Lattice QCDstatus and prospects

Christine DaviesUniversity of Glasgow, HPQCD collaboration

Hadrons 2007Frascati, Italy

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 flavor physics and CKM elements

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-consistent

Now (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.

2007 - improved staggered calculations have matured. Results using other formalisms now appearing

Future - looks good. Lots of analysis to be done ....

see http://www.physik.uni-regensburg.de/lat07/

(Quenched approx.)

0

0.05

0.1

0.15

0.2

0 0.005 0.01 0.015 0.02 0.025 0.03

m!

2m

in /

GeV

2

a2 / fm

2

MILC imp. staggered, 2+1RBC/UKQCD DW, 2+1

PACS-CS, clover, 2+1ETMC twisted mass, 2CERN-TOV, clover, 2

QCDSF, clover, 2

Choice of light quark formalism - issues are speed, disc. errors, chiral symmetry + technical issues. Can also mix e.g. dw valence on stagg. sea

imp.stagg.(asqtad)

domainwall

clover

twistedmass

fast

slow

fast

fast

Status of configs 2007

Extrapolate to physical point.Also need large L and good statistics.

speed chiralsymm. collab.

OK

good

bad

OK

MILC/HPQCD/FNAL

RBC/UKQCD

ETMC

PACS-CSQCDSFCERN-TOV

a2/ fm2m2 π,min/G

eV2≡mu/d

See Boyle, Urbach, Kuramashi, LAT07

no. of sea quarks

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

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

2007 resultslattexpt

Update of 2003 results using imp. stagg sea quarks.Fix QCD params from ϒ(2S−1S),mπ,mK,mηc,mϒ

QA is dead!

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$

mN

fK

f%

mBc

• Parameters of QCD are well-determined from lattice, again with imp. stagg sea quarks.

mq

αs ~1%~ few % , u,d,s, 10% for c,b - improvement shortly

!"# !"#$ !"#%

&'()*+(

,*-)./0123(45

6.7*)08(-29:;

9((<2:/(7*54012;1*44()0/+2=9:;>

!2-(1*?5

@2A0-4B

C)*+D(/4*40./

;<(14).51.<?2=E*4401(>

(<2('(/425B*<(5

6B.4.F<).-G140./

"2-(1*?

(H(F2)*4(5

#5=I

@>

1

10

100

1000

10000

Quark masses (MeV)

u

d

s

c

b

2006 lattice QCD2006 PDG

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25 0.3

f ! /

GeV

m!2 / GeV

2

HPQCD HISQ on asqtad sea 2+1MILC asqtad 2+1

NPLQCD DW on asqtad sea 2+1RBC/UKQCD DW 2+1

ETMC twsted mass 2 expt

Can compare results from different formalisms away from physical point if accurate enough.

Br(π→ µν) ∝V 2ud f 2πJ=Aµ

B Wπ

Need multiple a and mu/d for chiral and contnm extrapoln for final result.

Need large volumes for accuracy - test vol. effects vs. chiral pert. th. (MILC, ETMC, QCDSF, RBC)m2π/GeV 2 ∝ mu/d

fπ

GeV

leptonic decayπ

Fitted lines from HISQHPQCD 0706.1726[hep-lat]

no. of sea quarksquark formalisms with good chiral symmetry

McNeile, LAT07

Kaon physics (Kl2) and Vus

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

Γ(K→ µν)Γ(π→ µν)

→ V 2us f 2KV 2ud f 2π

2007 results for calcs with u, d, s sea quarks

HPQCD

see Jüttner, LAT07HPQCD: 0706.1726[hep-lat] , error budget

0.22 0.225 0.23

Vus

HPQCD HISQ

on asqtad sea

MILC asqtad

PDG 2006

PACS-CS clover

RBC/UKQCD dw

Error 0.6%

Vus competitive w. PDG

Kl3

K semileptonic decay (Kl3) and Vus

RBC/UKQCD, dw, 2+1, 2007

See Jüttner, LAT07

WJ = V0, Vi, A0, Ai

BK π

Vus

|Vus f Kπ+ (0)| = 0.21673(46)

Extrapoln gives

=f +

(0)

f+(0) = 0.9609(51)

Vus = 0.2257(15)Lattice error 0.5%, need to check disc. errors + other calcs.

