Cascades on the Lattice

Kostas OrginosCollege of William and Mary - JLab

Cascade Physics - Jlab 2005

LHP Collaboration

LHPC collaborators

• R. Edwards (Jlab)

• G. Fleming (Yale)

• P. Hagler (Vrije Universiteit)

• C. Morningstar (CMU)

• J. Negele (MIT)

• A. Pochinsky (MIT)

• D. Renner (UofA)

• D. Richards (Jlab)

• W. Schroers (DESY)

Summary

• What can the Lattice do for you?

• How will we make it happen?

• What has been done?

• Some very preliminary results from LHPC

Thanks to D. Richards

Particle Data Group

Lattice QCD

• Spectrum calculation: simplest thing to do

• Strange quarks don’t decay

• stable cascades

• Better signal than protons (strange quark is heavy)

• Can vary quark masses

• Quantum numbers easy to identify

Difficulties

• Strong decays: Unstable particles• Finite volume techniques• Heavy quark masses: above threshold

• Broken rotational symmetry• Angular momentum not a good quantum number

• Vacuum polarization effects• Inefficient algorithms • mass of up and down quarks too light!

• Chiral symmetry and lattice fermions

Lattice Operators

G1

H

G21/2

7/2

3/2

5/2 5/2

7/2

hep-lat/050801 Basak et al.

Particle Data Group

Lattice QCDLattice QCD

In continuous Euclidean space:

Z =!DqDq̄DAµ e!S[q̄,q,Aµ]

"O# =1

Z

!DqDq̄DAµ O(q̄, q, Aµ) e!S[q̄,q,Aµ]

The Lattice regulator:

U

q

µ

µ

!

µ!P

Uµ(x) = e!iaAµ(x+µ̂2)

Fermion doubling

8

Lattice QCD

In continuous Euclidean space:

Z =!DqDq̄DAµ e!S[q̄,q,Aµ]

"O# =1

Z

!DqDq̄DAµ O(q̄, q, Aµ) e!S[q̄,q,Aµ]

The Lattice regulator:

U

q

µ

µ

!

µ!P

Uµ(x) = e!iaAµ(x+µ̂2)

Fermion doubling

8

In continuous Euclidian space:

Lattice QCD

In continuous Euclidean space:

Z =!DqDq̄DAµ e!S[q̄,q,Aµ]

"O# =1

Z

!DqDq̄DAµ O(q̄, q, Aµ) e!S[q̄,q,Aµ]

The Lattice regulator:

U

q

µ

µ

!

µ!P

Uµ(x) = e!iaAµ(x+µ̂2)

Fermion doubling

8

Lattice regulator:

Lattice QCD

In continuous Euclidean space:

Z =!DqDq̄DAµ e!S[q̄,q,Aµ]

"O# =1

Z

!DqDq̄DAµ O(q̄, q, Aµ) e!S[q̄,q,Aµ]

The Lattice regulator:

U

q

µ

µ

!

µ!P

Uµ(x) = e!iaAµ(x+µ̂2)

Fermion doubling

8

Fermion doubling

Gauge sector:

Fermion sector:

The computation

QCD Lattice Action:

SLatQCD =

! !

NcReTr[1! Pµ,"(x)] + #̄!(U)#

with !(U) the fermionic matrix

Computers can’t do fermions !" Integrate them out

Z ="

dU det(!)nf e!# !

NcReTr[1!Pµ,"(x)]

Monte Carlo Integration:

• Produce configurations Ui with probability

P(U) =1

Zdet(!)nf e!

# !Nc

ReTr[1!Pµ,"(x)]

• Compute

#O$ ="

dU OP(U) = limN"%

1

N

N!

i

Oi

10

Chiral symmetry breaking

Spectrum• Correlation functions

• J an interpolating field for some state

C(t) = !J(t)J̄(0)"

Particle Masses

We can measure the function describing the propagation of a particlefrom point A to point B:

CAB = !JAJ̄B"where J is an appropriate combination of quarks that make a particle(e.g. proton)

For large time t

C(t) = Ze#Mt

26

Particle Masses

We can measure the function describing the propagation of a particlefrom point A to point B:

CAB = !JAJ̄B"where J is an appropriate combination of quarks that make a particle(e.g. proton)

For large time t

C(t) = Ze#Mt

26

C(t) = Z0e!M0t + · · ·

Proton

Need to do

• Continuum extrapolation

• Chiral extrapolation

• Infinite volume extrapolation

• In all cases use Effective field theory • Scale setting

• Heavy quark potential (Sommer scale)• Rho mass (bad choice)• Heavy quarkonia

• 2+1 Dynamical flavors

• 2 light (up down) 1 heavy (strange)

• charm bottom top (treated in HQET as extrernal)

• Light quark masses m < 400MeV

• Chiral extrapolations

• Finite volume corrections

• Numerical algorithm slows down (algorithm scaling )

• Continuum extrapolations

• compute at several lattice spacings (algorithm scaling )

π

!1

a7

!1

m2.5!

