Weak Charm Decays with Lattice QCD Weak Charm Decays with Lattice QCD Aida X. El-Khadra University...

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

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


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


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


“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.


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


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


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


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


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


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


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


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


Simulation parameters

Ensembles generated by MILC using the Asqtad action. Each point has400 – 800 configurations.


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


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)


From V. Pavlunin @ FPCP 2007:

Semileptonic Decays cont’d

The shape is determined more accurately than the normalisationin the Lattice QCD calculation


Kronfeld (Fermilab Lattice and MILC, 2005):



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)



Van de Water + Mackenzie (Fermilab Lattice and MILC, 2006/7):

B PRELIMINARY

q2 dependence fit using z-expansion



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)


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


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


Chiral fits

Example: FNAL/MILC fits with mvalence = msea at a = 0.09 fm

PREMLIMINARY


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


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


HPQCD

FNAL/MILC (Lattice 2007)

FNAL/MILC (PRL 2005)

Exp. Av. (Pavlunin @FPCP 2007)

PRELIMINARY

fD+ in comparison


HPQCD




PRELIMINARY

fDs in comparison


HPQCD




PRELIMINARY

fDs / fD+ in comparison


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.


nf = 2+1 ensembles used/available today


nf = 2+1 ensembles used/available today …. and soon available …


Back-up slides


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


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


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


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

Weak Charm Decays with Lattice QCD Weak Charm Decays with Lattice QCD Aida X. El-Khadra University...

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