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1Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Κ. Αλεξάνδρου

Hadron Structure from Lattice QCDStatus and Outlook

Πανεπιστήμιο Κύπρου

2Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Outline

• Lattice QCD

1. Introduction

2. Some recent results

• Hadron Deformation:

1. γN Δ matrix element on the lattice

Quenched results

Full QCD results with DWF

2. Charge density distribution

• Diquark dynamics

• Pentaquarks

• Status and perspectives for dynamical simulations

• Conclusions

3Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Introduction

Quantum ChromoDynamics is the fundamental theory of strong interactions. It is formulated in terms of quarks and gluons with only parameters the coupling constant and the masses of the quarks.

L a μνQCD

aμν

1=- F F +ψ D-4 m ψ

μ b bμ μ μν

a a a a a a cν ν μ μ ν

cμD =γ i∂ + T A F =∂ A - ∂ A + f A Ag g

Produces the rich and complex structures of strongly interacting matter in our Universe

Self energy diverges QCD + regularization renormalization

define a physical theory

4Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Fundamental properties of QCD

• Self interactive gluons unlike photons

• Interaction increases with distance Confinement

• Interaction decreases at large energies asympotic freedom

• Coupling constant αs >> αEM

QED

e eγ

q qg

QCD

5Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Solving QCD

• At large energies, where the coupling constant is small, perturbation theory is applicable has been successful in describing high energy processes

• At energies ~ 1 GeV the coupling constant is of order unity need a non-perturbative approach

Present analytical techniques inadequate

Numerical evaluation of path integrals on a space-time lattice

Lattice QCD – a well suited non-perturbative method that uses directly the QCD Langragian and therefore no new parameters enter

• At very low energies chiral perturbation theory becomes applicable

pQCDChPT

L a t t i c e Q C D

6Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Lattice QCD

Lattice QCD is a discretised version of the QCD Lagrangian finite lattice spacing a

many ways to discretise the theory all of which give the same classical continuum Lagrangian.

Physics results independent of discretization

Finite lattice spacing

must take a0 to recover continuum physics

All lattice quantities are measured in units of a dimensionless and therefore the scale is set by using one known physical observable

7Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Finite lattice spacing a

• Discrete space-time defines a non-perturbative regularization scheme: Finite a provides an utraviolet cut off π/a

In principle we can apply all the methods that we know in perturbation theory using lattice regularization.

In practice these calculations are more difficult than perturbative calculations in the continuum and therefore are not practical

What do we accomplish by discretising space-time?

• It can be solved numerically on the computer using similar methods to those used in Statistical Mechanics

Finite volume and Wick rotation into Euclidean time

must take the spatial volume to infinity

e e3 3i dtd - d d xL Hx

8Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Lattice QCD in short

ˆ ˆ

μ μ μ -μwμn

S = ψ(n)(γ -1)U (n)ψ(n+μ) - ψ(n)(γ +1)U (n)ψ(n-μ) + ψ(n)ψ(n)

= ψ(n ψ(k)D)

ˆ ˆ

g F

-βH

-S [U]-S [U,ψ,ψ]

O O

O[U,

1< > = Lim Tr[e (U, ψ, ψ)]Z

DU Dψ Dψ e= Lim

DU Dψ Dψ e

ψ, ψ]

Uμ(n)=e igaAμ(n)

ψ(n)

U3+

U4+

UP=U1 U2 U3+ U4

2 PgP

aμνaμνF

1S [U]= 1- ReTrU

6g 3

gauge invariant quantities

smallest possible

S=Sg+Sw c

1 1 1am = -2 κ κ

9Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Observables in LQCDIntegrating out the Grassmann variables is possible since FS =ψDψ

f g

n -S [U]n -S [U]

nn -S [U]n

-1-1

ODU detD e 1< >= = dU det[U,

D(U) eZDU detD

D ]O O[U,D

Sample with M.C.1. Generate a sample of N independent gauge fields U

2. Calculate propagator D-1 for each U

- 1

- S [U]+ln detDg U e

O[U1<O> , ]N D= - 1 - 1 - 1D (U)D (U)...D (U)

Easy to simulate since Sg[U] is local: use M. C. methods from Statistical Mechanics like the Metropolis algorithm

