Hadron05, Cyprus, page 1 Excited and Exotic States on the Lattice Introduction –How to extract excited states ? –The ′ ghost in quenched QCD Light baryons

Hadron05, Cyprus, page 1

Excited and Exotic States on the Lattice

• Introduction– How to extract excited states ?

– The ′ ghost in quenched QCD

• Light baryons– Nucleon, Roper, and S11(1535), (1405)

• Pentaquarks

• Magnetic moments and polarizabilities

Frank Lee, The George Washington University

Collaborators: K.F. Liu, N. Mathur, W. Wilcox, L. Zhou, R. KellyThanks also to: U.S. Department of Energy and computing resources from NERSC, PSC, and JLab


The Particle Zoo MesonsMesons

BaryonsBaryons

Excitation spectrum of QCD


The proton in QCD:

t

yz

udu

ud

u

qmDqFFL qQCD )(Tr 21

• confinement• chiral symmetry breaking • asymptotic freedom• mu, md ~ 5 MeV, ms ~ 100 MeV


Baryons on the lattice

• Parity separation is exact.

• For a certain spin-parity channel,

the entire mass spectrum is

contained in the correlation

function.

M1

M2

M3

…

0x

M1

M2

M3

…

½+ ½-


Curve-Fitting in Lattice QCD

• Variational Method

• Bayesian Priors

• Maximum Entropy Method

1

)(n

tMn

neAtGBasic Problem:

Given a finite set of measurements {G(t)}, how to extract {An, Mn} for n=1,2,3,… ?

Ground state is easy: look at large time (or the ‘plateau’ method)

What about excited states?


Bayesian Statistical Inference

)(

)()|()|(

YP

XPXYPYXP

Bayes’ Theorem: The conditional probability of X given Y is equal to the conditional probability of Y given X, multiplied by the probability distribution of X, divided by the probability distribution of Y. Or

Translation into our problem )|(

)|()|()|(

HGP

HAPAHGPGHAP

H represents all prior knowledge about our model A.P(G|AH) is likelihood probability of the dataP(A|H) is prior probabilityP(G|H) is a normalization factor independent of A.P(A|GH) is posterior probability

Rev. T. Bayes (1702-1761)

priorlikelihoodposterior

Basic idea: Find A by maximizing the posterior probability: 0)|(

A

GHAP


Likelihood

2/2

)|( eAHGP

For a large number of measurements, the data is expected to obey the Gaussian distribution according to the central limit theorem:

Average data

)()()()(1,

12jthj

N

jiijithi tGtGCtGtG

t

)()( )()( 1

)1(1

jjk

N

kiikNNij tGtGtGtGC

N

kiki tG

NtG

1

)(1

)(

Covariance matrix


Two types of priors

2/2

)|( eAHGPLikelihood (2)

For 2), maximize SQ

2

2

222prioraug For 1), minimize

2) Entropy prior:

1) Bayesian 2 prior:

SeHAP )|(

2/2

)|( prioreHAP

)|()|()|( HAPAHGPGHAP Maximize:

In practice


Constrained Curve Fitting with Bayesian Priors222 minimize prioraug

priors)(Bayesian parametersinput are }~~,~~

{nn EnAn EA

n E

nn

n A

nnprior

nn

EEAA2

2

2

22

~)

~(

~)

~(

n

tEninn

ineAtGEA )(in parametersfit are },{ th

2) Fit as many terms in Gth.

3) Use prior knowledge, like

0~~

,0~

1 nnn EEA

4) Un-constrain the term of interest to have conservative error bars.

(See heplat/0208055)

1) All data points are used.


Example: fitting all time slices

Un-constrained fit Constrained

1

th )( n

tEn

neAtG

(Lepage, heplat/0110175)


Reconstructing Artificial Data

(heplat/0405011, Y. Chen et el)


A ‘double blind test’

• We could not reproduce the input initially.– Reason: error too big

• After reducing the error by a factor of 2, we could, but we found two solutions close to each other, one of them wrong.

• After reducing the error by a factor of 10, we could reproduce the input unambiguously.

• The details of this experience are in heplat/0405001

with relative error 1.1% at t=1 and 12% at t=16


Pion excited state

(Nucleon channel later)


Maximum Entropy Method (MEM)

)(

)(log)()()(

0

m

AAmAdSunbiased

entropy

m() is called the default model (real and positive) which incorporates our prior knowledge on the functional dependence of A(). For example, m()=m0 n, where n=2 for mesons, n=5 for baryons.

