Prospects for Lattice Phenomenology Chris Sachrajda 1st LHCb Collaboration Upgrade Workshop
Prospects for Lattice Phenomenology
Chris Sachrajda
School of Physics and Astronomy
University of Southampton
Southampton
UK
1st LHCb Collaboration Upgrade Workshop
Edinburgh, January 11 – 12 2007
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Outline of Talk
In this talk I ask the question
Will we be able calculate hadronic parameters with a 1% precision by
2015?
1.
Introduction
2.
Required Parameters for 1% Precision.
3.
Scaling Laws and Costs of Simulations.
4.
B -Physics.
5.
Conclusions.
I have made extensive use of
1.
V.Lubicz – CKM Fit and Lattice QCD, presented at SuperB IV, Monte
Porzio Catone, 2006.
2.
S.Sharpe – Weak Decays of Light Hadrons, presented at Lattice QCD,
Present and Future, Orsay, 2004.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Introduction
Lattice Simulations of QCD, in partnership with Experiments and Theory, play a central rôle in the determination of the fundamental parameters of the Standard Model
(e.g. CKM matrix elements, quark masses); in searches of signatures of New Physics and potentially in understanding the structure of the new physics.
The principal reason for performing lattice simulations in Flavour Physics is, of course, to quantify non-perturbative QCD effects.
There is a very large community of theorists working in lattice QCD, improving our understanding of the systematics, developing new theoretical and computational techniques to reduce the uncertainties and extending the range of quantities which can be computed in lattice simulations ⇒ I expect that there will be many improvements which I cannot predict today.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Introduction - Continued
Lattice results include both statistical and systematic errors.
Statistical errors arise from the Monte Carlo evaluation of the functional integrals.
I
I
Statistical errors are relatively easy to estimate.
A rule-of-thumb is that O ( 100 ) independent configurations are needed for about a 1% statistical error. The precise numbers however, depend on the quantities being studied, the volume of the lattice and the formulation of lattice QCD being used.
Sources of systematic uncertainty include:
I
I
I
I
I
Chiral Extrapolation: m u , d
→ m phys u , d
;
Discretization errors (the lattice spacing a = 0 );
Finite Volume Effects: (the dimensions of the lattice L =
∞
);
Heavy quark extrapolation: m
Renormalization Constants:
H
O
R
→ m b
, m c
.
( µ ) = Z ( a
µ , g ) O
B ( a ) . I will not talk about the calculations of the normalization constants Z , since in most cases we have non-perturbative techniques to compute them with better that 1% precision.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Precision of Lattice Results in CKM-Physics
B
Measured
B
Quantity
K
→
B → K ∗
→
→
D
π
/
π
/
`
ε
K
B → `
ν
∆ m d
D
ρ
∗
ν
∆ m d
/
∆ m s
`
`
ν
ν
Element
|
Matrix
V
CKM
|
|
| V
V
V us cb ub
|
Im V
2 td
| V ub
|
| V td
| td
/ V
|
| ts
/
ρ
(
γ
, `
+
` − ) | V td
/ V ts
|
|
Hadronic
Matrix
Element f f
B d
+
K
π
B f
K
B p
B
B d
ξ
( 0 )
F
B
→
D / D ∗ `
ν f
+
B
π
T
,
1
···
Current
Error
0.9%
22% on 1
− f
+
K
π
( 0 )
11%
14%
14%
5%
25% on
ξ
−
1
4%
40% on
ξ
−
1
11%
13%
V.Lubicz
Estimated
Error in
2015
???
???
???
???
???
???
???
???
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Projected CPU Power
I will assume that on the appropriate time-scale there will be 1-10
PFlops available for the major lattice simulations (today 1-10 TFlops).
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Assumptions and Comments
I will not assume any further improvements of the algorithms [very conservative indeed].
Indeed, I believe that most of my assumptions err on the conservative side.
For some quantities, notably B → H
1
H
2 decay amplitudes, we do not know how to formulate lattice computations.
{For K →
ππ decays, there are a number of technical difficulties but the matrix elements can be computed.}
We continue to learn how to calculate new quantities (e.g. the first lattice computations of f
+
K
π
( 0 ) were performed in 2004).
