s and light b Heavy quark physics with NRQCD dynamical ... · Heavy quark physics is an important part of the Standard Model and place where lattice QCD can make key calculations.

Heavy quark physics with NRQCD bs and lightdynamical quarks

Christine Davies

University of Glasgow

HPQCD and UKQCD collaborations

Key aim of HPQCD collabn: accurate calcs in lattice QCD,

emphasising heavy q physics. Requires a whole range of lattice

systematic errors to be simultaneously minimised - critical one has

been inclusion of light dynamical quarks.

• Current results on heavyonium, αs etc

• Developments for calculations for next 1-2 years - moving

NRQCD, HISQ

Japan Dec 2004

1

People involved in various aspects of this work:

I. Allison, S. Collins, CD, A. Dougall, K. Foley, E. Follana, E. Gamiz,

A. Gray, E. Gulez, A. Hart, P. Lepage, Q. Mason, M. Nobes, J.

Shigemitsu, H. Trottier, M. Wingate

HPQCD/UKQCD

C. Aubin, C. Bernard, T. Burch, C. DeTar, S. Gottlieb, E. Gregory, U.

Heller, J. Hetrick, J. Osborn, R. Sugar, D. Toussaint,

MILC

M. Di Pierro, A. El-Khadra, A. Kronfeld, P. Mackenzie, D. Menscher,

M. Okamoto, J. Simone

HPQCD/Fermilab

2

Heavy quark physics is an important part of the Standard Model

and place where lattice QCD can make key calculations.

• Form many ’gold-plated’, well-characterised heavy-heavy bound

states whose masses can be calculated accurately in lattice QCD.

New states being discovered there currently - predictions possible.

• Heavyonium states test the b and c quark actions for use in

calculations for heavy-light mesons and baryons.

Heavy-light bound states are critical to understanding CKM

unitarity triangle.

Problem on lattice is mQa not small → special techniques needed.

3

The Unitarity triangleImportant objective of current particle physics: accurate determination

of elements of CKM matrix.

-1

0

1

-1 0 1 2

sin 2βWA

∆md

∆ms & ∆md

εK

εK

|Vub/Vcb|

ρ

η

CK Mf i t t e r

B factory prog. needs small

2-3% reliable lattice QCD

errors for Bs/d oscillations,

B → D or π decay.

CLEO-c will test lattice pre-

dictions for D physics in next

2 years.

Requires all systematic er-

rors to be small simulta-

neously. Precise quenched

calcs are no good!

4

HPQCD/MILC spectrum results 2003

MILC collab. have used improved staggered quark formalism (+ highly

improved gluon action) to generate ensembles of configurations which

include 2+1 flavours of dynamical quarks.

2 = u, d degenerate with masses down to ms/8.

1 = s (can ignore heavy c, b, t dynamical qs.)

3 values of lattice spacing, a ≈ 0.087 fm and 0.12fm and 0.18fm.

Fix 5 free parameters of QCD (bare mu = md, ms, mc, mb, and

a ≡ αs) using

mπ,mK ,mDs,mΥ and ∆EΥ(2S − 1S). These are ‘gold-plated’

quantities (e.g. stable hadron masses).

Compute other ‘gold-plated’ quantities as a test of (lattice) QCD.

5

Lattice QCD/Experiment (no free parameters!):

Before Now

0.9 1 1.1quenched/experiment

Υ(1P-1S)

Υ(3S-1S)

Υ(2P-1S)

Υ(1D-1S)

ψ(1P-1S)

2mB

s − mΥ

mΩ

3mΞ − mN

fK

fπ

0.9 1.0 1.1(n

f = 2+1)/experiment

Tests:

light mesons and

baryons

heavy-light mesons

heavyonium

Find agreement with

expt (at last!) when

correct dyn. quark

content is present.

Quenched approx.

has syst. errors

10% and internal

inconsistency.