0.225

0.23

0.97 0.975Vud

V us

0.225

0.23

0.97 0.975

FlaviaNetKAON DECAY WG

Vud (0+ → 0+)

Vus (Kl3)

fit withunitarity

fit

Vus/Vud (Kµ2)

unitarity

Charm physics, new method, 2007

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

Key issue is discretisation errors, because mca ~ 0.5HISQ - much improved control of disc. errorscan use for charmonium and D - 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]

HPQCD

2.8

3

3.2

3.4

3.6

3.8

4

mass

/G

eV

charmonium masses, HISQ on fine MILC

!c(1S)

"(1S)

!c(2S)

"(2S)

#c0

#c1

- lattice errors 6 MeV

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• em corrns could be important ..

Dx→ lν

2 v. different lattice calcs

Lattice inc u,d,s sea vs expt

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

,

To others using lattice results: QA is dead!

Harder hadron physics

Baryon masses with a2tad staggered quarks D. Toussaint

Figure 3a: Nucleon correlators at a= 0.06 fm

for light quark masses 0.4ms and 0.2ms.

Figure 3b: Nucleon mass fits for ml = 0.2ms.

Figure 4a: Nucleon masses versus quark

mass, (red blue and green symbols), and fits

to the nucleon mass (cyan and magenta).

Figure 4b: The same, expanded to show the

small quark mass region. The points with

ml = 0.2ms are from the fits indicated with

arrows in Fig. 3b.

We show two forms for extrapolating the nucleon mass to the physical quark mass and contin-

uum limit. The first (cyan) includes only the lowest order nonanalytic correction, order m3! .

mNr1 =M0r1+A1(m!r1)2+B1a

2"+ r1−3g2A32! f 2!

m3! (2.1)

In these fits we just set f! , gA and M#−MN to their physical values. The cyan and magenta

symbols in Figs. 4a and 4b are the data points with the fitted lattice spacing corrections, B1a2" ,

subtracted, i.e. at a= 0, and the cyan and magenta lines are the fitting function at a= 0.

5

Baryon masses with a2tad staggered quarks D. Toussaint

fixing strange quark mass. Using data available at conference time, the continuum and chiral ex-

trapolated value is M! ∗ r1 = 2.679+.025−.056 . Converting to physical units using r1 = 0.318(7) and

adding errors, this is M!(MeV ) = 1660+40−50 , where the experimental value is 1672 MeV.

This work is supported by the US Department of Energy and National Science Foundation.

Computations were done at the NSF Teragrid, NERSC, USQCD centers and computer centers at

the Universities of Arizona, Indiana, Utah and California at Santa Barbara.

Figure 5a: !− mass fits for ml = 0.2ms. In

these plots the strange quark mass has not

been tuned to the correct ms; it is one of the

masses at which correlators were computed.

Figure 5b: !− masses and fitting (see text).

The error on the experimental point is from

the error in r1.

References

[1] T. Blum et. al [MILC], Phys. Rev. D 55, 1133 (1997); K. Orginos and D. Toussaint, Phys. Rev. D 59

(1999) 014501; K. Orginos, D. Toussaint and R.L. Sugar, Phys. Rev. D 60 (1999) 054503;

G.P. Lepage, Phys. Rev. D 59 (1999) 074502; J.F. Lagäe and D.K. Sinclair, Nucl. Phys. (Proc. Suppl.)

63, 892 (1998); C. Bernard et. al [MILC], Phys. Rev. D 58 (1998) 014503.

[2] Claude Bernard et. al, Phys. Rev. D 64 (2001) 054506 [hep-lat/0104002].

[3] C. Aubin et. al, Phys. Rev. D 70 (2004) 094505. [hep-lat/0402030]

[4] C. Davies and G.P. Lepage, HPQCD collaboration, private communication; M. Wingate et al., Phys.

Rev. Lett. 92 (2004) 162001 [hep-ph/0311130]; A. Gray et al., Phys.Rev. D72 (2005) 094507

[hep-lat/0507013].

[5] Poster by C. Bernard, this conference.

[6] R. Sommer, Nucl. Phys. B411, 839 (1994).

[7] C. Bernard et. al [MILC], Phys. Rev. D 62, 034503 (2000) [hep-lat/0002028].

