Weak Charm Decays with Lattice Q C D
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Transcript of Weak Charm Decays with Lattice Q C D
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Weak Charm Decays with Lattice QCD
Weak Charm Decays with Lattice QCD
Aida X. El-Khadra
University of Illinois
Charm 2007 workshop, Aug. 5-8, 2007
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A. El-Khadra, Charm 2007, Aug 5-8, 2007 2
Outline
1. Introduction
2. Light Quark Methods
3. Heavy Quark Methods
4. Semileptonic Decays
5. Leptonic Decay Constants
6. Conclusions and Outlook
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Collaborations
Fermilab Lattice collaboration:
Kronfeld, Mackenzie, Simone, Di Pierro, Gottlieb, AXK,Freeland, Gamiz, Laiho, Van de Water, Evans, Jain
Computations at FNAL Lattice QCD clusters
HPQCD:Davies, Hornbostel, Lepage, Shigemitsu, TrottierFollana, Wong
MILC:Bernard, DeTar, Gottlieb, Heller, Hetrick, Sugar, ToussaintLevkova, Renner
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Introduction
2 20| | ( ), ( )J D f q f q
parameterize the matrix element in terms of form factors
2 2 2( ) | | | ( ) |cd
dΓknown V f q
dE
c d
u_
D
e
e
Example: Semileptonic D meson decay
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“Gold-Plated” Quantities orWhat are the “easy” lattice calculations ?
For stable (or almost stable) hadrons, masses and amplitudes with no more than one initial (final) state hadron,for example:
• , K, D, Ds, B, Bs mesons masses, decay constants, weak matrix elements for mixing, semileptonic and rare decays
• charmonium and bottomonium (c, J/, hc, …, b, (1S), (2S), ..) states below open D/B threshold masses, leptonic widths, electromagnetic matrix elements
This list includes most of the important quantities for CKM physics. Excluded are mesons and other resonances.
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B lK l
D lD l
D K lDs l
B D, D* l
00 KK
00 BB ss BB mixing
Vud Vus
Vcd
Vtd
Vub
Vcs Vcb
Vts Vtb
Lattice QCD program relevant to many CKM elements
K
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Introduction to Lattice QCD
fermion field lives on sites: (x)L
a
x
… discretize the QCD action (Wilson)
e.g. discrete derivative
in QCD Lagrangian
where c(a) = c(a;s,m) depends on the QCD parameters calculable in pert. theory
)]ˆ()ˆ([2
1 axaxa
)()( 432 aOaca
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Introduction to Lattice QCD, cont’d
in general: n 1
errors scale with the typical momenta of the particles, e.g. (QCDa)n for gluons and light quarks. keep 1a QCD
QCD ~ 200 – 300 MeV typical lattice spacing a 0.1 fm 1/a ~ 2 GeV
in practice: need to consider a range of a’s.
Improvement: add more terms to the action to make n large
napO )(contlat OO
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errors, errors, errors, …
finite lattice spacing, a: napO )( contlatOO
a (fm)
Ltake continuum limit:
computational effort grows like ~ (L/a)6
ml dependence: chiral extrapolation
nf dependence: sea quark effects: results with nf = 2+1 exist
statistical errors: from monte carlo integration
finite volume
perturbation theory latlatcont JZJ
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systematic errors, cont’d
• chiral extrapolation, ml dependence:
In numerical simulations, ml > mu,d because of the computationalcost for small m. use chiral perturbation theory to extrapolate to mu,d
need ml < ms/2 and several different values for ml (easier with staggered than Wilson-type actions)
ml
f
ms/2Decay constants, form factors: chiral logs contribute ~ )log( 22
mm
Staggered chiral perturbation theory: (Bernard, Sharpe, Aubin,…)
remove leading O(a2) errors in fits
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Light Quark Methods• Asqtad (improved staggered): (Kogut+Susskind, Lepage, MILC)
errors: ~ O(sa2), O(a4), but large due to taste-changing interactions has chiral symmetry; uses square root of the determinant in sea computationally efficient
tested against experiment at the few percent level
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lattice QCD/experiment
Testing the rooted Asqtad action
works quite well!
