Mass dependence of the heavy quark potential and its effects on - - PowerPoint PPT Presentation
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
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
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 mv2 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
Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 3
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)
0.2 0.3 0.4 0.5 0.6 [V(r)-V(r0)]r0 r/r0 β = 6.0 β = 6.2 fit to r > 0.4 r0 κ = 0.1560 κ = 0.1565 κ = 0.1570 κ = 0.1575 fit to r > 0.4 r0 κ = 0.1580 0.10 0.15 0.20 0.25 0.30
r [fm]
1.0 0.8 0.6 0.4 0.2 0.0
V0(r) [GeV]
1.0 0.8 0.6 0.4 0.2 0.0
r [fm]
β = 6.0 β = 6.3
Sea quark effects important at small distances
Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 4
full QCD quenched
Perturbative sector: static potential is known at three-loop order
Three-loop: C. Anzai, Y. Kiyo, Y. Sumino, Phys.Rev.Lett.104 (2010),
Momentum space
˜ V (0)(| q |) = − 4πCF αs(| q |)
q |) 4π a1 + αs(| q |) 4π 2 a2 + αs(| q |) 4π 3 a3 + 8π2C3
A ln µ2 IR
a1 = 7, a2 ≈ 268.8, a3 ≈ 5199.8 (nf =3)
At N3LO (three-loop order): infrared divergences (µ2
IR) from ultrasoft gluons
Avoid expansion of αs(| 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
PS scheme with numerical Fourier transform Evaluate numerically (with a low-momentum cutoff µf)
V (0)( r, µf ) = −4πCF
q|>µf
d3 q (2π)3 ei
q· r αs(|
q|)
⇓ LO
+ αs(| q|) 4π a1
⇓ NLO
+ αs(| q|) 4π 2 a2
⇓ NNLO
+ . . .
0.1 0.2 0.3 0.4 5 4 3 2 1 r [fm] (V (0)− const ) [GeV] µf = 0.7 GeV µf = 1.0 GeV µf = 1.5 GeV 0.0 0.1 0.2 0.3 0.4 5 4 3 2 1 r [fm] (V (0)− const ) [GeV] LO NLO NNLO NNNLO µf = 1.0 GeV
No free scale parameter µ Unknown constant is moved into the definition of mPS:
2mpole + V (0)(r) = 2mPS(µf ) + V (0)(r, µf )
Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 6
Perturbative potential (here NNLO) and lattice potential matched
0.1 0.2 0.3 0.4 0.5 4 3 2 1
r [fm] V (0)(r) − V (0)(0.5 fm) [GeV]
matching position
lattice QCD
Differentiable quark-antiquark potential for distances up to ∼1 fm Matching at 0.14 fm gives µf = 0.9+0.3
−0.2 GeV
(for charmonium and bottomonium) Grey band: uncertainty of lattice calculation and uncertainty of αs Dot-dashed curve: continuation of the “Coulomb-plus-linear” fit
Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 7
Solve the Schr¨
9.2 9.4 9.6 9.8 10.0 10.2 10.4 10.6
1S0 3S1 1P1 3Pj
Mass [GeV]
B¯ B-threshold
ηb(1S) Υ(1S) χbj(1P) Υ(2S) χbj(2P) Υ(3S) Υ(4S)
Experiment Model
Single parameter mPS(0.908 GeV) = 4.78 GeV Can be converted to the MS scheme
MS masses [GeV] mMS PDG 2010 bottom quark 4.20±0.04 4.19+0.18
−0.06
charm quark 1.23±0.04 1.27+0.07
−0.09
Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 8
static potential
Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 9
Expansion in inverse powers of the heavy quark mass m
V (r) = V (0)(r) + V (1)(r) m/2 + V (2)(r) (m/2)2 + . . .