What does it take

!1

m2.5q

1

The Berlin Wall

Motivation

Overlap versus Twisted Mass

Accelerating the HMC

Conclusion and Outlook

Multiple Time Scale Integration and Preconditioning

Numerical Results

Simulation Cost

Cost for 1000 independent configurations,

a ! 0.08 fm, 243 " 40 lattices.

Tflops · years

Ukawa

run D

run A, B, C

0.00

mPS/mV

10.50

1

0

run D

run A,B, CTflops · years

mPS/mV

10.50

0.05

0.04

0.03

0.02

0.01

0

C. Urbach Algorithmic Challenges in Lattice QCD

Urbach (ILFTN 3)

Queching

Quenched “Approximation”

The simulation is very simple if we ignore the quark loop e!ects, i.e.:

det(") = 1

This is called the Quenched approximation

Simulations with internal fermion loops are called dynamical fermionsimulations.

These simulations are much harder.9

The computation simplifies if we ignore the fermion loops

Uncontrolled “approximation”

The computation

QCD Lattice Action:

SLatQCD =

! !

NcReTr[1! Pµ,"(x)] + #̄!(U)#

with !(U) the fermionic matrix

Computers can’t do fermions !" Integrate them out

Z ="

dU det(!)nf e!# !

NcReTr[1!Pµ,"(x)]

Monte Carlo Integration:

• Produce configurations Ui with probability

P(U) =1

Zdet(!)nf e!

# !Nc

ReTr[1!Pµ,"(x)]

• Compute

#O$ ="

dU OP(U) = limN"%

1

N

N!

i

Oi

10

Quenched spectrumHadron Spectrum

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

m (G

eV)

K input! inputexperimentK

K*

!N

"#

$

%

#*

$*

&

[CP-PACS]

27

CP-PACS

Quenched Spectrum

FIG. 7. The quenched light hadron spectrum computed in the O(a) improved and its compari-

son to the results of ref. [4], obtained using the unimproved Wilson action (full circles). The levelsof the experimental points are denoted by the solid lines.

36

UKQCD

Recent Developments

• Cheap dynamical fermions (Kogut-Susskind)

• “Taste” breaking

• Improved KS action (Asqtad ) [KO, Sugar, Toussaint ‘99]

• MILC has generated lattices: Ready to milk the MILC

• Chiral symmetry on the lattice ( errors)

• Domain wall fermions [Kaplan -- Shamir]

• Overlap fermions [Neuberger, Narayanan]

• Costly for dynamical: RBC now starting

• Improvements:

• Improved gauge actions [KO with RBC ‘02]

• Mobius fermions [Brower, Neff, KO ‘04]

• Big Computers!

Recent Developments

• Cheap dynamical fermions (Kogut-Susskind)Improved KS action (Asqtad) [KO,Sugar,Toussaint, ’99]O(a4, g2a2) errorsSignificant reduction in flavor breakingVery good scaling [MILC, ’00]MILC has generated lattices: Ready to milk the MILC

• Chiral symmetry on the LatticeDomain Wall Fermions [Kaplan,Shamir]Overlap Fermions [Neuberger,Narayanan]Improved scaling: O(a2) errors, full flavor symmetry

Renormalization and mixingCostly for dynamical: Now starting =! RBC

• Big Computers!

Very promising for the next few years

16

Recent Developments

• Cheap dynamical fermions (Kogut-Susskind)Improved KS action (Asqtad) [KO,Sugar,Toussaint, ’99]O(a4, g2a2) errorsSignificant reduction in flavor breakingVery good scaling [MILC, ’00]MILC has generated lattices: Ready to milk the MILC

• Chiral symmetry on the LatticeDomain Wall Fermions [Kaplan,Shamir]Overlap Fermions [Neuberger,Narayanan]Improved scaling: O(a2) errors, full flavor symmetry

Renormalization and mixingCostly for dynamical: Now starting =! RBC

• Big Computers!

Very promising for the next few years

16

Domain Wall Fermions for QCD

Formulate the 5D Wilson fermions with mass M != 0 in s ! [1, Ls]

1 2 Ls/2 Ls... ...

q(R)

For "2 < M < 0, light chiral modes are bound on the walls.Only one Dirac fermion without doublers remains.