before 2004
HPQCD+MILC+FNAL, C. Davies, et al, Phys. Rev. Lett. 92:022001,2004
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Light Quark Methods• Asqtad (improved staggered): (Kogut+Susskind, Lepage, MILC)
errors: ~ O(sa2), O(a4), but large due to taste-changing interactions has chiral symmetry; uses square root of the determinant in sea computationally efficient
• HISQ (Highly Improved Staggered Action): (Follana, Hart, Davies, Follana et. al)
errors: ~ O(sa2), O(a4), ×1/3 smaller than Asqtad comp. cost: efficicient, ×2 Asqtad
• improved Wilson (Clover, …): (Wilson, Sheikholeslami + Wohlert, etc …)
errors: ~ O(sa), O(a2) if tree-level (tadpole) imp.; O(a2) if nonpert. imp. Wilson term breaks chiral symmetry comp. cost: ×4 Asqtad for mlight ~ mstrange , but inefficient at small quark masses
• Domain Wall Fermions: (Kaplan)
errors: ~ O(a2), O(mresa) almost exact chiral symmetry; breaking ~ mres ~ 3 ×10-3
comp. cost: ×L5 Asqtad, L5 ~ 16 - 20
• Overlap Fermions: (Neuberger)
errors: ~ O(a2) exact chiral symmetry comp. cost: ×5-10 DWF
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Simulation parameters
Ensembles generated by MILC using the Asqtad action. Each point has400 – 800 configurations.
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mQ QCD and amQ 1:
Heavy Quark Methods
• rel. Wilson action has the same heavy quark limit as QCD
• add improvement: preserve HQ limit• smoothly connects light and heavy mass limits, valid for all amQ
• errors: s(a, (a)2 or s /mQ, , (/mQ)n
• good for for charm and beauty
Fermilab (Kronfeld, Mackenzie, AXK):
lattice NRQCD (Lepage, et al., Caswell+Lepage) :
• discretize NRQCD lagrangian: valid when amQ > 1
• errors: (ap)n , (p/mQ)n • good for b quarks, but not charm
HISQ (Follana, Hart, Davies, Follana et. al):
• errors: ~ s (amc)2, (amc)4
• good for charm, but not beauty
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Semileptonic Decays: D l
• p (q2) dependence: 0p
lat cont| | | | ( )nV D V D O ap
p 1GeV improved actions help (keep n large)
• finite volume (L):
for a = 0.1 fm, L = 20, pmin = 620 MeV
2ip n
L
• Lattice Result with nf = 2+1 from Fermilab Lattice and MILC collaboration (C. Aubin et al, PRL 2005).
using MILC coarse ensembles with msea = 1/8 ms, …., ¾ ms Asqtad action for light valence quarks Fermilab action for charm quarks staggered chiral pert. thy (mvalence, msea, a2)
but: q2 dependence parameterized using BK model only one lattice spacing (a 0.12 fm)
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From V. Pavlunin @ FPCP 2007:
Semileptonic Decays cont’d
The shape is determined more accurately than the normalisationin the Lattice QCD calculation
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Kronfeld (Fermilab Lattice and MILC, 2005):
Semileptonic Decays cont’d
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Kronfeld (Fermilab Lattice and MILC, 2005):
Semileptonic Decays cont’d
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Semileptonic Decays: Improvements
add more lattice spacings to analysis: reduce discretisation errors
q2 dependence: z-expansion (Arnesen et al, Becher+Hill, Flynn+Nieves, …)
based on unitarity and analyticity
model independent analysis of the shape (fig)
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Semileptonic Decays: Improvements
Van de Water + Mackenzie (Fermilab Lattice and MILC, 2006/7):
B PRELIMINARY
q2 dependence fit using z-expansion
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Semileptonic Decays: Improvements
add more lattice spacings to analysis: reduce discretisation errors
q2 dependence: z-expansion (Arnesen et al, Becher+Hill, Flynn+Nieves, …)
based on unitarity and analyticity
model independent analysis of the shape (fig)
finite volume: twisted boundary conditions (Tantalo, Bedaque, Sachrajda,….)
arbitrarily small momenta pi < 2/L
further technical improvements:• random wall sources (MILC):
improve statistics
• double ratios to extract form factors (FNAL, RBC/UKQCD, Becirevic, Haas,
improve statistics Mescia)
reduce systematic errors…..