- 1

- S [U]g U eO[U,D ]1<O>=N

Quenched approximation: set det D=1 omit pair creation

For a lattice of size 323 x 64 we have 108 gluonic variables

10Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

The quenched light quark spectrum from CP-PACS, Aoki et al., PRD 67 (2003)

• Lattice spacing a 0

• Chiral extrapolation

• Infinite volume limit

Precision results in the quenched approximation

Included in the quenchedapproximation

Not included in thequenched approximation

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Computational aspects

•Lattice spacing a small enough to have continuum physics

•Quark mass small to reproduce the physical pion mass

• Lattice size large:

• Fermion determinant – Full QCD

4 πLm 5

L (fm) mπ (MeV)

1.6 560

2.5 360

4.5 200

6.4 140

most quenched calculations reaching ~300 MeV pions

Computational cost ~ (mq)-4.5 ~ (mπ)-9

Currently we are starting to do full QCD by including the fermionic determinant but we are still limited to rather heavy pions understand dependence on quark mass

Physical results require terascale machines

Lmπ-1

12Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

M. Lüscher Lattice 2005: Schwartz-preconditioned HMC

Within errors and with mπ=500 ΜeV the data are compatible with 1-loop ChPT

Fπ =80(7) MeV

New algorithm: Numerical simulations with Wilson fermions much less expensive than previously thought e.g. a lattice of size 483x96 with a=0.06fm and mπ=270 ΜeV can be simulated on a PC cluster with 288 nodes

Fermion Discretizations•Dynamical Wilson:

1. conceptually well-founded 2. many years of experience

but expensiveat least so we thought until Lattice 2005

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• Chiral fermions

New fermion formulation:

Ginsparg-Wilson relation: D-1γ5 + γ5D-1=aγ5 where D is the fermion matrix

There are two equivalent approaches in constructing a D that obeys the Ginsparg-Wilson relation

1. Domain wall fermions:

Define the Wilson fermion matrix in five dimensions

2. Overlap fermions

Infinite number of flavors in four dimensions

Left-handed fermionright-handed fermion

Kaplan, PLB 288 (1992)L5

Narayanan and Neubeger PLB 302 (1993)

In the limit the domain wall and overlap Dirac operators are the same5L → ∞

Review: Chandrasekharan &Wiese, hep-lat/0405024

PRD 25 (1982)

Expensive to simulate next stage full QCD calculations

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• Dynamical Kogut-Susskind (staggered) fermions

• More fermions than in the real theory – extra flavors complicate hadron matrix elements• Approximation – fourth root of the fermion determinant

Advantages

Disadvantages

• fastest to simulate • Have a residual chiral symmetry giving zero pions at zero quark mass• MILC collaboration simulates full QCD using improved staggered fermions. The

dynamical 2+1 flavor MILC configurations are the most realistic simulations of the physical vacuum to date.

Just download them!

Hybrid calculationA practical option: use different sea and valence quarks

A number of physical observables are calculated using MILC configurations and different valence quarks e.g. LHP and HPQCD collaborations to check consistency

staggered DWF

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Precision results in Full QCD

“Gold plated” observables (hadron with negligible widths or at least below 100 MeV decay threshold and matrix elements that involve only one hadron in the initial and final state

C. Davies et al. PRL 92 (2004) Simulations using staggered fermions, a~0.13 and 0.09 fm

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Recent progress• Hadronic decays: In a box of finite spatial size L the two-body energy spectrum is discrete : For two

non-interacting hadrons h1 and h2 the energy is given by

1 2 1 2

2 22 2

h h h hn 2π 2πE = m + + m +

L Ln n

In the interacting case there is a a small L-dependent energy shift due to the interaction. From these energy shifts close to the resonance energy one can determine the phase shift and from those the resonance mass and width (Lüscher 1991). However one needs very accurate data.