1 2 Ls/2 Ls... ...

mf

q(R) Fermion mass is introduced byexplicitly coupling mf of thewalls. [Shamir,Furman & Shamir]

8

Why Domain Wall Fermions • Excellent chiral properties

at finite lattice spacing:

• Ls → ∞ Exact chiral symmetry

• Ls finite: Exponentially small chiral symmetry

breaking

• Gauge action affects chiral symmetry [KO with RBC hep-lat/0211023]

• Chiral extrapolations

• Simpler renormalization due to symmetry

• Can work close to the chiral limit

• Have O(a2) errors

• Excellent scaling properties

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14(a/r0)

2

1.1

1.2

1.3

1.4

1.5

m/m

ρ

N/DBW2N/WilsonK*/DBW2

N

K*

Scaling

MILC and RBC data

Dynamical 2+1 flavors

FIG. 16: The “big picture”. Crude continuum and chiral extrapolations of hadron masses and

splittings compared with experimental values. The upsilon and charmonium columns are di!erences

from the ground state masses, from work of the HPQCD and Fermilab groups[16, 19]. Here the !

and K masses fix the light and strange quark masses, and the " 1P-1S mass splitting is used to

fix the lattice spacing.

PHY99-70701 and NSF–PHY00–98395.

[1] The MILC Collaboration, C. Bernard et al., Phys. Rev. D 64 (2001) 054506;

40

MILC

f!

fK

3M! ! MN

2MBs ! M"

!(1P ! 1S)!(1D ! 1S)!(2P ! 1S)!(3S ! 1S)!(1P ! 1S)

LQCD/Exp’t (nf = 0)1.110.9

LQCD/Exp’t (nf = 3)1.110.9

MILC, HPQCD, UKQCD

Quenched vs Dynamical

• Domain wall fermions for valence (with hyp smeared links)• Chiral symmetry• Ward Identities

• Kogut-Susskind 2+1 Dynamical flavors

• Improved KS action (Asqtad: O(a4, g2a2)) [KO, Sugar, Toussaint ‘99]

• MILC has generated lattices: Ready to milk the MILC

• Light quark masses: Lightest pion mπ ~ 250MeV

• Volumes: 2.6 to 3.2 fm

• Future: Continuum extrapolation• MILC lattice spacings: a=0.125fm, 0.09fm• a=0.06fm in 1 - 2 years

The LHPC program

The DWF quark masses

• Domain wall fermions for valence (hyp smeared links)

• We tune the DWF quark mass to the staggered Goldstone pion

• Unitarity violation

• Baer et.al.: tune to the taste singlet for mπ

• Not clear it helps for other quantities ( ex. fπ )

• Unitarity is restored in the continuum in any case

0 5 10 15 20−1

0

1

2

3

4

5

6

7

t

C(t)

0 5 10 15 20−2

0

2

4

6

8

t

C(t)

5 10 15 20−0.05

0.00

0.05

0.10

0.15

0.20

t

C(t)

0 5 10 15 20−2

0

2

4

6

8

t

C(t)

6 8 10 12 14 16 18 20−0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

t

C(t)

0 5 10 15 200

2

4

6

8

t

C(t)

IsoVector scalar correlator

m=0.05

m=0.01 m=0.02

m=0.03

χPT calculation:Prelovsek LAT 05

Pion decay constant• Fit the lower 4 points

• Scale used a = 0.125 fm

• One loop χPT extrapolation: 130.6(1.8)MeV

• Systematic error: chiral extr. 3 MeV 2% from scale setting

• χ2/d.o.f. ~ 2

• Need mixed χPT of Baer et.al.

LHPC dwf on MILC

0 0.1 0.2 0.3 0.4 0.5

Mπ

2 (GeV2)

0

0.5

1

1.5

2M

ass (

GeV

)

N1/2+

Ξ1/2+

Ξ3/2+

Δ3/2+

Ξ1/2−

Σ1/2+

Σ3/2+

N

ΔΞ1/2

Ξ3/2+

LHPC Preliminary

D. Richards

LHPC data vs Experiment

Prelim

inary

Cascade - Nucleon mass splitting

• Mild quark mass dependence

• Small systematic error due to chiral extrapolation

• Other systematice errors cancel

• Scale used a = 1588 MeV

• Latt./Exp. = 1.006(8)

Prelim

inary

f!

fK

3M! ! MN

2MBs ! M"

!(1P ! 1S)!(1D ! 1S)!(2P ! 1S)!(3S ! 1S)!(1P ! 1S)

LQCD/Exp’t (nf = 0)1.110.9

LQCD/Exp’t (nf = 3)1.110.9

MILC, HPQCD, UKQCD

Quenched vs Dynamical

LHPC: preliminary (no errorbars)

Conclusions

• Lattice QCD can be very helpful in studying the cascade spectrum

• Not much has been done up to now

• The “gold plated” observable 3MΞ - MN is well reproduced (MILC and LHPC).

• LHPC: Need to work on statistics and extrapolations

• Finer lattice spacing is on the way