repeat calculation with other valence quarks, e.g. HISQ (HPQCD)
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Leptonic D and Ds Meson Decay Constants
• important test of LQCD methods
• results from two groups (FNAL/MILC & HPQCD) using MILC ensembles at a = 0.09 fm, 0.12 fm, 0.15 fm
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FNAL/MILC
Fermilab action for charmAsqtad light valence quarks
a = 0.09 fm: msea = 1/10 ms, 1/5 ms, 2/5 ms a = 0.12 fm: msea = 1/8 ms, 1/4 ms, 1/2 ms, ¾ ms
a = 0.15 fm: msea = 1/10 ms, 1/5 ms, 2/5 ms, 3/5 ms
+ 8-12 valence quark masses/ensemble
partial nonpert. renormalisation(~1.5% error)
staggered chiral PT fits to all valence and sea quark masses and lattice spacings together
blind analysis for Lattice 2007
HPQCD
HISQ action for charm and lightvalence quarks
a = 0.09 fm: msea = 1/5 ms, 2/5 ms
a = 0.12 fm: msea = 1/8 ms, 1/4 ms, 1/2 ms
a = 0.15 fm: msea = 1/5 ms, 2/5 ms
+ mvalence = msea
nonpert. renormalization from PCAC(no error)
cont. chiral PT + O(a2) terms, fit to alllattice spacings and masses together
Comparison of parameters
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Chiral fits
Example: FNAL/MILC fits with mvalence = msea at a = 0.09 fm
PREMLIMINARY
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Chiral fits
FNAL/MILC: staggered chiral PT fit to all data with extrapolation to physical masses and removal of O(a2) errors (left most point) compared to chiral fits at each lattice spacing.
PREMLIMINARY
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FNAL/MILC Error Budget
PRL 2005 Lattice 2007 (PRELIMINARY)
fD fDs/fD fD fDs/fD
HQ disc. 4.2% 0.5% 2.7% 0.2%
Light quark + Chiral fits 4-6% 5% 1.3% 1.3%
stat + fits 1.5% 0.5% ~1%
Inputs (a, mc, ms) 2.8% 0.6% 2.4% 1.0%
+ finite volume, PT matching ≤1.5% each
Total 8.5% 5.0% 4.3% 1.7%
PRELIMINARY
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HPQCD
FNAL/MILC (Lattice 2007)
FNAL/MILC (PRL 2005)
Exp. Av. (Pavlunin @FPCP 2007)
PRELIMINARY
fD+ in comparison
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HPQCD
FNAL/MILC (Lattice 2007)
FNAL/MILC (PRL 2005)
Exp. Av. (Pavlunin @FPCP 2007)
PRELIMINARY
fDs in comparison
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HPQCD
FNAL/MILC (Lattice 2007)
FNAL/MILC (PRL 2005)
Exp. Av. (Pavlunin @FPCP 2007)
PRELIMINARY
fDs / fD+ in comparison
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Conclusions and Outlook
Charm physics important test bed for LQCD methods
leptonic decay constants:
HPQCD: ~1-2 % error FNAL/MILC: ~4% error (PRELIMINARY)
semileptonic decay form factors:
existing result from FNAL/MILC can be improved to allow a quantitative comparison of the shapes with experiments:
first fit LQCD and exp. separately to test LQCD then fit together for best determination of CKM elements
HPQCD plans to use HISQ on MILC ensembles
Outlook:
nf = 2+1 ensembles using other sea quark actions are currently being generated. Important test.
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nf = 2+1 ensembles used/available today
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nf = 2+1 ensembles used/available today …. and soon available …
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Back-up slides
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Introduction to Lattice QCD, cont’d
the fermion doubling problem
naïve lattice action
in momentum space: sin p zeroes at p = (0,0,0,0), (/a,0,0,0), …
16 degenerate particles (“tastes”)
)( mL
Kogut-Susskind: keep the naïve action, and “stagger” 4 tastes on a hypercube and combine into one quark flavor
still leaves 4 degenerate “tastes” O(a2) errors are large due to taste changing interactions remove perturbatively: improved staggered action (Lepage, MILC) chiral symmetry is preserved light quarks computationally efficient
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Light Quark Methods
the fermion doubling problem
naïve lattice action
in momentum space: sin p zeroes at p = (0,0,0,0), (/a,0,0,0), …
16 degenerate particles (“tastes”)
)( mL
Wilson: add a dim 5 term to the action: 22 a
breaks the doubling degeneracy introduces O(a) error remove with improvement (Sheikholeslami+Wohlert) breaks chiral symmetry light quarks computationally inefficient
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systematic errors
• finite lattice spacing, a:
)(contlat naO OO
a (fm)
L
take continuum limit:
improved actions are much better …
•by brute force:
computational effort grows like ~ (L/a)6
•by improving the action:
computational effort grows much more slowly
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mixing:00 BB b d
d_
b_
W W
u,c,t
u,c,t
B0
_B0
02
02* ||||)( BBVVknownx Btbtdd O
BBB Bfm 2238
b
u_
WB-
_
fB:
Introduction, cont’d