Charm physics with Highly Improved Staggered Quarks
Eduardo Follana, CTHD, Peter Lepage, Kit Wong.
Charm physics has been ‘poor relation’ of b physics but right now
provides a good chance to predict results ahead of CLEO-c expt.
Predictions needed for fD , fDs , D → Klν, D → πlν form factors.
Ratios of these independent of CKM elements.
c also has cc bound states for further tests.
Current Status
• Best results from FNAL/MILC. Use FNAL action on MILC
ensembles at 3 a values (but most coarse and supercoarse since slow).
Action has O(αs a) and O(a2 ) errors in principle.
Some reduction using non-rel. interpretn and ratios of corrln fns.
Errors in fD etc quoted at around 10%.
Don’t do a great job on cc e.g. fix c from Ds instead of ψ.
• Also have results (I. Allison) using NRQCD on MILC coarse and
supercoarse.
Limit to accuracy is pert. matching since αs large when mc a > 0.7.
Current Status - decay constants
1.40
a = 0.121 fm
staggered χPT fit (to 60 points)
taste violations removed
mq = ml
CLEO-c LP05
s
1.20
s
1/2
fD mD /fDmD
1/2
1.30
1.10
FNAL/MILC/HPQCD
1.00
0.0
0.2
0.4
0.6
mq/ms
0.8
1.0
1.2
100
150 200 250
fD (MeV)
300
Both exptl and theory errors 8% in fD from 2005. Ratio fDs /fD
quoted at 1.24(7) by FNAL (hep-lat/0506030).
SL form factors
2
2
qmax/mD*
s
2.5
D → Klν
1.5
2
f+(q )/f+(0)
2
1
experiment [FOCUS, hep-ex/0410037]
lattice QCD [Fermilab/MILC, hep-ph/0408306]
1σ (statistical)
2σ (statistical)
0.5
0
0
0.05
0.1
0.15
0.2
0.25
2
/mD*
s
2
q
0.3
0.35
0.4
0.45
Comparison to FOCUS here.
New
results
on
D → K and D → π
expected over next
year from CLEO-c.
Ratios independent
of CKM with decay
constants.
Quark masses
Quark masses (MeV)
2005 lattice QCD
2004 PDG
10000
b
c
1000
s
100
10
d
u
1
mc (mc ) = 1.10(13) GeV
(Nobes, Trottier, LAT05)
One-loop matching.
Relativistic charm
Conclusion: we need a better (and faster) c quark action. How
improved must this be for few % errors?
With mc a ≈ 0.33, (mc a)2 ≈ 0.1, αs (mc a)2 ≈ 0.02.
→ an action with tree-level a2 errors removed will have few % errors
at a ≈ 0.06fm.
There is only one such action in use : improved staggered.
Have developed Highly Improved Staggered (HISQ) action that has
smaller taste-changing than asqtad. Also very fast.
Test this on MILC fine lattices. Simplest to test this on cc and
provides worst case scenario on syst. errors.
cc pseudoscalar and vector
’ηc’ masses, fine, ma=0.43
3.2
space-split pion
time-split pion
rho
expt
mass /GeV
3.15
3.1
local ψ
3.05
3
2.95
3 link taste-s
1 link
goldstone
2 link
Stat. errors tiny.
16 ηc and ψ.
Taste-splittings not visible (¡
2 MeV) for ψ.
10 MeV total for ηc . This is
as expected since taste-split
at fixed a ∝ 1/M .
Speed of light
2
c , fine, ma=0.43, coarse, ma=0.65
2
fine goldstone
coarse goldstone
c
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2
p
2.5
3
3.5
4
c2 is ratio of E 2 to p2
term in dispn reln.
i.e.
mkin /m = 1/c2 .
Find c2 = 1.02(2) on fine
MILC, 1.15(2) on coarse
MILC.
Tree-level analysis gives
9/20(ma)4 , pert. analysis
gives < 0.1αs (ma)2 in
agreement with this.
Tree-level a2 errors give 20%
error even on fine.
Error budget for charm mass using HISQ and ψ
Fine Superfine
Errors in a
0%
Speed of light 2%
0%
<1%
p4 /8m3 errors 1.5% <1%
αs3 errors
3%
2%
4%
2%
in matching
Total
Good
check
is
2mDl/s − mψ .
Need high β MC
for mass renormln.
Looks do-able.
Hyperfine splitting very sensitive to systematic errors.
hyp vs asq for hisq inc. syst errors
fine
coarse
fnal
expt
140
Prop.
to inverse kinetic
mass, so must use this if
c2 6= 1.
a error appears twice, because of this inverse reln.
FNAL results have ambiguity
(at least for coarse) whether
fix c from HH or HL.
hyp/MeV
120
100
80
60
0
0.005
0.01
2
a (fm)
0.015
2
0.02
Matrix elements
fD /fDs and ψ leptonic width in progress on fine MILC. Can be done
with conserved currents so no renormalisation.
Leaves at worst αs (ma)2 errors for fD and fDs = 5% on fine, 2% on
superfine. Ratio more accurate.
Conclusions
We can get 3% errors for charm on superfine lattices using Highly
Improved Staggered Quarks.
Not possible with any other existing method.