Recently the π+π+ scattering length was calculated by measuring the energy shifts. Domain wall fermions with smallest pion mass ~300 MeV were used and the result obtained in the chiral limit is in agreement with experiment (R.S. Beane et al. hep-lat/0506013)

C. Michael, Lattice 2005: ρ 2π using dynamical Wilson fermionsEvaluate the correlation matrix of a rho meson of energy m-Δ/2 and a ππ state with energy m+ Δ/2

1/2 1/2

< ρ(0) | π(t)π(t) > = xt + const< ρ(0) | ρ(t) > < π(0)π(0) | π(t)π(t) >

Use linear t-dependence to extract transition amplitude x=<ρ|ππ>

• Pion form factor at large Q2 within the hybrid approach G. Flemming, Lattice 2005

17Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

q 2 2 n-1

n0ib .ΔA (-Δ ) = d b dx x q(x,b )e

q20 q

A (0) =< x > = dx x q (x) + q (x)

From W. Schroers

Generalized form factors (LHPC)

Quenched DWF

Nf=2 unquenched DWF

Ohta and Orginos hep/lat0411008

Still too high <x>exp~0.15

In the forward limit:Ph. Hägler et al. PRL 2004

1q20 q

A (0) =< x > = dx x q (x) + q (x)

q 2 2 n-1

n0ib .ΔA (-Δ ) = d b dx x q(x,b )e

•Nucleon structure

M. Burkardt, PRD62 (2000)

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Hadron Deformation

19Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Hadron Deformation

Spin defines an orientation hadron can be deformed with respect to its spin axis

• Deformation of the Δ by evaluating 2 2z z

3 3Δ,J = J = J = (3z - r ) Δ, J = J = J =2 2

Q20 quadrupole operatorExtract the wave function squared from current-current correlators

• For spin 1/2 : Quadrupole moment not observable

20 0(2J -1)JQ = Q

(J + 1)(2J + 3)

observable

intrinsic (with respect to the body fixed frame)

Oblate: Q20 < 0 Prolate: Q20 > 0Sphere: Q20=0

Make direct connection to experiment: γNΔ(1232)

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γN Δ

•Spin-parity selection rules allow a magnetic dipole, M1, an electric quadrupole, E2, and a Coulomb quadrupole, C2, amplitude.

Three transition form factors, M1, E2 and C2

• Experimentally the measured quantities are given usually in terms of the ratios E2/M1 or C2/M1 known as EMR (or REM) and CMR (or RSM):

Δ M1

|q|CMR=-2m

M1EMR=- G

Gand

non-zero E2 and 2C multipoles signal deformation

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Deformed nucleon?

C.N. Papanicolas, “Nucleon Deformation” Eur. Phys. J. Α18, 141 (2003)

OOPS collaboration

Deformed

Spherical

MAID

DMT

0.0 0.2 0.4 0.6 0.8 1.0

-1

-2

-3

-4

-5

R %

Q2 (GeV/c)2

Bates

0.0 0.2 0.4 0.6 0.8 1.0

-5

-10

-15

-20

R %

Q2 (GeV/c)2

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γN Δ matrix element

1/2μΔ N

μΔ N

μ μ μ μM

2 2 2M1 1 E2 C2 22 CE

m m2Δ(p') | j | N(p) = i u (p')O u(p)

3 E E

O = K + K +) (q ) (q )K

G G G

M1 E2 C2

• At the lowest value of q2 the form factors were evaluated in both the quenched and the unquenched theory but with rather heavy sea quark masses. The main conclusion of this comparison is that quenched and unquenched results are within statistical errors.

C.A., Ph. de Forcrand, Th. Lippert, H. Neff, J. W. Negele, K. Schilling. W. Schroers and A. Tsapalis, PRD 69 (2004)

• Quenched results are done on a lattice of spatial volume 323 at β=6.0 (~3.2 fm)3 using smaller quark masses (smallest gives mπ/mρ = 1/2) up to q2 of 1.5 GeV2

C. A., Ph. de Forcrand, H. Neff, J. W. Negele, W. Schroers and A. Tsapalis, PRL 94 (2005)

Δ(p’) N(p)

γμ

Sach´s form factors

• MILC configurations and DWF

C. A., R. Edwards, G. Koutsou, Th. Leontiou, H. Neff, J.W. Negele, W. Schroers and A. Tsapalis, Lattice 2005

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Three-point functions

( )

