Charm physics from lattice QCD

Christine DaviesUniversity of Glasgow, HPQCD collaboration

EINN 2007Milos

QCD is key part of SM but quark confinement trickyLattice QCD enables calcn of QCD effects “from first principles”. Done by numerical evaln of Path Integral in a 4-d vol. of space-time defined as a lattice

RECIPE• Generate sets of gluon fields that contribute most to the PI• Calculate averaged “hadron correlators” on these and fit to obtain masses and simple matrix elements

ZdAµe−LQCDO(Aµ)

a• Fix and determine to get physical results

amq

Where can lattice QCD have most immediate impact?Precision calculations of electroweak decay rates for gold-plated hadrons CKM physics

Vud Vus Vubπ→ lν K→ lν B→ πlν

K→ πlνVcd Vcs Vcb

D→ lν Ds→ lν B→ DlνD→ πlνD→ KlνVtd Vts Vtb

〈Bd|Bd〉 〈Bs|Bs〉

WJ = V0, Vi, A0, Ai

B π

Vub

I will concentrate on results relevant to this programme...

expt=(CKM)x(lattice calc.)

1

|VudV ∗ub|

|VcdV ∗cb|

|VtdV ∗tb|

|VcdV ∗cb|

Unitarity triangle - test this!

Why is lattice QCD so hard?Handling light u,d, s quarks is a big headache

Quarks must be ‘integrated out’by inverting Dirac matrix M

Lq,QCD = ψ(γ ·D+m)ψ≡ ψMψ

valence quarks, calculate M−1

sea quarks, includein importance sampling gluons

det(M)

Cost inc. as mq→ 0a→ 0 L→ ∞and also as

The story so far ....Early days (before 2000) - u, d, s sea quarks omitted or inc. with u/d masses 10-20x too big. Systematic errors 10-20 % and theory not self-consistentNow (since 2000) - possible to inc. u/d sea quarks with masses only 3-5x too large and extrapolate to real world. Improved staggered quark formalism first to do this since numerically very fast. Uses to reduce 4 “tastes” to 1 flavor. 2007 - improved staggered calculations have matured, many values of and from MILC collaborn.Results using other formalisms now appearingFuture - looks good. Lots of analysis to be done ....

a mu/d

(det(M))0.25

Essential to check how lattice QCD is doing vs well-known gold-plated experimental quantities

0.9 1 1.1

nf=2+1

!(3S-1S)

!(1P-1S)

!(2P-1S)

!(1D-1S)

2mBs,av-m!

"(1P-1S)

m" - m#c

mD*

s - mDs

mD

mDs

m$

3m% - mN

fK

f&

mBc New (HPQCD): Highly improved staggered quarks (HISQ) - improves disc. errors further over asqtad. Allows use for c quarks.

2007 resultslattexpt

Update of 2003 results (MILC/FNAL/HPQCD) using improved staggered sea quarks.Fix QCD params from:ϒ(2S−1S),mπ,mK,mηc,mϒQuenched Approx. is dead!

Heavy (b,c) valence quarks in lattice QCD

For light quarks, discretisation errors set by

mQa

ΛQCDaFor a≈ 0.1 fm (ΛQCDa)2 ≈ 0.06

For b, c quarks discretisation errors set bya≈ 0.1 fm mca≈ 0.6, mba≈ 2.5

Discretise Dirac equation onto lattice

Lq,QCD = ψ(γ ·D+m)ψ≡ ψMψsyst. errors

Multiple values of a allows fit and extrapolation to a= 0

For b quarks, use the fact that is not a dynamical scale to write down an effective theory in which it is removed. Possibilities: HQET, NRQCD, FNAL heavy quarks

Now disc. errors set by e.g (mom. in bound state)a

handles and

mba

ϒ B

For c quarks, have a choice of effective theory or beating down disc. errors in relativistic theory

FNAL/MILC use FNAL heavy (clover) quarks

NEW - HPQCD use Highly Improved Staggered Quarks as a relativistic method

How good a disc. needed for precise charm results?