Davies et al, hep-lat/0304004 + Toussaint,Davies, LAT04

6

These results needed:

• Large ensembles to get good statistical errors. Long length in the

time direction gives good π mass.

• Large physical volume.

• Very light u and d quark masses so chiral extrapolation is not far.

• Good control of discretisation errors with a highly improved gluon

and quark action.

7

In fact discretisation errors are largest source of remaining uncertainty.

Disc. errors are worse unquenched than they were quenched. (a few %

vs zero)

Is this from glue or from a2 errors in dynamical quarks (handled by

staggered chiral pert. th.)?

8

Future calculations will improve further on this in two ways:

• Run at even finer lattice spacing values - a = 0.06fm with

ml/ms = 0.2 costs 3 Tflopyrs. (483 × 144). This halves all

discretisation errors compared to MILC fine lattice set. May be

done by UKQCD+MILC.

• Run with more highly improved gluon and quark action. Gluon

Nfαsa2 corrections being calculated (Mason and Horgan). Highly

Improved Staggered Quarks have half the taste-changing errors of

asqtad, (Follana).

9

NonRelativistic QCD (NRQCD)Discretisation errors are naively a worse problem for heavy quarks

because mQa is large. However, their non-relativistic nature saves us.

NRQCD good for heavy quarks - can match order by order in vQ and

αs to continuum full QCD.

L is ψ†(Dt +H)ψ, where ψ is a 2-spinor.

H0 = −∆(2)

2M

δH = −c4g

2M~σ · ~B + c2

ig

8M2(∇ · ~E − ~E · ∇)

− c3g

8M2~σ · (∇× ~E − ~E ×∇)

− c1(∆(2))2

8M3(1 +

Ma

2n) + c5

a2∆(4)

24M+ . . .

Fast to solve on one pass thru lattice. All Us tadpole-improved.

For b quarks this is an excellent action. For c quarks more problematic.

10

Υ(bb) spectrumLattice NRQCD for bs on MILC configs. Tests/tunes action for Bs.

2S-1S fixes a and 1S fixes amb.

1-loop matching gives mb,MS(mb,MS)=4.3(3) GeV.

! "# $% &' (

) * +,- .% &' ( /!0 1 2354 0 6 798 /!: 4 ;

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.05 0.1 0.15 0.2

Ene

rgy

Split

ting

(Gev

)

mlight(GeV)

∞∼ ∼

MILC nf=3: 1P-1S 3S-1S 2P-1S

MILC nf=0: 1P-1S 3S-1S 2P-1S

experiment: 1P-1S 3S-1S 2P-1S

Gray, Davies et al, HPQCD, hep-lat/0310041, Gulez,Shigemitsu, hep-lat0312017.

11

Prediction of Bc mass.

From difference between mass of Bc (NRQCD b, Fermilab c) and

average of Υ and J/ψ, get 6.304±12+18-0 GeV.

6.2

6.25

6.3

6.35

6.4

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Mas

s G

eV c

-2

sea quark mass ratio ml/ms

’light’ u 1:2:3’heavy’ u 1:2:3

’u0’ u 1:2:3’fine lattice’ u 1:2:3

New experimental result from CDF (Glasgow, FNAL and Texas Tech)

6287(5) MeV.

Allison, Davies, Gray, Kronfeld, Mackenzie, Simone (HPQCD), LAT04

12

Precise determination of αs.

Mean value of various Wilson loops and their ratios calculated to 3rd

order in lattice pert. th. and on the lattice.

Results available at

3 values of a (inc.

MILC super-coarse)

allows higher order

terms to be esti-

mated.

Preliminary re-

sult: αs(Mz) =

0.1181(15).

Improves on PDG.