SQ

2

2

Maximize

Key features of MEM:• It makes no a priori assumptions on the input parameters.• For given data, an unique solution is obtained if it exists.• statistical uncertainty can be quantified.

hep-lat/0011040Y. Asakawa et al


fm085.0

36.0,)(

MeV10,GeV6

)()(

02

0

max

0

2max

mmm

deG ii

Testing MEM:Testing MEM: mock data mock data

30,001.0

)(

)(

:noisegaussian add

t

i

Nb

iGb

G

30,001.0

)(

)(

:noisegaussian add

t

i

Nb

iGb

G

fm085.0

36.0,)(

MeV10,GeV6

)()(

02

0

max

0

2max

mmm

deG ii


• Reducing the errors is more effective than having more time steps.

b=0.1

Nt=10 Nt=20 Nt=30

b=0.01

b=0.001Testing MEM: sharp peak

+broad peak


Testing MEM: pole+continuum


Testing MEM:Testing MEM: 3 Peaks 3 Peaks

width = 0.01, 0.02, 0.03 GeVspacing = 0.5 GeV

spacing = 0.1 GeVspacing = 0.2 GeVspacing = 0.3 GeV

spacing = 0.4 GeV


MEM fitting: nucleon channel, JP=1/2+-

(heplat/0504020, Sasaki et al)


We use overlap fermion action• It’s a particular way of putting D+mq on the

lattice that preserves exact chiral symmetry (H. Neuberger, PLB417 (1998) 141)– No O(a) error, O(a2) is small– Numerically checked that there is no additive quark mass

renormalization– Critical slowing down is gentle all the way to low pion mass– No exceptional configurations.

• Our results are based on– 163 X 28, a = 0.200(3) fm. L = 3.2 fm (80 configurations analyzed, 300

more being added)– 123 X 28, a = 0.200(3) fm. L = 2.4 fm (80 configurations, 200 more being

added)– 203 X 32, a ~ 0.171 fm. L ~ 3.4 fm (100 configurations) – pion mass down to about 180 MeV – quenched approximation ghosts


The ′ ghost in quenched QCDQuenched QCDFull QCD

(hairpin)

… ....

• It becomes a light degree of freedom– with a mass degenerate with the pion mass.

• It is present in all hadron correlators G(t).• It gives a negative-metric contribution to the G(t)

– It is unphysical (thus the name ghost)– A pathology of the quenched approximation

′(958)


W > 0

W<0

-- --η η

Evidence of ′ N ghost state in S11

• The effect of the ghost state decreases as pion mass increases.• It has a sensitive volume dependence.• First time seen in a baryon channel.

• Phys. Lett. B605 (2005) 137-143


Decoupling of ′ N ghost state

• The ′N ghost state decouples from the nucleon correlator around m 300 MeV.


Baryon Two-point Function

])[1( ])[1(

v|)0()(|v)(

)()(4

)()(4

11

0000 ttmttNmttNmttm

x

eAebAebAeA

acxTactG

tt

cbaTabc udCu )( 51 Nucleon interpolating field:

...)1()(

tmR

tEN

tmN

RNN ewetEwewtG

Positive-parity channel: N + ′ N (p=2/L) + Roper + …

2222 m m E ' pp NN

Negative-parity channel: ′ N (p=0) + ′ N (p=2/L) + S11 +…

...)1()1()(

2

111

11

'' tm

S

tEtmSNN ewetEwetmwtG

Hadron05, Cyprus, page 25Cross-over occurs close to chiral limit.

Nucleon, Roper and S11

0.5

1.0

1.5

2.5

2.0

Mas

s (G

eV)

N(938) 1/2+

P11(1440) 1/2+

S11(1535) 1/2-

Phys.Lett. B605 (2005) 137

QCDin 2qmm

Lowest m ~ 180 MeV

Phys.Lett. B605 (2005) 137


Hyperfine Interaction of quarks in BaryonsAt higher quark massAt higher quark mass ( (above 300 MeV pion above 300 MeV pion massmass))

2121 .. cc

Color-Spin Interaction (one-gluon-exchange) Color-Spin Interaction (one-gluon-exchange) dominates.dominates.

Isgur and Karl, PRD18, 4187 (1978)At lower quark massAt lower quark mass ( (below 300 MeV pion massbelow 300 MeV pion mass))

2121 .. FF

Flavor-Spin interaction (meson-exchange) Flavor-Spin interaction (meson-exchange) dominates.dominates.Chiral symmetry plays major roleChiral symmetry plays major roleGlozman and Riska, Phys. Rep. 268,263 (1996)

(1600)1/2+


What about Hyperons? The (1405)?