2 1

1 2

iq.x- ip.x2 2 1 1

βα12

βμασx x

Δσ

j N

<G (t ,t ;p,p; ) =

e e <0| T J (x ,t ) j x ,t J (0

0)Γ , |0>

Matrix elements like <Δ| jμ |N>:

q =p’-p

D-1(x2 ;0)

D-1(x1;0)D-1(x2;x1)

quark line with photon needs to be evaluated sequential propagator

all to all propagator

Projects momentum p’ to Δ sum first: fixed sink technique

Projects to momentum transfer q for the photon sum first: fixed current technique

2 1

1 2

-ip .x iq.x

x ,x

e e

Since we worked on a finite lattice the momentum takes discrete values 2πk/L, k=1,…,N

Use fixed sink technique obtained all q2 values at once

24Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

ˆ ˆ

2 1 1

2 1 N 1

2 11 1

-E (t -t ) -E t+σ 2 1 J J

m,n

-E (t -t ) -E tJ Jt -t 1

n m

< TJ (t )O(t )J (0) >= <

< Δ | O | N

ψ |n >< n | O | m >< m | >e e

< ψ | Δ >< N | e e>>≫

≫

γN Δ matrix element on the Lattice

N 1n 1

-2E t-2E t+ 2 21 J Jt 1

n< TJ(2t )J (0) >= |< | n >| e |< | N >| e

μΔ(p') | j | N(p)

M1, E2, C2

Fit in the plateau region to extract

Trial initial state with quantum numbers of the nucleon

Trial final state with quantum numbers of the Delta

J| >= J | 0 >

J |=< 0 | J

Normalize with

-2E t+ -2E t2 21 J Jt 1

n< TJ (2t )J (0) >= |< | n >| e |< | >| e

time dependence cancels in the ratio

1/2

Rμσ ~

Δ Δ0t2

0Δ

2t1

0N N

2t1

0t2

25Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Technicalities

The form factors are extracted from the ratio

2 1

μΔj N NN2 1 4 4 2 42 1

ΔΔ NN NN2 1 4 1 4 2 4ii 2 4

t - t ≫1, t ≫12

1/2ΔΔ ΔΔG G Gii iiΔΔG G Gii

t - t ;p;Γ t ;p;Γ t ;p;ΓG t ,t ;p,p;Γt - t ;p;Γ t ;p;Γ t ;p;ΓG t ;p;Γ

p,p;Γ;μ

where σ is the spin index of the Δ field and the projection matrices Γ are given by

ii 4Γ Γ

0 I 01 1= = 0 00 02 2

What combination of Δ index σ , current direction μ and projection matrices Γ is optimal?

Using the fixed sink technique we require an inversion for each combination of σ index and projection matrix Γi whereas we obtain the ratio Rσ for all current directions μ and momentum transfers q=p’-p without extra inversions.

R t ,t ; , ;Γ;μ2 1p p

We consider kinematics where the Δ is produced at rest i.e. p=0 q=-p

26Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Optimal choice of matrix elements Πσ

The form factors, M1 , E2 and C2, can be extracted by appropriate choices of the current direction, Δ spin index and projection matrices Γj

For example the magnetic dipole can be extracted from 2σ4μj j

M1σ 4 QΠ (q;Γ ;μ)=iBε ( )p GThis means there are six statistically different matrix elements each requiring an inversion.

e.g. If σ=3 and μ=1 then only momentum transfers in the 2-direction contribute.

kinematical factor

1 1 4 2 4 3 4

μ1 μ2 μ33 2 3 1 2

2M11

S =Π Π ΠQ

(q; ) (q;Γ ; )+ (q;Γ ; )+ (q;Γ ; )iB (- p +p )δ +(p - p )δ +(- p )+ ) (p δ G

all lattice momentum vectors in all three directions, resulting in a given Q2, contribute

This combination requires only one inversion if the appropriate sum of Δ interpolating fields is used as a sink.

However if we take the more symmetric combination:

Smallest lattice momentum vectors: 2π/L {(1,0,0) (0,1,0) (0,0,1)} contributing to the same Q2

27Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Lowest value of Q2

Linear extrapolation

We perform a linear extrapolation in mπ2.

Chiral logs are expected to be suppressed due to the finite momentum carried by the nucleon .

Systematic error due to the chiral extrapolation?

28Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Overconstrained analysis

G G*m M12

2N Δ

1 1=3 Q1+

m +m

Results are obtained in the quenched theory for β=6.0 (lattice spacing a~0.1 fm) on a lattice of 323x64 using 200 configurations at κ=0.1554, 0.1558 and 0.1562 (mπ/mρ=0.64, 0.58, 0.50)

Linear extrapolation in mπ2 to

obtain results at the chiral limit

(0)

G G 2* 2 * 2m m 2 2

1(Q )= 1+ Q exp(- γQ )Q1+0.71

Fits to

Proton electric form factor

29Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Results: EMR

Linear extrapolation in mπ2 to

obtain results at the chiral limit

Sato and Lee model with dressed vertices

Sato and Lee model with bare vertices

2QEMR 1

?pQCD:

30Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Results: CMR

Linear extrapolation in mπ2 to

obtain results at the chiral limit

Discrepancy at low Q2 where pion cloud contributions are expected to be important

2QCMR constant pQCD:

31Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Chiral extrapolation

NLO chiral extrapolation on the ratios using mπ/Μ~δ2 , Δ/Μ~δ. GM1 itself not given.

V. Pascalutsa and M. Vanderhaeghen, hep-ph/0508060

32Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Domain wall valence quarks using MILC configurations

• Fix the size of the 5th dimension: we take the same as determined by the LHP collaboration

• Tune the bare valence quark mass so that the resulting pion mass is equal to that obtained with valence and sea quarks the same.

• The spatial lattice size is 203 (2.5 fm)3 as compared to 323 (3.2 fm)3 used for our quenched calculation.

Larger fluctuations

• We use ψγμ ψ for the current which is not conserved for finite lattice spacing renormalization constant ZV which can be calculated

Left-handed fermionright-handed fermion

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125 confs75 confs38 confs (283 lattice)

First results in full QCD

Use MILC dynamical configurations and domain wall fermions

EMR and CMR too noisy so far to draw any conclusions

The lightest pion mass that we would like to consider is ~250 MeV. This would require large statistics but it will be close enough to the chiral limit to begin to probe effects due to the pion cloud

34Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

They provide information on the spatial distribution of quarks in a hadron

Deformation can be studied

non-relativistic limit |wave function|2

charge 0 0C (y)= d x<Δ|j (x+y) j (x)|Δ>

Two- and three- density correlators

chargeC (y,z)

Δ Δx

γ0

Δ Δx

γ0

j0 (x) = : u(x) γ0 u(x) :

Can we improve this calculation?

smaller quark mass, larger lattice, chiral fermions, momentum projection

rho is elongated along its spin axis prolate

Δ+ is flatten along its spin axis oblate

sphere

Deformation in full QCD: a feasibility study mπ=760 MeV

C.A., Ph. de Forcrand and A. Tsapalis

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y,z

Momentum projectionRequires the all to all propagator

We use stochastic techniques to calculate it. We found that the best method is to use the stochastic propagator for one segment and sum over the spatial coordinates of the sink by computing the sequential propagator.

3d y

charge 0 0C (x)= <M|j (x)j (x+y)|M>

j0 (x) = : u(x) γ0 u(x) :

quark propagator G(x+y;0)

t=Τ/2 t=0

j0(x)

j0(x+y)

G(z;x+y)

ρ meson distribution is broader

C. A., P. Dimopoulos, G. Koutsou and H. Neff, Lattice 2005

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Rho meson asymmetry

quenched

unquenched

Opens up the way of evaluating other physically interesting quantities like polarizabilites.

37Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Diquark Dynamics

38Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Diquark dynamics

Originally proposed by Jaffe in 1977: Attraction between two quarks can produce diquarks:

qq in 3 flavor, 3 color and spin singlet behave like a bosonic antiquark in color and flavor :scalar diquark

andq q

Soliton model Diakonov, Petrov and Polyakov in 1997 predicted narrow Θ+(1530) in antidecuplet

A diquark and an anti-diquark mutually attract making a meson of diquarks

tetraquarks

A nonet with JPC=0++ if diquarks dominate no exotics in q2q2

ff3 3ff⇒ 8 1D D

pentaquarks

Exotic baryons?

