeV] m MS PDG 2010 4.19 +0 . 18 bottom quark 4.20 ± 0.04 − 0 . 06 1.27 +0 . 07 charm quark 1.23 ± 0.04 − 0 . 09 Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 8
Outline Static quark-antiquark potential Heavy quark potential at order 1 /m Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 9
Quark-antiquark potential at order 1 /m Expansion in inverse powers of the heavy quark mass m V ( r ) = V (0) ( r ) + V (1) ( r ) + V (2) ( r ) ( m/ 2) 2 + . . . m/ 2 Non-perturbative expression for 1 /m potential is known N. Brambilla et al., Phys.Rev.D63 (2001) 014023 Lattice simulations Efficient method from M. & Y. Koma and H. Wittig Quenched simulation, renormalization issues ( ≈ 15% error estimated) 1.0 Contains a non-perturbative contribution β=5.85 2 ] quenched V (1) (r) - V (1) (r = 0.8r 0 ) [1/r 0 β=6.0 0.5 β=6.2 Fit function ( r ) = − A 2 0.0 V (1) r 2 + B 2 ln r + C 2 ln -0.5 Effective string theory suggests Fit β=6.0 ( 0.53 < r/r 0 < 1.26) logarithmic shape: V (1) ∝ ln r + C -1.0 pert. + linear pert. + ln G. Perez-Nadal, J. Soto, Phys.Rev.D79 (2009) -1.5 0.0 0.5 1.0 1.5 2.0 r / r 0 Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 10
Quark-antiquark potential at order 1 /m Perturbative potential at order 1 /m ( C F = 4 3 , C A = 3 ) q | ) = C F π 2 α 2 s ( | � q | ) ˜ V (1) ( | � � � ( − C A ) + O ( α s ) 2 | � q | Restricted numerical Fourier transform V (1) ( r ) V (1) ( r ) 0.0 � � � � � � � 0.0 � � � � � pert. QCD � V (1) ( r ) − V (1) ( 0.5 fm ) [GeV 2 ] lattice QCD ( V (1) − const ) [GeV 2 ] � 0.5 � 0.5 � 1.0 � 1.0 f = 0.7 GeV µ ′ � 1.5 � 1.5 f = 1.0 GeV µ ′ matching f = 1.5 GeV µ ′ position � 2.0 � 2.0 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.5 r [fm] r [fm] Differentiable quark-antiquark potential for distances up to ∼ 1 fm Matching at 0.14 fm gives µ ′ f = 1.6 +0 . 5 − 0 . 8 GeV (for charmonium) f = 1.9 +0 . 4 µ ′ − 0 . 6 GeV (for bottomonium) Grey band: uncertainty of lattice calculation and uncertainty of α s Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 11
Heavy quark masses at order 1 /m PS ( µ f , µ ′ PS mass needs redefinition m PS ( µ f ) → m � f ) PS ( µ f , µ ′ 8 m C F C A α 2 1 s µ ′ 2 m � f ) ≡ m PS ( µ f ) − f Quark masses from comparison with empirical quarkonium states MS masses [GeV] static static + 1 /m PDG 2010 4.18 +0 . 05 4.19 +0 . 18 bottom quark 4.20 ± 0.04 − 0 . 04 − 0 . 06 1.28 +0 . 07 1.27 +0 . 07 charm quark 1.23 ± 0.04 − 0 . 06 − 0 . 09 Error estimates include: uncertainties in the potentials (static and order 1 /m ) uncertainties from matching to experimental spectra Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 12
Spectroscopy 10.6 B -threshold Bottomonium spectrum B¯ 3 S 1 10.4 3S Υ(3S) 1 S 0 Tightly bound η b (1S) and Υ (1S) 10.2 Mass [GeV] states are most sensitive to 3 S 1 Υ(2S) 2S 10.0 1 /m -effects 1 S 0 9.8 Hyperfine effects (h.f.) added 9.6 phenomenologically (one-gluon 1S 3 S 1 Υ(1S) exchange) with α eff η b (1S) s = 0.3 9.4 1 S 0 1 S 0 3 S 1 9.2 . . . (work in progress) static + h.f. experiment +1 /m to be substituted by the full 10.6 1 /m 2 potential B¯ B -threshold Mass [GeV] 10.4 3 P j χ bj (2P) String tension σ = 1.01 GeV/fm 2P 10.2 3 D j Υ(1D) 1D 10.0 Different strategies needed above 3 P j χ bj (1P) 1P BB threshold 9.8 3 P j 3 D j static + h.f. experiment +1 /m Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 13
Spectroscopy 3.8 Charmonium spectrum 3 S 1 D¯ D -threshold 2S 3.7 ψ (2S) η c (2S) 3.6 1 S 0 Downward shift from V (1) in the 3.5 1S states ( η c and J / ψ ) to large Mass [GeV] 3.4 3.3 1 /m 2 effects significant 3.2 1S J / ψ (1S) 3.1 3 S 1 Hyperfine effects (h.f.) added 3.0 η c (1S) phenomenologically (one-gluon 2.9 1 S 0 exchange) with α eff 2.8 s = 0.3 1 S 0 3 S 1 2.7 static + h.f. experiment +1 /m . . . (work in progress) to be substituted by the full 3.8 D¯ D -threshold 1 /m 2 potential 3.7 Mass [GeV] 3.6 1 P 1 , 3 P j h c (1P) χ cj (1P) 1P 3.5 String tension σ = 1.01 GeV/fm 3.4 3.3 Different strategies needed above 1 P 1 3 P j 3.2 static + h.f. experiment +1 /m DD threshold Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 14
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