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
2. Target Simulation - How small does the lattice spacing have to be?
After S.Sharpe, Lattice QCD: Present and Future, Orsay, 2004
Heuristic Estimate: Let O be an observable and imagine that we are using an O ( a ) improved action:
O latt
= O phys n
1 + c
2
( a
Λ ) 2 + c n
( a
Λ ) n + ··· o where we assume that c
2 and c n are O ( 1 )
For some actions n = 3 for others n = 4 .
For light quark physics
Λ
∼
Λ
QCD
.
For heavy quark physics in general Λ ∼ m
Q and so we have to work to avoid the lattice artefacts being large (see below).
Imagine that we simulate at a min
Relative error is
ε
=
∆
O phys
O phys and
' c n
√
( 2
2 a min n / 2
− and linearly extrapolate in
2 ) ( a min
Λ
) n a
2
.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Target Simulation - How small does the lattice spacing have to be?
ε
=
∆
O phys
O phys
' c n
( 2 n / 2
− 2 ) ( a min
Λ ) n
For c n
= 1 and taking
ε = 1% , we have:
Λ
0.5 GeV 1.5 GeV 4.5 GeV a min
; ( n = 3 ) 0.091 fm 0.030 fm 0.010 fm a min
; ( n = 4 ) 0.105 fm 0.035 fm 0.012 fm
Current simulations typically have a ' 0 .
05 – 0 .
13 fm .
For light quark physics (or in some effective theories for the heavy quarks) the required value of a min is not so daunting.
Anticipating that the spatial extent will have to be several fermi we begin to get a feel for the number of points which will be required.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Target Simulation - How small do the quark masses have to be?
Again we present a heuristic estimate.
χ
PT ⇒ that approximately
O latt
= O phys
(
1 + c
1 m m
π
ρ
2
+ c
2 m m
π
ρ
4
+ ···
) where again we might expect that c
1 and c
2 are O ( 1 ) .
Imagine that we perform simulations with two values of ( m π / m ρ ) ,
( m π / m ρ ) min and 2 × ( m π / m ρ ) min say.
The resulting relative error is then
ε
=
∆
O phys
O phys
' 2 c
2 m m
π
ρ
4 min
.
Taking c
2
= 1 and
ε = 1% gives m π m ρ min
' 0 .
27 or
ˆ min m s
1
'
12 or m π
, min
' 200 MeV .
( ˆ / m s
) phys
' 1 / 25 .
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Target Simulation - How large does the volume have to be?
The finite-volume effects can be estimated using
χ
PT.
The leading effects come from pion loops and (for quantities without final state interactions) are typically of the form:
ε =
∆
O
O
= ce − m π L , where c is O ( 1 ) .
(The value of c depends on the quantity O being studied.)
If we require
ε
= 1% and take c = 1 then m π L ' 4 .
5 .
For a 200 MeV pion m π L ' 4 .
5 ⇒ L ' 4 .
5 fm .
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Target Simulation
Thus, with the generic aim of 1% precision, a target simulation might be:
I
I
I
I
120 Independent Configurations; a = 1 / 20 fm ( a −
1 ' 4 GeV); m π = 200 MeV;
L = 4 .
5 fm ( V = 90
3 × 180 ) .
The possible inclusion of the scale m b will be considered below.
How much would such a simulation cost?
This does depend on the formulation of lattice QCD being used.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
3. Cost of the Simulations
At the 2001 Lattice Conference in Berlin, Ukawa presented an estimate of the CPU cost of generating 100 independent configurations for
N f
= 2 , O ( a ) improved Wilson quarks on a L
3 × 2L lattice:
5
0 .
2
ˆ / m s
3
L
3 fm
5
0 .
1 fm a
7
TFlops-Years .
This has become known as the Berlin Wall.
Cost @ TFlops
-
Year D
2
1.5
1
0.5
0.1
0.2
0.3
0.4
0.5
0.6
m ms
L = 2 .
5 fm , a = 0 .
08 fm ,
V = 32
3
× 64 .
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Cost of the Simulation – Cont.
The CERN Group using the DD-HMC algorithm find that the corresponding cost is
1 5 6
.
05
0 .
2
ˆ / m s
L
3 fm
0 .
1 fm a
TFlops-Years .
L.Del Debbio et al., [hep-lat/0610059]
This is a huge improvement on the Berlin Wall:
3 5 7
5
0 .
2
ˆ / m s
L
3 fm
0 .
1 fm a
TFlops-Years .
Cost @ TFlops
-
Year D
2
1.5
1
0.5
0.1
0.2
0.3
0.4
0.5
0.6
m ms
L = 2 .