[8] E. Jenkins, Nucl. Phys. B368,190 (1992); V. Bernard, N. Kaiser and U.G. Meissner, Z. Phys. C60,

111 (1993).

[9] J. Bailey, Phys. Rev. D 75 114505 (2007) [hep-lat/0611023].

7

2=

3

signalnoise

= exp− (mp

1.5×mπ)

Baryons have exponentially growing signal/noise and harder to do chiral/contnm extrapoln

Toussaint, MILC, LAT07

mΩ = 1660(50)MeV

mN = 942(60)MeV

mπ = 275MeV mπ = 500MeV

Even harder hadron physics

Flavour singlet PS meson: η2With Nf = 2 degenerate quarks, the flavour singlet pseudoscalar meson

(called η2) is related to the experimental η′(958) and is expected to have

mass around 800 MeV. hep-lat/0006020

Here we summarise results with light pions and a < 0.1 fm (r0/a > 4.5).

CP-PACS r0 = 4.49UKQCD r0 = 5.04ETMC r0 = 5.22 L = 24ETMC r0 = 5.22 L = 32UKQCD r0 = 5.32ETMC r0 = 6.6

(r0mπ)2

r 0m

η

210

3

2

1

0

η2 mass is consistent with a constant behaviour in the chirallimit with m(η2) ≈ .88 GeV (r0m(η2) = 2).

Neutral mesons and disconnected diagrams in Twisted Mass QCD – p.7/13

850MeV

Flavor singlet hadronsrequire calculation of disconnected diagrams

Extremely noisyNew results this year from ETMC for 2-flavor case - 2+1 is harder

Flavor singlet PS η2

C. Michael, ETMC, LAT07

Conclusions• Lattice calculations inc. sea quarks are in excellent shape. Calcs. with staggered quarks continue to improve and good results appearing now from other quark formalisms.

• Significant new results this year in s and c physics

• I have not mentioned hadron structure, phase structure at finite temp. etc etc. - see LAT07 website

Future:In next two years, errors on CKM constraints should halve.More checks will be done against other gold-plated decays.Harder hadron physics calcs will improve.

Round Table discussion

Christine DaviesUniversity of GlasgowHPQCD collaboration

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$

mN

fK

f%

mBc

Lattice QCD has made hugeprogress towards precision calculations for ‘gold-plated’ hadrons, particularly mesons, since 2003.Gold-plated = stable, well away from decay thresholds, accurately measured exptlly. These can be used for precision test of QCD.Excited states are noisy Unstable hadrons associated with multihadron states etcPrecision will not be as good

Need to decide what question you are answering ......

latt/expt

!"#$"%&"'()*+(,**- ./0(!#"1$'2%(3/4(56'7879:$;'< =>

?21@"67(:#"1$'2%?21@"67(:#"1$'2%

! ,**(A2"71B"C(167D89:+(),=E>F(;78:6$'6#81(G8@:67 @;$$81"+(;:H*4)(D%+(

;:I;$H=+(%!H-**(5"J

LHPC collaborn

plans to include sea quark effects- multihadron states a problem?

0 0.05 0.1 0.15 0.2

(a/r0)2

1

2

3

4

5

m0

++ r

0

Quenched

Wilson, Wilson gauge, SESAM

Tad clover, Wilson gauge, UKQCD-I

NP clover, Wilson gauge, UKQCD-II

Tad clover, Iwasaki gauge, UKQCD-II

f0(600)

f0(980)

f0(1370)

f0(1500)

f0(1710)

m!=600 MeV

m!=775 MeV

m!=520 MeV

m!=780 MeV

Flavor singlets/glueballs are also much harder

C. McNeile, LAT07, 0710.0985[hep-lat]

Key will be very high statistics, i.e. fast sea quark formalisms, and a good operator basis

Meanwhile, high precision results for gold-plated hadrons will continue to improve.

• Simultaneous calcns of heavy-heavy, light-light and heavy-light spectrum to a few MeV is a stringent test of QCD, not possible for models. - extend this to low-lying baryons, ‘silver-plated’ mesons • Ditto for electroweak decays. Now haveto 2%. Calcn in progress for Requires understanding QED corrections.Strong impact on flavor physics programme.- expect accurate form factors, structure function moments for baryons.

fD, fDsΓe+e−(J/ψ) Γe+e−(φ)

• Many more calculations with different quark formulationswill appear in next few years ...