mca≈ 0.4, (mca)2 ≈ 0.2, αs(mca)2 ≈ 0.06, (mca)4 ≈ 0.04Remove all these errors by improved disc. and check this

Improved staggered quarks remove all tree-level a2 errors

+ + + + =

=(Naik)

c5’c1 (Fat link)

c3’

c5 c7c3

Highly Improved Staggered Quarks (HISQ) reducefurther ‘taste-changing’ errors in staggered formalismvery accurate

Excellent statistical accuracy from random wall sources(as used by MILC for light mesons)

0.016

0.018

0.02

0.022

0.024

0.026

0.028

0.03

0 10 20 30 40 50

t

∑aie−Mit +(−1)taipe−Mipt

+(t→ T − t)correlator/(fit ground state)

Ds

Can even study the noise in the correlator and see what mass it has ...

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35 40

effe

ctiv

e m

ass

t

Fine D signal and noise

signalnoise

D mass0.5*[!c + "]mass

Important tests e.g. using charmonium

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0 1 2 3 4 5

c2

p2 in units of (2!/L)

2

hisq, ma=0.43, N=0.79 on MILC fine

c2 =E2−m2p2

is sensitive test of a2

errors.We adjust Naik coeff. so

c2 = 1

Removes errors at Follana et al, hep-lat/0610092

αs(mca)2, (mca)4

Compare charmonium and D spectrum

MD

MDs

Exp’t

1.9

2

Mas

s(G

eV)

Mas

s(G

eV)

0.1 0.2 0.3 0.4 0.5

mu/d/msmu/d/ms

Another key test of disc. errors since charmonium and D have different dynamics more tests of lattice QCD possible. Same action as for u,d,s.

Fix mc

test D massesvs expt.

E.Follana et al, 0706.1726[hep-lat]

lattice errors 6 MeV

using MILC configs with sea quarks

Will lead to a very accurate value for mc

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1

bar

e m

c/m

s

sea mu/d/ms

HPQCD mc/ms PRELIM AUG 07

MILC v. coarseMILC coarse

MILC fineoverall uncertainty

PDG06

Since using same formalism for s and c can get accurate ratio of bare lattice quark masses. For accurate ratio need a 1-loop lattice perturbative calcn.Lattice nonperturbativemethods also underway.

MS

J=Aµ

B Wπleptonic decayπ

Br(π→ µν) ∝V 2ud f 2π

Γ(K→ µν)Γ(π→ µν)

→ V 2us f 2KV 2ud f 2π

HPQCD

Leptonic decay rate gives CKM element or test vs expt.

Using HISQ for valence u/d,s on MILC gives

E.Follana et al, 0706.1726[hep-lat]

1

1.05

1.1

1.15

1.2

1.25

1.3

0 0.2 0.4 0.6 0.8 1

f K/f!

mu,d / ms

very coarsecoarse

fineextrapoln

Vus = 0.2262(14)competitive with PDG from Kl3

2007 New Results for D, Ds decay constants,for comparison to experiment

Next two years: more lattice results and improved semileptonic form factors also ...

fDs

Exp’t

0.25

0.3

fD0.2

0.25

Dec

ayC

onst

.(G

eV)

Dec

ayC

onst

.(G

eV)

HPQCD, HISQ

GeV

GeV

CLEO

250 300 350

fDs

HPQCD HISQ

on asqtad sea

0706.1726[hep-lat]

FNAL/MILC asqtad

LAT07 prelim.

CLEO

0704.0437[hep-ex]

BaBar

hep-ex/0607094

Belle

EPS2007

fDs

• expt uses Vcs = Vud • error will improve to few % in 2 yrs• not all em corrns calculated ..

