s and light b Heavy quark physics with NRQCD dynamical ... · Heavy quark physics is an important...
Transcript of s and light b Heavy quark physics with NRQCD dynamical ... · Heavy quark physics is an important...
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
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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
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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.
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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!
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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.
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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
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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.
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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.)?
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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).
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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.
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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.
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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
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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
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Mason, Trottier, Lepage et al (HPQCD), LAT04
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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.
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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)
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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〉
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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
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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.
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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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
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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
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Future First QCDOC machine being tested in Edinburgh.
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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.
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