5 fm , a = 0 .
08 fm ,
V = 32
3 × 64 .
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Cost of the Target Simulation
For the target simulation of 120 configurations on an L = 4.5 fm, a = 1 / 20 fm lattice, with m π ' 200 MeV we would estimate a cost of about
0.07 PFlops-Years .
Overhead for N f
= 2 + 1 and including some lattices at larger values of and ˆ is about a factor of 3.
a
There is a bigger overhead (perhaps a factor of 10-30) for using quarks with the Ginsparg-Wilson symmetry and therefore good chiral properties.
The main message of this talk is that such simulations will certainly be feasible.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Current Status
The current reach of pion masses in lattice simulations is approximately:
SU ( 3 ) Limit
Currently Typical m q
/ m s m π (MeV) m π / m ρ
1 690 0.68
1/2 490 0.55
Impressive
MILC
Physical
1/4
1/8
1/25
340
240
140
0.42
0.31
0.18
The MILC collaboration (and collaborating groups using their data) using
Improved Staggered Fermions have calculated many quantities with small quoted errors.
There is a major theoretical question mark about the use of staggered fermions ⇒ most of the community uses other formulations.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Rooted Staggered Fermions
Staggered fermions ⇒ 4 tastes of quark for each flavour.
In practice this problem is tackled by rooting:
Z
[ dU ] e −
S gauge det [ D ( m u
)] det [ D ( m d
Z
)] det [ D ( m s
[ dU ] e −
S gauge { det [ D ( m u
)] det [ D ( m d
)] det
)]
[ D (
→ m s
)] }
1
4
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Rooted Staggered Fermions
Staggered fermions ⇒ 4 tastes of quark for each flavour.
In practice this problem is tackled by rooting:
Z
[ dU ] e −
S gauge det [ D ( m u
)] det [ D ( m d
Z
)] det [ D ( m s
[ dU ] e −
S gauge { det [ D ( m u
)] det [ D ( m d
)] det
)]
[ D (
→ m s
)] }
1
4
Is this legitimate?
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Rooted Staggered Fermions
Staggered fermions ⇒ 4 tastes of quark for each flavour.
In practice this problem is tackled by rooting:
Z
[ dU ] e −
S gauge det [ D ( m u
)] det [ D ( m d
Z
)] det [ D ( m s
[ dU ] e −
S gauge { det [ D ( m u
)] det [ D ( m d
)] det
)]
[ D (
→ m s
)] }
1
4
Is this legitimate?
Steve Sharpe reviewed this question at Lattice 2006 asking whether this procedure was:
I
I
I
Good – correct continuum limit without any complications;
Bad – wrong continuum limit;
Ugly – correct continuum limit, but with unphysical contributions present for a = 0 requiring theoretical understanding and complicated fits.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Rooted Staggered Fermions
Staggered fermions ⇒ 4 tastes of quark for each flavour.
In practice this problem is tackled by rooting:
Z
[ dU ] e −
S gauge det [ D ( m u
)] det [ D ( m d
Z
)] det [ D ( m s
[ dU ] e −
S gauge { det [ D ( m u
)] det [ D ( m d
)] det
)]
[ D (
→ m s
)] }
1
4
Is this legitimate?
Steve Sharpe reviewed this question at Lattice 2006 asking whether this procedure was:
I
I
I
Good – correct continuum limit without any complications;
Bad – wrong continuum limit;
Ugly – correct continuum limit, but with unphysical contributions present for a = 0 requiring theoretical understanding and complicated fits.
Rooted staggered fermions cannot be described by a local theory with a single taste per flavour.
Bernard, Golterman & Shamir, hep-lat/0604017
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Rooted Staggered Fermions
Staggered fermions ⇒ 4 tastes of quark for each flavour.
In practice this problem is tackled by rooting:
Z
[ dU ] e −
S gauge det [ D ( m u
)] det [ D ( m d
Z
)] det [ D ( m s
)] →
[ dU ] e −
S gauge { det [ D ( m u
)] det [ D ( m d
)] det [ D ( m s
)] }
1
4
Is this legitimate?
Steve Sharpe reviewed this question at Lattice 2006 asking whether this procedure was:
I
I
I
Good – correct continuum limit without any complications;
Bad – wrong continuum limit;
Ugly – correct continuum limit, but with unphysical contributions present for a = 0 requiring theoretical understanding and complicated fits.