Non-perturbative expression for 1/m potential is known
Lattice simulations Efficient method from M. & Y. Koma and H. Wittig Quenched simulation, renormalization issues (≈15% error estimated)
0.0 0.5 1.0
V(1)(r) - V(1)(r = 0.8r0) [1/r0
2]
2.0 1.5 1.0 0.5 0.0
r / r0 β=5.85 β=6.0 β=6.2 Fit β=6.0 ( 0.53 < r/r0 < 1.26)
Contains a non-perturbative contribution Fit function
V (1)
ln
(r) = − A2 r2 + B2 ln r + C2
Effective string theory suggests logarithmic shape: V (1) ∝ln r+C
Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 10
quenched
Perturbative potential at order 1/m (CF = 4
3 , CA = 3)
˜ V (1)(| q|) = CF π2 α2
s(|
q|) 2 | q|
V (1)(r)
0.0 0.1 0.2 0.3 0.4 2.0 1.5 1.0 0.5 0.0 r [fm] (V (1)− const ) [GeV2] µ′
f = 0.7 GeV
µ′
f = 1.0 GeV
µ′
f = 1.5 GeV
V (1)(r)
0.1 0.2 0.3 0.4 0.5 2.0 1.5 1.0 0.5 0.0 r [fm] V (1)(r) − V (1)(0.5 fm) [GeV2]
matching position
lattice QCD
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)
µ′
f = 1.9+0.4 −0.6 GeV (for bottomonium)
Grey band: uncertainty of lattice calculation and uncertainty of αs
Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 11
PS mass needs redefinition mPS(µf) → m
PS(µf, µ′ f)
m
PS(µf, µ′ f) ≡ mPS(µf) − 1 8mCF CAα2 s µ′2 f
Quark masses from comparison with empirical quarkonium states MS masses [GeV] static static + 1/m PDG 2010 bottom quark 4.20±0.04 4.18+0.05
−0.04
4.19+0.18
−0.06
charm quark 1.23±0.04 1.28+0.07
−0.06
1.27+0.07
−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
Bottomonium spectrum Tightly bound ηb(1S) and Υ(1S) states are most sensitive to 1/m-effects Hyperfine effects (h.f.) added phenomenologically (one-gluon exchange) with αeff
s = 0.3
. . . (work in progress) to be substituted by the full 1/m2 potential String tension σ= 1.01 GeV/fm Different strategies needed above BB threshold
9.2 9.4 9.6 9.8 10.0 10.2 10.4 10.6
static +1/m +h.f. experiment
1S0 3S1
Mass [GeV]
B¯ B-threshold ηb(1S) Υ(1S) Υ(2S) Υ(3S) 1S 2S 3S
1S0 1S0 1S0 3S1 3S1 3S1
9.8 10.0 10.2 10.4 10.6
static +1/m +h.f. experiment
3Pj 3Dj
Mass [GeV]
B¯ B-threshold χbj(1P) χbj(2P) Υ(1D) 1P 2P 1D
3Pj 3Pj 3Dj
Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 13
Charmonium spectrum Downward shift from V (1) in the 1S states (ηc and J / ψ) to large 1/m2 effects significant Hyperfine effects (h.f.) added phenomenologically (one-gluon exchange) with αeff
s = 0.3
. . . (work in progress) to be substituted by the full 1/m2 potential String tension σ= 1.01 GeV/fm Different strategies needed above DD threshold
2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8
static +1/m +h.f. experiment
1S0 3S1
Mass [GeV]
D¯ D-threshold ηc(1S) J / ψ(1S) ηc(2S) ψ(2S) 1S 2S
1S0 1S0 3S1 3S1
3.2 3.3 3.4 3.5 3.6 3.7 3.8
static +1/m +h.f. experiment
1P1 3Pj
Mass [GeV]
D¯ D-threshold hc(1P)χcj(1P) 1P
1P1,3Pj
Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 14
Heavy quark-antiquark potential from QCD (perturbative QCD ↔ lattice QCD) Excellent matching in r-space up to order 1/m Spectroscopy at order 1/m
Works well for bottomonium Less successful for charmonium (1/m2 effects sizeable: work in progress)
Quark masses can be extracted
MS masses [GeV] static static + 1/m charm quark 1.23±0.04 1.28+0.07
−0.06
bottom quark 4.20±0.04 4.18+0.05
−0.04
See for details: A. Laschka, N. Kaiser, W. Weise, Phys.Rev.D83 (2011) 094002
Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 15
Alexander Laschka Mass dependence of the heavy quark potential Hadron 2011 16