…different story!!

0.5

1.0

1.5

2.5

2.0

Mas

s (G

eV)

(1115) 1/2+

(1405) 1/2 -

(1600) 1/2+


S11(1535)1/2- and (1405)1/2-

Puzzle: the two states have the same spin-parity, but why (1405)(uds), having a

strange quark, is lighter than S11(1535)(uud)?

Answer: it’s the flavor structure. The (1405)1/2-

was constructed as a flavor-singlet state.


Conclusions on light excited baryons• The combination of overlap fermion action and Bayesian fitting

algorithm has allowed access deep into the chiral regime and excited states in quenched QCD . – The ′ ghost is clearly seen.

– As along as the ′ ghost is removed, useful physics can be explored in the chiral regime even in quenched QCD.

• Exploratory studies have shown that the basic ordering of low-lying baryons can be reproduced on the lattice with standard interpolating fields built from three QCD quarks.– cross-overs in the region around m ~ 300 MeV where chiral dynamics starts to

dominate.

– This supports the notion that there is a transition from color-spin to flavor-spin in the hyperfine interactions between quarks.

– a node is observed in Roper’s radial wavefunction.

Hadron05, Cyprus, page

30

PentaquarksPentaquarks

New Topic


31

(from J. Negele)

See talk by A. Williams and F. Csikor at this workshop


32

Five-quark mass spectrumFive-quark mass spectrum

Pentaquark correlation function contains the entire 5-quark spectrum: KN scattering states + genuine pentaquark states, of both parities.

M1

M2

M3

…

0x

On the lattice parity can be separated exactly:

M1

M2

M3

…

uu

sud

uud u

s½+ ½-


33

How to identify a pentaquark on the lattice?How to identify a pentaquark on the lattice?

1.431.54

GeV

2m

0

Negative-parity channel

• Pentaquark, if it exists, is entangled with KN scattering states.• Positive-parity is easier to identify than negative-parity.

-KN threshold raised for positive-parity (p=n*2/L)-at least two states have to be isolated for negative-parity.

• The nature of extracted states must be further tested.

1.431.54

GeV

2m

p=1

p=3

0

Positive-parity channel

pentaquark

p=2

p=4

p=1

p=3p=2

p=0


P-wave (1/2+, I= 0)

• Propagators turn negative: ground state is KNη' ghost state. In fitting function this ghost state, pentaquark and KN p-wave scattering state are the first three states. We find ghost and scattering states, but not pentaquark near 1.54 GeV.

• The volume dependence in

EK(pL) + EN(pL) due to

the P-wave nature is seen. Near chiral limit the scattering length is close to zero which is consistent with experiment.


S-wave (1/2¯, I = 0)

• No need to consider ghost state (propagators are positive).

• Near the chiral limit ground state mass is consistent with the threshold KN scattering state. Identification of this ground state with the scattering state implies vanishing scattering length, which is consistent with the experiment. •The next state is an average of p=1, 2 and perhaps 3.


Test 1: volume dependence (squeeze it)For bound state, the spectral weight, as defined in G(t)=We -m t , will

not show very weak volume dependence.

For two particle scattering state, the spectral weight will show

inverse volume dependence (1/V)

We see (12)/W(16)=163/123=2.37, so the observed states are KN scattering states in both channels.

Negative-parity channelPositive-parity channel


Test 2: hybrid boundary condition (twist it)

Plateau raised, suggesting KN scattering state.

Change b.c.: anti-periodic for u and d quarks, periodic for s quark

Consequence: If KN state, mass will rise. If bound state, mass will not change

(Ishii et al)


Test 3: KN open jaw (bite it)

tm

KN

q ecctG

tG )(10

5

)(

)(

No sign of pentaquark of positive-parity near 1.54 GeV.

yx

x

l)exponentia (falling 0m then no, Ifl)exponentia (rising 0m then yes, If

*

KNKN

KN

EEEm


Conclusions on pentaquarks on the lattice• We found no evidence for a pentaquark near 1.54 GeV for either

parity. The states we found are all consistent with KN-states.• The presence of KN scattering states is a major complication in the

isolation of a true pentaquark signal, if it exists.– Positive-parity

• Almost consensus among different groups. • The adjustable P-wave KN threshold is a great advantage.