Rooted staggered fermions cannot be described by a local theory with a single taste per flavour.
Bernard, Golterman & Shamir, hep-lat/0604017
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Are Rooted Staggered Fermions Bad or Ugly?
Steve Sharpe, based on theoretical arguments using perturbation theory, the renormalization group and
χ
PT (as well as on the circumstantial evidence from many numerical results), argues that it is
plausible that the correct choice is ugly, i.e. that rooted staggered fermions are unphysical for a = 0 , yet the results go over to those of a single-taste theory in the continuum limit.
S.Sharpe, [hep-lat/0610094]
Physical results must be extracted using complicated forms of rS
χ
PT ⇒ unphysical effects included in the fit.
In any case we need to check the results against those obtained with other actions.
As we have seen, the challenge set by the MILC Collaboration is being taken up by other collaborations using different lattice actions.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Clark’s Updated Berlin Wall Plot (Lattice 2006)
N configs
= 1000
L = 1 .
9 fm , a = 0 .
08 fm ,
V = 24
3
× 40 .
M.A.Clark hep-lat/0610048
I anticipate that, in the medium term, the theoretical uncertainties and complications due to the four tastes, ⇒ other formulations of lattice QCD will dominate the simulations.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
4. B-Physics - Introductory Comments
We have seen that for a relativistic b-quark in order to have 1% errors we would need a lattice spacing of about 0.01 fm.
I
I
Recall that the cost of simulations scales as a −
6 or a −
7 .
Such a fine lattice is therefore prohibitively expensive even for
PFlop computers.
Charm Physics is feasible with Wilson fermions.
I
If we take a = 0 .
033 fm, then the scaling law ⇒ the cost of generating 120 configurations is about 0.9 PFlop-Years.
Three complementary approaches to b -Physics:
I
I
I
Simulate relativistic heavy quarks in the charm-mass region and extrapolate to the b -quark mass.
Use effective theories:
I
I
HQET - Substantial progress has been made by Rainer Sommer and his collaborators at DESY-Zeuthen, Rome II and Berlin to develop a strategy to renormalize the HQET non-perturbatively (including the
O (
Λ
QCD
/ m b
) corrections.) R.Sommer [hep-lat/0611020]
NRQCD or the Fermilab/Tsukuba Actions.
The finite-volume and step-scaling approach of the Rome-II group within QCD.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Comments – Continued
It is likely that the best results in 2015 will come from a combination of those extrapolated from QCD with m
Q
' m c and those in effective theories (including power corrections in
Λ
QCD
/ m b
).
To get to O ( 1% ) precision, non-perturbative renormalization of the effective theories will be necessary. This is being done for the HQET.
In the medium term we will be looking for the results from the different approaches (HQET, NRQCD and Fermilab/Tsukuba) to converge and studying the theoretical foundations.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
5. Summary and Conclusions
Access to PFlops CPU power + current knowledge and techniques ⇒
O ( 1% ) precision on hadronic parameters.
It is likely that further theoretical and technical improvements will improve the precision still further.
For B -Decays into two-hadron states (or states with a higher numbers of hadrons) we need new ideas before we can formulate the problem of the numerical evaluation of the amplitudes.
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
Estimates of Lattice Errors in 2015
B
Measured
Quantity
K →
π
`
ε
K
B → `
ν
∆ m d
ν
∆ m d
/ ∆ m s
→
B → π / ρ ` ν
B → K ∗
D / D ∗ `
ν
/
ρ
(
γ
, `
+
CKM
Matrix
Element
| V us
|
Im V
| V
|
|
|
|
V
V td
V
V ub td
/ V cb ub
|
2 td
|
|
| ts
|
` − ) | V td
/ V ts
|
Hadronic
Matrix
Element f f
B d
+
K
π
B f
B
(
K
0 ) p
B
B d
ξ
F
B
→
D / D ∗ `
ν f
+
B
π
T
,
1
···
Current
Error
0.9%
22% on 1
− f
+
K
π
( 0 )
11%
14%
14%
5%
25% on
ξ
−
1
4%
40% on 1
−
F
11%
13%
V.Lubicz
Estimated
Error in
2015
<0.1%
2.4% on 1
− f
+
K
π
( 0 )
1%
1-2%
1-2%
0.5-0.8%
3–4% on
ξ
−
1
0.5%
5% on 1
−
F
2–3%
3–4%
LHCb Upgrade Workshop, Edinburgh, January 11th 2007