Will get better with

even finer lattices.

log(1 × 1)

log(1 × 2)

log(BR)

log(CE)

log(1 × 3)

log(2 × 2)

log(1 × 4)

log(2 × 3)

log((1 × 1)(2 × 2)/(1 × 2)2)

log(1 × 3/2 × 2)

log((CE × BR)/(1 × 1)3)

log(CE/BR)

log(1 × 4/2 × 3)

log((1 × 1)(2 × 3)/(1 × 2)(1 × 3))

Lattice Results Compared With PDG-04

αMS

(MZ)

0.105 0.110 0.115 0.120

1

Mason, Trottier, Lepage et al (HPQCD), LAT04

13

Matrix elements in heavyoniumAccurate calculation of Υ leptonic width is good check.

Renormln calc. by Horgan in progress. Meanwhile take ratio of 2S to

1S to cancel leading piece.

Clear that discretisation errors are main problem here - improve

operators, action etc.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.02 0.04 0.06 0.08 0.1

(Γee

(2s)

/Γee

(1s)

)(M

2s/M

1s)2

mlight(GeV)

∞∼ ∼

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5

(Γee

(2s)

/Γee

(1s)

)(M

2s/M

1s)2

a2 (GeV-1)

14

Gold-plated quantities for the CKM matrix

Gold-plated decays (i.e at

most one hadron in final

state) exist for almost every

element (+ K −K mixing).

Can now calculate these ac-

curately in lattice QCD.

Important for lattice calcs to

have extensive cross-checks

for error calibration: Υ, B,

ψ, D, etc.

Vud Vus Vub

π → lν K → lν B → πlν

K → πlν

Vcd Vcs Vcb

D → lν Ds → lν B → Dlν

D → πlν D → Klν

Vtd Vts Vtb

〈Bd|Bd〉 〈Bs|Bs〉

15

Unquenched results for B → π form factorsExtrapoln to physical mπ is done at fixed Eπ and is not far (lightest

mπ = 260 MeV).

|ub|V0 0.002 0.004 0.006

FNAL04

HPQCD

) w/ full-recon.2,qX(m

-recon.ν) w/ 2,qX(m

end-pointeP

w/ LQCD (unquenched, preliminary)νlπBelle

νluBelle X

Used by Belle in new Vub de-

termn. (hep-ex/0408145)

Shigemitsu+Gulez, HPQCD, LAT04

16

These results are great but still suffer from one problem - they are

unable to cover the full q2 range.

B → π at small q2 corresponds to π at large lattice momentum. This

gives:

• increased statistical errors since signal/noise set by splitting to

zero momentum π.

• increased systematic errors from discretisation errors at large pπa.

Can solve these problems by moving B instead provided that we do not

increase systematic errors in B system.

In fact we can treat large B momentum accurately since it is mostly b

momentum and b momentum we can treat exactly with moving

NRQCD.

17

Moving NRQCDWrite Pb = mbu+ k. u = γ(1, ~v).

Now remove mbu from the action in the same way that mb was

removed for NRQCD.

For NRQCD, do FWT in rest frame of b ≡ lattice frame.

For moving NRQCD rest frame of b boosted wrt lattice, and must

boost back to get L in lattice frame.

Early work by Hashimoto and Sloan.

We have extended action, tested heavy-heavy and heavy-light and

done O(αs) pert. th.

Foley, Lepage, Davies, Dougall, HPQCD, LAT04

18

Moving NRQCDIn b rest frame:

L = ψ†(iDt +D2

2m+σ · B2m

+ . . .)ψ

Ψ = Te−imγ0t

ψ

χ

In lattice frame:

L = ψ†(iDt + iv · D +D

2

2γm+ − (v · D)2

2γm+σ · B2γm

. . .)ψ

Ψ =Λ(v)√γTe−imu·xADt

ψv

χv

; Λ =1

√

2(1 + γ)

1 + γ σ · vσ · v 1 + γ

19

Tests of Moving NRQCDUse simplest action (with no spin-dependence).

Take quenched lattices at β=5.7 as cheap.

Do HH and HL. For L use clover propagators at κs.

Antiquark G is G∗q with v → −v

Check v dependence of e.g. kinetic mass for fixed ma. Extract this

from:

Ev(k) + C(v) =√

(ZpP0 + k)2 +M2kin

P0 = γmv (twice this for HH).