– Negative-parity• The S-wave KN threshold is just below 1.54 GeV and is fixed. • Very hard to tell apart a pentaquark from a KN-state.• At least two lowest states in this channel need to be isolated reliably. • Variational method based on multiple operators holds promise.

• After a state is isolated, it should be tested to reveal its true character.– Test 1: Squeeze it (inverse volume dependence of spectral weights) – Test 2: Twist it (hybrid boundary condition)– Test 3: Bite it (remove the KN scattering states explicitly from the correlation

function, and examine the rest)


Pentaquarks

• Lesson 1: It’s very hard to establish its existence– pentaquark or KN scattering states?– Reliable isolation of two states in each parity channel– Must be put through tests (squeeze it, twist it, bite it)

• Lesson 2: It’s very hard to rule it out– One has to exhaust all possibilities.– operators– quenched approximation– …


Magnetic moments and Magnetic moments and polarizabilties in thepolarizabilties in the

background field methodbackground field method

New Topic


Introduction of a static magnetic field on the lattice

• Minimal coupling in the QCD covariant derivative

qAgGD

• This suggests multiplying a U(1) phase factor to the SU(3) link variables :

(Quark propagators are generated in the new background)

BUUU '

• To apply B in the z-direction, one can choose the vector potential , then the phase factor is BxAy

)exp( 2BxiqaU By

)exp( 2aigGU


Lattice details• Standard Wilson gauge action

– 244 lattice, =6.0 (or a ≈ 0.1 fm)

– 150 configurations

• Standard Wilson fermion action =0.1515, 0.1525, 0.1535, 0.1540, 0.1545, 0.1555

– Pion mass about 1015, 908, 794, 732, 667, 522 MeV

– Strange quark mass corresponds to =0.1535 (or m~794 MeV)

– Source location (x,y,z,t)=(12,1,1,2)

– Boundary conditions: periodic in y and z, fixed in x and t

• The following 5 dimensionless numbers ≡qBa2 =+0.00036, -0.00072,

+0.00144, -0.00288, +0.00576 correspond to 4 small B fields

eBa2 = -0.00108, 0.00216, -0.00432, 0.00864 for both u and d (or s) quarks.– Small in the sense that the mass shift is only a fraction of the proton mass: B/m ~ 0.6 to 4.8% at the smallest pion mass. In absolute terms, B ~ 1013 tesla.

x

z

B


Polarizabilities on the LatticePolarizabilities on the Lattice Polarizability is a measure of how tightly a hadron is bound in response to

external probes. In particular, they are related to the quadratic response in the interaction

energy

where is electric polarizability, is magnetic polarizability

On the lattice, we determine the polarizabilities directly by calculating the mass shift of hadrons in the presence of progressively small E and B fields

33

221)0()()( BcBcBcmBmBm

33

221)0()()( EbEbEbmEmEm

22b

22c

22

2

1

2

1BEH


A computational trick• We generate two sets of quark propagators, one with the

original set of fields, one with the fields reversed.• The mass shift in the presence of small fields is

• At the cost of a factor of two, – by taking the average, [m(B) + m(-B)]/2 , we get the leading

quadratic response with the odd-powered terms eliminated. (magnetic polarizability)

– by taking the difference, [m(B) - m(-B)]/2, we get the leading linear response with the even-powered terms eliminated. (magnetic moment)

• Our calculation is equivalent to 11 mass calculations.– 5 original fields, 5 reversed, plus the zero-field to set the baseline

44

33

221)0()()( BcBcBcBcmBmBm


Magnetic Moment: two methods

• Form factor method: GM(Q2=0)– Since the minimum momentum on the lattice is non-zero

(p=2/L), extrapolation to zero momentum transfer is required.

– Three-point function calculations

• Background field method– direct access

– Two-point function calculations


Magnetic moment• For a Dirac particle of spin s in small fields,

where upper sign means spin-up and lower sign spin-down, and

BmE

sm

eg

2

• g factor is extracted from

(linear) )()(2

eBs

mEmE

mm

mmg

(square1) 2

)()( 2222

eBs

mEmEg

• Look for the slope on the straight line of the form: )(eBgm

(square) )()(

eBs

mEmmEmg


Proton mass shifts

• We use the 2 smallest fields to fit the line.


Neutron mass shifts


Proton magnetic moment


Neutron magnetic moment


Octet Sigma magnetic moments


Delta magnetic moments


Proton and + magnetic moments


Baryon M

agnetic M

oments

Phys. L

ett. B, F

.X. L

ee et al


PolarizabilitiesPolarizabilities

New Topic


Experimental Information on Polarizabilities Experimental Information on Polarizabilities

(a theorist’s summary)(a theorist’s summary) Proton electric polarizability () is around 12 in units of

10-4 fm3.