Dx→ lν

2 v. different lattice calcs

Lattice inc u,d,s sea vs expt

241(3)MeV

b physics - B0 mixing and CKM constraint

B0 B0 =

HW

VtdV ∗tb

f 2BBBParameterise with where is decay constant.

fB

ΔMx =G2FM2

W6π2 |V ∗

txVtb|2ηB2S0(xt)MBx f 2BxB̂Bx

Take exptl ratio from oscillation rates for Bs and Bd

|VtdVts

| = ξ

√ΔMdMBsΔMsMBd

ξ=fBs

√BBs

fB√BB

,

calculate in lattice QCD,renormln cancels

0.1 0.2 0.3 0.4 0.5 0.6

md / m

s

1

1.05

1.1

1.15

1.2

1.25

1.3 HPQCD (no 1/M correct.)

MILC

HPQCDFNAL/MILC

ξ=fBs

√BBs

fB√BB

Gamiz, LAT07

2007 New results for

Can compare to easier lattice calc. of fBs/ fBHPQCD, hep-lat/0507015 1.20(3)FNAL/MILC, Simone, LAT07 1.26(4)

ξ

√MBsMB

PRELIMINARY

inc. u, d, s sea quarks

fBs√B̂Bs = 0.281(21)GeV

HPQCD hep-lat/0610104

one value of a so far, main error is renormln of lattice 4-q operator

NRQCD for bFNAL for b

cf fDs/ fD = 1.164(11)

Semileptonic decay rates/form factors WJ = V0, Vi, A0, Ai

B

!"#$%&'()*+,%-+.%/0%1223%%%4+*%&5)67'. 89

!"##$%&'()&*$%#$+,'-."/& and "0-+12#&',+)3"1$4"#$+,'56789:

;0-+12#&'-."/&'",*',+)3"1$4"#$+,'<"1$*"#$+,'

=+)3"1$4"#$+,'()&%$-$+,>''&?/&)$3&,#'9@''!"##$%&'AB@

#77:()*;%<=7>2?93@A

!"#$B-4$CB

DEFCG

C:,H'%=I:,J'7.%

#*K,'++7

L,I*M'NK

D K

VcsMuch harder “3-pt” calculation

q2 = m2D+m2K−2mDEKD rest frame

dΓdq2

∝ |Vcs|2| f+(q2)|2

FNAL/MILC results compared to expt - theory error ~ 10%

Better precision needed for good lattice testShipsey: EPS07,

FNAL/MILC: hep-ph/0408036

B excl. semileptonic decay and CKM constraintDiscretization errors

0 0.005 0.01 0.015 0.02 0.025 0.03

a2

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

hA1

(1)

Lat ‘07, August 2, 2007 – p.21/24

Laiho, LAT07

FNAL/MILCW

J = V0, Vi, A0, Ai

BB π,D,D∗, . . .

Vub,Vcb

B→ D∗lν

∝ |VcbF(1)|2rate at zero recoil

F(1)

a2/ fm2

FNAL/MILC with u, d, s sea quarks, 3 values of aF(1) = 0.930(12)(19) HFAG Vcb = 38.7(0.7)(0.9)×10−3

stat syst expt latt

B→ πlν Flynn+Nieves 0705.3553[hep-ph] combine lattice (HPQCD, FNAL/MILC) with LCSR, get Vub = 3.47(29)×10−3work underway to extend lattice results

Use lattice results with u, d, s sea quarks + experiment for constraints on unitarity triangle.

!-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

"-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1CKM 2007 LATTICE QCD

BBBfsBB

sBf

KB

#cbV##ubV#

$%

PDG06

CKM 2007 LATTICE QCD

Lattice inputs (2+1 sea quarks):BKfK/ fπ f+(K→ πlν)F(B→ D∗lν)f+(B→ πlν)fBs

√BBs

fB√BB

Errors on lattice results should halve over next two years

,

Improve checks in D physics

Conclusions• Lattice calculations inc. sea quarks are in excellent shape.

• Significant new results this year in charm physics that now give accuracy similar to that for light hadrons

Future:In next two years, lattice errors on CKM constraints should halve.More checks will be done against gold-plated decays of D.Precise values of mc and mb expected soon.

• Precision continues to improve for the staggered quarkformalism

• More work needed to improve b physics further.