Working with different k dirns wrt to P0 can extract Zp and Mkin.

Noise grows as v and k grow as again set by zero total momentum

states. Ameliorate with smearing.

20

Tests of Moving NRQCDFind HH Mkin not strongly v-dependent for fixed ma i.e. will not need

to change ma rapidly as a function of v to tune.

Find shift between χ and Ev(k = 0) (per quark) is same for HL and

HH.

21

Tests of Moving NRQCDCan calculate heavy quark self-energy perturbatively as done in

NRQCD.

Working to O(αs):

G−1 = Q−1 − aΣ(k)

= −ik4a− αsΩ0 + αsik4aΩ1 + v · ka− αsv · kΩv + . . .

= Zψ(−ik4a+k

2a2

2γRmRa+

PR · kγRmRa

+ . . .)

with Ωv = 1vx

∂Σ∂kx

etc.

Remnant of reparameterisation invariance keeps renormln of P0 small.

Physics is same if shift momentum between k and P0.

22

Tests of Moving NRQCDTest pert.th. against non-perturbative extraction of e.g. Zp =

renormln of P0.

Also calculate HL binding energy as γR(Ev(k = 0) −E0) and find

v-independent.

23

Tests of moving NRQCD - HHCompare ”decay constant” of ηb at rest and moving and from spatial

and temporal axial vector current.

J0 = χ†vψv and Jk = χ†

vvkψv.

Relativistic corrns will fix Ak case to 1.

24

Tests of moving NRQCD - HLDitto for HL. Now have two pieces to currents at leading order coming

from boost operator Λ.

e.g. Jk ∝ (1 + γ)χ†vσkq34 + γχ†

vσkσ · vq12.

Now makes a big difference to

getting fB right from spatial

axial current.

For non-moving case, sub-

leading current is same size as

leading current because Ak ∝ k.

Renormln Zv not yet calculated.

25

Moving NRQCD - future

• Need to start calculations on real MILC configs with staggered

light quarks for appropriate 3-pt functions for B → π.

• Need to calculate renormalisation of current operators, preferably

to 2-loops.

26

Further improving the staggered formalism

Limit to precision with asqtad improved

staggered quarks is still taste-changing in-

teractions associated with high-momentum

gluon exchange.

p=0

p=0

p=π/a

p=-π/a

Improve action further by repeating the ‘Fat7’ smearing. Add Naik and

Lepage terms (x2) as before to keep an action with αsa2 errors only.

This is the Highly Improved Staggered Quark action (HISQ).

27

Discretisation errorsHISQ shows v. good behaviour on taste-changing and dispersion reln.

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0 1 2 3 4 5

c2 β=5.7, mπ=0.55

asqtadhisq

unimprovedhyp

0

0.005

0.01

0.015

0.02

0.025

0.03

0 1 2 3 4 5

β=5.93, ∆mπ2, 3-link - NambuGoldstone

asqtadhisqhyp

Follana, Mason, Davies, HPQCD, in preparation

28

Future: Use HISQ for charm

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5

mH

Ha

HISQ charm

exptunimp ηcHISQ ηc

unimp J/ψHISQ J/ψ

Unimproved calcs

(JLQCD hep-

lat/9411012) had

problems with taste-

changing in π ≡ ηc.

This is much im-

proved for HISQ.

Plan: try this on

MILC fine (and

planned super-

fine) lattices where

αs(mca)2 = a few

%.Allison, Davies, Follana, Lepage, Mason HPQCD

29

Future First QCDOC machine being tested in Edinburgh.

30

Conclusions

• Calculations with 2+1 flavors of light dynamical quarks have made

first high precision lattice prediction of a hadron mass, that of the

mass of Bc meson.

• Gold-plated matrix elements for CKM determinations are in

progress (more in Andreas and Junko’s talks).

• Future work based on some new techniques which have lots of

promise.

31