Proton magnetic polarizability () is around 2 in units of 10-4 fm3.

Neutron is about the same as proton

Neutron is about the same as proton

44

33

2210)( cccccf

They are extracted from low-energy expansion of Compton scattering amplitude


Polarizabilities on the LatticePolarizabilities on the Lattice

33

221)0()()( BcBcBcmBmBm

33

221)0()()( EbEbEbmEmEm

22b

22c

For polarizability (quadratic response) , we average over mass shifts from B and –B to eliminate the odd-power terms. So we are expecting parabolas going through the origin.

For magnetic moment (linear response), we take the difference in mass shifts from B and –B to eliminate the even-power terms. So we are expecting straight lines going through the origin.

Mass shifts in response to background fields:


Effective-mass plot for neutron mass shiftsEffective-mass plot for neutron mass shifts


Neutron Mass Shifts in Magnetic FieldsNeutron Mass Shifts in Magnetic Fields 2

2)( BcBm


Proton and NeutronProton and Neutron


Octet SigmasOctet Sigmas


DeltasDeltas

-


PionsPions


KaonsKaons


RhosRhos


Baryon Octet Magnetic PolarizabilitiesBaryon Octet Magnetic Polarizabilities


Baryon Decuplet Magnetic PolarizabilitiesBaryon Decuplet Magnetic Polarizabilities


Meson Magnetic PolarizabilitiesMeson Magnetic Polarizabilities


Conclusion• Using standard technology, we obtained the magnetic moments and

polarizabilties, sweeping through 30 hadrons.– For magnetic moment, most of our results are consistent with experiments

where available. Others are predictions.

– For polarizabilities, most of our results are predictions.

• The background field method provides a good tool for to the form factor method for obtaining the magnetic moments using mass shifts. – The method is robust. No extrapolation to Q2=0 is needed.

– For 4 non-zero field values, the cost is about a factor of 11 that of a standard mass spectrum calculation.

– Polarizabilties can be obtained from the same data set with little overhead

• Further study: quantify systematic errors– Continuum extrapolation

– Push deeper into the light mass region (tmQCD, or overlap fermions)

– Chiral extrapolation (guidance from ChPT)

– Quenching effects


71

Reserve SlidesReserve Slides

New Topic


72

Electric Polarizabilities of Neutral ParticlesElectric Polarizabilities of Neutral Particles


Scattering state and its Scattering state and its volume dependencevolume dependence

),,|,,| spnVE

mspn

xxx

sq

xqi

x

xpix

xpiNN

VOVEV

m

EV

m

sqsqxee

xTeG

since ),1(1

(...)...)(

0|)0(|,,|)(|0

0|))0()((|0

,

..

.

Box normalization on the lattice:

Two point function :

VVEV

m

EV

m

EV

m

EV

mG

xx

11........

222

2

22

2

11

1

11

112

Lattice Continuum

For one particle bound state there is no explicit volume dependence in the spectral weight W.

For two particle state :

mtWetG )(Fitting fuction :

Therefore, for two particle scattering state the spectral weight has inverse volume dependence.


The issue of interpolating fields

KN type: 5 51

d)(u)dγs(u)bdCγTa(uabcεχ eec

d)(u)cdγes(eu)bdCγTa(uabcεχ

5

52

},1{A}{S,

1 5

1 5

1 5

3

5

TesC)cdCTe(u)bdCγTa(uabcε

TesC)edCTc(u)bdCγTa(uabcε

TesC)hdCTf(u)bdCγTa(uabcεgfhεgceεχ

Diquark-diquark-antiquark type:

(minus sign for I=0, plus sign for I=1)

However, the two types are related by a Fiertz re-arrangement.


Pentaquark interpolating fields

• The two types couple to the same physical spectrum, albeit with different strengths.

AχSχ χχγ

332

1215

)sother term(operator]antiquark -diquark-diquark[2

1operator] KN[5

The two types are explicitly related by a Fiertz re-arrangement:

• The complete set, including non-local operators, contains 19 operators.

• Need correlation matrix method to select the best operators.


Comparison of different operators

At large time, they all project to the same state (KN scattering state).

preliminary

Documents

Hadron05, Cyprus, page 1 Excited and Exotic States on the Lattice Introduction –How to extract excited states ? –The ′ ghost in quenched QCD Light baryons