8 D Dff f3 3 3f ff fq ⇒ 10 8 1

39Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Can we study diquarks on the Lattice?

Define color antitriplet diquarks in the presence of an infinitely heavy spectator:

f f f f3 3 = 63

Flavor antisymmetric spin zero

Flavor symmetric spin one

R. Jaffe hep-ph/0409065

JP color flavor diquark structure

1+ 6

qTCγ5q, qTCγ5γ0q

qTCγiq, qTCσ0i q

3 attraction: MA

MS>MA

1ΔMm

t t=0

light quark propagator G(x;0)

Static quark propagator

Baryon with an infinitely heavy quark

Models suggest that scalar diquark is lighter than the vector

In the quark model, one gluon exchange gives rise to color spin interacion:

cs s ij i j i ji,j

H = -α M σ .σ λ .λ

MS -MA ~ 2/3 (MΔ-MN)= 200 MeV and

40Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Mass difference between ``bad`` and ``good`` diquarksΔM

(GeV

) β=6.0 κ=0.153

K. Orginos Lattice 2005: unquenched results with lighter light quarks

• First results using 200 quenched configurations at β=5.8 (a~0.15 fm) β=6.0 (a~0.10 fm)

• fix mπ~800 MeV (κ=0.1575 at β=5.8 and κ=0.153 at β=6.0)

• heavier mass mπ ~1 GeV to see decrease in mass (κ=0.153 at β=5.8)

C.A., Ph. De Forcrand and B. Lucini Lattice 2005

β κ ΔΜ (MeV)

5.85.86.0

0.1530.15750.153

70 (12)109 (13)143 (10)

41Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Diquark distribution

Two correlators : provide information on the spatial distribution of quarks inside the heavy-light baryon

charge 0 0C (x,y)=<B| j (x)j (y)|B>

j0(x)

j0(y)j0 (x) = : u(x) γ0 u(x) :

Study the distribution of d-quark around u-quark. If there is attraction the distribution will peak at θ=0

quark propagator G(x;0)

42Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Diquark distribution

``Good´´ diquark peaks at θ=0

43Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Linear confining potential

A tube of chromoelectric flux forms between a quark and an antiquark. The potential between the quarks is linear and therefore the force between them constant.

Flux

tube

forms

between

linear potential

G. Bali, K. Schilling, C. Schlichter, 1995

44Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Static potential for tetraquarks and pentaquarks

C. Α. and G. Koutsou, PRD 71 (2005)

Main conclusion: When the distances are such that diquark formation is favored the static potentials become proportional to the minimal length flux tube joining the quarks signaling formation of a genuine multiquark state

45Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Pentaquarks

46Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

The Storyteller, like a cat slipping in and out of the shadows. Slipping in and out of reality?

Θ+

47Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

First evidence for a pentaquark

SPring-8 : γ 12C Κ+ Κ- n

High statistics confirmed the peak

CLAS at Jlab: γD K+ K- pn

48Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Experiment Reaction Mass (MeV)

Width (MeV)

LEPS γ C12K- K+ n 1540(10) <25

DIANA K+ Xe KS0 pXe’ 1539(2) <9

CLAS γ d K- K+ npγ p K- K+ nπ+

1542(5) <21

SAPHIR γ pKS0 K+ n 1540(6) <25

COSY ppΣ+ KS0 p 1530(5) <18

SVD pA KS0 pX 1526(3) <24

ITEP νAKS0pX 1533(5) <20

HERMES e+ d KS0pX 1528(3) 13(9)

ZEUS e pKs0 p X 1522(3) 8(4)

Experiment Reaction

CDF p pPX

ALEPH Hadronic Z decays

L3 γγΘΘ

HERA-Β pA PX

Belle KN PX

BaBar e+ e- Y

Bes e+ e- J/ψ

HyperCP (K+,π+,p)CuPX

SELEX (p,Σ,π)p PX

FOCUS γp PX

E690 pp PX

DELPHI Hadronic Z decays

COMPASS μ+(6Li D) PX

ZEUS ep PX

SPHINX pC ΘK0C

PHENIX AuAuPX

Summary of experimental results

Positive results Negative results

P=pentaquark state (Θs,Ξ,Θc)A. Dzierba et al., hep-ex/0412077

49Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Properties

• Parity: Chiral soliton model predicts positive

Naïve quark model negative

• Isospin: probably I=0 as predicted by soliton model

• Width: Why so narrow?

