Heavy Quark Physics without
Perturbation Theory
International Lattice Field Theory Workshop
Edinburgh --- March 8,2005
HueyWen Lin
Peter Boyle
Norman H. Christ
Outline
• Limitations of perturbation theory.
• Review of NPR for light quarks.
• Problems for NPR with heavy quarks.
• A proposal.
ILFT/Edinburgh 2
Perturbation Theory?
• Promise of LGT is accurate control of all errors.
• QCD perturbation theory errors are hard to control:
– Lima0 g2(a2) 1/ln(1/a)
Decreases too slowly.
– Limnh g2n(a2) [1/ln(1/a)]n Not practical/meaningful.
– Limnh g2(a2n) 1/n · ln(1/a) Requires step scaling.
• Ideally:
– Use QCD perturbation theory only at Λ=100 GeV for
perturbatively defined physical quantities.
– Connect with scale of normal lattice calculations by step scaling.
ILFT/Edinburgh 3
Off-Shell NPR
• Off-shell momentum space RomeSouthampton scheme works on few % level.
• For example consider DWF axial current:
ILFT/Edinburgh 4
Calculation of ZA
Evaluate ZA two ways:
1. Directly compute matrix elements of both
currents:
2. Evaluate the off-shell
vertex:
ILFT/Edinburgh 5
ZA from hadronic correlators.
ZA /Zq from off-shell vertex.
0.82
1
0.81
0.98
0.8
Bare
SI
0.96
0.79
0.94
0.78
0.92
ZA
0.7732(14)
0.77
0.9
0.76
0.7555(3)
0.88
0.75
0.86
0.74
1
0.5
0.72
1.5
2
2.5
2
0.73
(ap)
0
4
8
12
16
20
24
28
32
36
t
Hadronic correlators:
ZA = 0.7555(3)
Off-shell vertex:
ZA /Zq = 0.934 (2)(10)
Zq from off-shell conserved
current vertex.
1.1
mf = 0.02
mf = 0.04
1
0.9
0.8
Zq
= 0.753 (16)(30)
ZA = 0.703 (16)(32)
0.7
0.6
0.5
0
0.5
1
1.5
2
2
(hep/lat-0102005 / C. Dawson)
(ap)
ILFT/Edinburgh 6
Relativistic Heavy Quarks
•
In the rest system of a heavy quark:
–
A lattice theory with mHa ≥ 1 can be described
(on-shell) by a Symanzik effective Lagrangian:
–
Thus, we need only adjust 4 lattice parameters to achieve
a physical effective theory.
(El-Khadra, Kronfeld, Mackenzie, hep-lat/9604004)
–
As mHa 0 this will approach the usual fermion action.
ILFT/Edinburgh 7
Off-shell Heavy Quark NPR
• Static approximation:
– The requirement p2-m2>> ΛQCD2 is not possible
since the static quark is on-shell.
– Perturbative matching between the lattice and
continuum relies on identical wrong perturbative
results for p2 ≅ ΛQCD 2.
ILFT/Edinburgh 8
Off-shell Heavy Quark NPR
• Relativistic heavy quarks:
– Constraints p2 –mH2 >> ΛQCD 2 and p2 –mH2 << mH2 require
that one work at Minkowski momenta.
– Off-shell effective Lagrangian contains 8 parameters.
ILFT/Edinburgh 9
On-shell Heavy Quark NPR?
• Step-scaling in the static approximation
– Hietger and Sommer, hep-lat/0310035
– Match heavy-light spectrum computed with
standard quark and HQET actions:
• mH >> ΛQCD and 1/L (required by HQET)
• mH << 1/a (required by standard quark action)
– Used to determine mH from B meson spectrum.
• Generalize to RHQ action?
ILFT/Edinburgh 10
On-Shell RHQ NPR
• Use step scaling to connect
DWF’s at small a to RHQ’s at
larger, more practical values of a.
• Match on-shell, finite-volume
meson energies.
• To determine four parameters,
m0H, ζ, cE, cB, compute four
masses: mJ=0HH, mJ=1HH, mJ=0HL,
For fixed physics match
a and 2/3 a:
mJ=1HH.
ILFT/Edinburgh 11
How can be m0H, ζ, cE, cB determined?
• m0H will directly effect all masses.
• ζ (the speed of light) look at states with p 0,
compute E(p) m1 + p2/2m2, require m1 = m2.
• cB will directly effect the spin-spin splittings,
δm = mJ=1 - mJ=0.
• cE less obvious: perhaps it will effect the
splittings: δm = mJHH – mJHL?
ILFT/Edinburgh 12
Determination of cE
• Experiment with a β = 6.0, 163 x 32 lattice with
Wilson quarks. Determine HH and HL masses for
four sets of parameters, mπLL = 500 MeV, 252
configurations:
mJ=0HH
mJ=0HL
m0H = 0.2
cE = 1.0
1.5384(14)
0.9714(20)
m0H = 0.4
cE = 1.0
1.7914(14)
1.1102(20)
m0H = 0.4
cE = 1.769
1.6866(14)
1.0502(20)
• Are mJ=0HH and mJ=0HL independent functions of m0H
and cE?
ILFT/Edinburgh 13
Mass determination
• Use two hydrogenic
sources.
• Perform a two-cosh fit.
Heavy Heavy Pseudoscalar Meson
1.9
1
Ex-Local
Local-Local
Gnd-Gnd
Ex-Ex
Wall-Local
Wall-Wall
Gnd-Local
PT
GND
EX
1.85
0.8
1.8
|Psi|
a Meff
0.6
1.75
0.4
1.7
0.2
1.65
0
0
2
4
6
8
|R|
10
12
14
1.6
0
2
4
6
8
10
12
14
t
ILFT/Edinburgh 14
16
Fitted mass versus tmin
•Lowest mass of fit to two J=0 states.
•Fitting range from tmin O t O 16
ILFT/Edinburgh 15
Dependence on m0H and cE
• Compute the Jacobean:
• The largest effects of m0H and cE
are the same: a shift in the quark
mass.
0.007
• Result:
0.0065
0.006
0.0055
Jacobean
• An accurate determination
mJ=0HH and mJ=0HH of allows
m0H and cE to be disentangled.
0.005
0.0045
0.004
J = 0.09870- 0.09459
= 0.00412(27)
0.0035
0.003
0
2
4
6
8
10
t_fit_min
ILFT/Edinburgh 16
12
Next steps
• Perform a 3-level quenched calculation:
RHQ
163x32
mHa = 0.3
DWF
β = 6.351
Ls = 0.9Fm mH = 1.2GeV 1/a=3.6GeV
RHQ
163x32
mHa = 0.45
β = 6.075
Ls = 1.3Fm mH = 1.2GeV 1/a=2.4GeV
243x48
mHa = 0.2
β = 6.638
Ls = 0.9Fm mH = 1.2GeV 1/a=5.4GeV
now underway
RHQ
243x64
mHa = 0.3
β = 6.351
Ls = 1.3Fm mH = 1.2GeV 1/a=3.6GeV
RHQ
243x64
mHa = 0.45
β = 6.075
Ls = 2.0Fm mH = 1.2GeV 1/a=2.4GeV
ILFT/Edinburgh 17
Conclusion
• The errors will grow with each matching step.
Likely error 3 x error for above project.
• Can be done for full QCD as well but each step
will be more costly.
• A first attempt could use quenched lattices for
all but the coarsest a.
• First results by Lattice 2005?
ILFT/Edinburgh 18