• Other multiplets (Ξ ?)

World average

M. Karliner, Durham, 2004

50Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Does lattice QCD support a Θ+?

Objective for lattice calculations: to determine whether quenched QCD supports a five quark resonance state and if it does to predict its parity.

51Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Pentaquark mass

Time evolution

Correlator: C(t) ~ w exp(-m t)

Initial state |φtrial> with the quantum numbers of Θ+ at time t=0

u d u d

Θ at a later time t>0

u d u d

C(t) ~ w0exp(-mKN t)+w1 exp(-mΘ t) +…

mΘ-mKN~100 MeV

If one state dominates then

efft>>1C(t)m (t)=- log mC(t- 1)Effective mass:

Fit in plateau regiont

meff

52Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Interpolating fields for pentaquarks

What is a good |φtrial> for Θ+? All lattice groups have used one or some combinations of the following isoscalar interpolating fields:

Results should be independent of the interpolating field if it has reasonable overlap with our state

Both local and smeared quark fields were considered :

q(x,t)= f(x,y,U(t))q(y,t)

• Motivated by KN strucutre:

N K

a T

c 5 c 5b

5K ba u sγ d - d sγ uJ u Cγ d=ε

• Motivated by the diquark structure:

a aeb bgh Tg 5diquar

Tec ε uC ε u CJ s Cd γε d=

L=0

L=1

u d

a Te e e e

c5 c 5 caNK 5 u su Cγ γ d - d s γ udJ =ε

• Modified NK

Jaffe and Wilczek PRL 91 (2003)

53Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Scattering states

Correlator: ...ΘKN - m t- m t0 1C(t)=w e +w e +

Dominates if w1>>w0 and (mΘ-mKN) t <1 t<10 GeV-1 assuming energy gap~100MeV or t/a<20

If mixing is small w0~L-3 suppressed for large L

S-wave KN

Θ+

Contributes only in negative parity channel

The two lowest KN scattering states with non-zero momentum

2 22 2N s K sE = m +n 2π/L + m +n 2π/L

E (G

eV) n=1 n=2

Spacing between scattering states~1/ Ls2

54Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Study the two-pion system in I=2

π π π1 1 1 1 5

π π π2 2 2 2 5 0

π π π3 3 3 3 5 μ μ

3ρ ρ ρ

4 0 0 0 ii=1

J (x) = J (x) J (x), J (x) = d(x)γ u(x)

J (x) = J (x) J (x), J (x) = d(x)γ γ u(x)

J (x) = J (x) J (x), J (x) = d(x)γ e γ u(x)

J (x) = J (x) J (x), J (x) = d(x) γ

ρ ρ ρ5 i i i i

i=1

u(x)

J (x) = J (x) J (x), J (x) = d(x)γ u(x)

C.A. and A. Tsapalis, Lattice 2005

jk j kx

C (t) = < 0 | J (x)J (0) | 0 >

Correlation matrix

total momentum=0

2mπ

2mρ

Ε12π

55Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Project on good relative momentum p: π(p)π(-p)

ip.(x-y) s s s+ s+jk j j k k

x,yC (t,p) = e < 0 | J (x)J (y)J (0)J (0) | 0 > s = π,ρ

Check taking p=0 on small lattice (163x32)

56Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Scaling of spectral weights

Lowest eigenstate

1. Consider the ratio

2. Fit the correlators on different volumes to extract the spectral weights and check or scaling. Fix upper fit range and look at the ratio of spectral weights WL1/WL2 and vary the lower fit range ti

Upper fit range 26

Upper fit range 56 • Use largest possible range

• Need accurate results

• same results even if 2x2 matrix with Jπ and Jρ

m teff,L1L1

L :L m t1 2 eff,L2L2

C (t)eR =

C (t)e

Mathur et al. PRD 70 (2004)

57Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

PentaquarkPerform a similar analysis as in the two-pion system using Jdiqaurk and JKN

Negative parity

58Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Spectral weights for pentaquark

Negative parity

Different from two pion system can not exclude a resonance

Ratio WL1/WL2 ~1 for ti/a up to 26 which is the range available on the small lattices

59Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Identifying the Θ+ on the Lattice

Alexandrou & Tsapalis (2.9 fm)Lasscock et al. (2.6 fm)

Mathur et al. (2.4 fm)

Ishii et al. (2.15 fm)

Mathur et al. (3.2 fm)

Csikor et al. (1.9 fm)Sasaki (2.2 fm)

Negative parity

There is agreement among lattice groups on the raw data but the interpretation differs depending on the criterion used

From Lassock et al. hep-lat/0503008

60Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

How do we identify a resonance?

Method used:

• Identify the two lowest states and check for volume dependence of their mass

• Extract the spectral weights and check their scaling with the spatial volume

• Change from periodic to antiperiodic boundary condition in the spatial directions and check if the mass in the negative parity channel changes

• Check whether the binding increases with the quark mass

61Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Summary of lattice computationsGroup Method of analysis/criterion Conclusion

Alexandrou and Tsapalis Correlation matrix, Scaling of weights

Can not exclude a resonance state. Mass difference seen in positive channel of right order but mass too large

Chiu et al. Correlation matrix Evidence for resonance in the positive parity channel

Csikor et al. Correlation matrix, scaling of energies

First paper supported a pentaquark , second paper with different interpolating fields produces a negative result

Holland and Juge Correlation matrix Negative result

Ishii et al. Hybrid boundary conditions Negative result in the negative parity channel

Lasscosk et al. Binding energy Negative result

Mathur et al. Scaling of weights Negative result

Sasaki Double plateau Evidence for a resonance state in the negative parity channel.

Takahashi et al. Correlation matrix, scaling of weights

Evidence for a resonance state in the negative parity channel.

J. Negele, Lattice 2005 Correlation matrix, scaling of weights

Evidence for a resonance state?

62Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Dynamical calculations-status and perspectives

63Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Dynamical calculations-status and perspectives

• No continuum limit– Two lattice spacings

or fewer• Autocorrelations ignored

– Depend on the physics anyhow

• Naïve volume scaling– V for R algorithm– V5/4 for HMC

• Dynamical quarks– Two or three flavours

A. Kennedy, Lattice 2004

64Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Three choices:

• Commercial supercomputers : general purpose, rather expensive

• PC Clusters: general purpose, cheaper than commercial but scalability a problem

• custom-design machines (APEnext, QCDOC): optimized for QCD

•Commercial supercomputers

-SGI Altrix (LRZ Munich)

16.4 TFlop/s peak 2006-07

34.6 TFlop/s peak 2006-07

-BlueGene/L very good scalability

11.2 TFlop/s peak at Jülich

5.6 TFlop/s peak at Edinburgh, BU, MIT

T. Wettig, Lattice 2005

• PC clusters

65Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

• Custom-design machinesAPEnext QCDOC

Planned Installations ( 1 rack=512 nodes =0.66 TFlop/s peak)

• INFN 6 144 nodes (12 racks)

• Bielefeld 3072 nodes (6 racks)

• DESY 1576 nodes (3 racks)

• Orsay 512 nodes (1 rack)Cost: $0.6 per peak MFlop/s

Planned Installations ( 1 rack=1024 nodes =0.83 TFlop/s peak)

• UKQCD 14 720 nodes

• DOE 14 140 nodes

• RIKEN-BNL 13 308 nodes

• Columbia 2 432 nodes

• Regensburg 448 nodes

Cost: $0.45 per peak MFlop/s

Tens of Teraflop/s will be available for lattice QCD

66Summer Workshop Hall C Jefferson Lab Aug. 18, 2005

Conclusions

γN Δ : M1, E2 and C2 calculated in quenched approximation in the range of Q2 where recent measurements are taken. Results in full QCD are within reach. Non-zero values deformation of the nucleon/Δ

Diquark dynamics

Studies of exotics and two-body decays

• Lattice QCD is entering an era where it can make significant contributions in the interpretation of current experimental results.

• A valuable method for understanding hadronic phenomena

• Computer technology and new algorithms will deliver 10´s of Teraflop/s in the next five years

Provide dynamical gauge configurations in the chiral regime

Enable the accurate evaluation of more involved matrix elements