hadron05, cyprus, page 1 excited and exotic states on the lattice introduction –how to extract...
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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
Hadron05, Cyprus, page 2
The Particle Zoo MesonsMesons
BaryonsBaryons
Excitation spectrum of QCD
Hadron05, Cyprus, page 3
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
Hadron05, Cyprus, page 4
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-
Hadron05, Cyprus, page 5
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?
Hadron05, Cyprus, page 6
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
Hadron05, Cyprus, page 7
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
Hadron05, Cyprus, page 8
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
Hadron05, Cyprus, page 9
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.
Hadron05, Cyprus, page 10
Example: fitting all time slices
Un-constrained fit Constrained
1
th )( n
tEn
neAtG
(Lepage, heplat/0110175)
Hadron05, Cyprus, page 11
Reconstructing Artificial Data
(heplat/0405011, Y. Chen et el)
Hadron05, Cyprus, page 12
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
Hadron05, Cyprus, page 13
Pion excited state
(Nucleon channel later)
Hadron05, Cyprus, page 14
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
Hadron05, Cyprus, page 15
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
Hadron05, Cyprus, page 16
• 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
Hadron05, Cyprus, page 17
Testing MEM: pole+continuum
Hadron05, Cyprus, page 18
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
Hadron05, Cyprus, page 19
MEM fitting: nucleon channel, JP=1/2+-
(heplat/0504020, Sasaki et al)
Hadron05, Cyprus, page 20
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
Hadron05, Cyprus, page 21
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)
Hadron05, Cyprus, page 22
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
Hadron05, Cyprus, page 23
Decoupling of ′ N ghost state
• The ′N ghost state decouples from the nucleon correlator around m 300 MeV.
Hadron05, Cyprus, page 24
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
Hadron05, Cyprus, page 26
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+
Hadron05, Cyprus, page 27
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+
Hadron05, Cyprus, page 28
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.
Hadron05, Cyprus, page 29
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
Hadron05, Cyprus, page
31
(from J. Negele)
See talk by A. Williams and F. Csikor at this workshop
Hadron05, Cyprus, page
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-
Hadron05, Cyprus, page
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
Hadron05, Cyprus, page 34
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.
Hadron05, Cyprus, page 35
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.
Hadron05, Cyprus, page 36
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
Hadron05, Cyprus, page 37
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)
Hadron05, Cyprus, page 38
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
Hadron05, Cyprus, page 39
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)
Hadron05, Cyprus, page 40
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– …
Hadron05, Cyprus, page 41
Magnetic moments and Magnetic moments and polarizabilties in thepolarizabilties in the
background field methodbackground field method
New Topic
Hadron05, Cyprus, page 42
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
Hadron05, Cyprus, page 43
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
Hadron05, Cyprus, page 44
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
Hadron05, Cyprus, page 45
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
Hadron05, Cyprus, page 46
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
Hadron05, Cyprus, page 47
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
Hadron05, Cyprus, page 48
Proton mass shifts
• We use the 2 smallest fields to fit the line.
Hadron05, Cyprus, page 49
Neutron mass shifts
Hadron05, Cyprus, page 50
Proton magnetic moment
Hadron05, Cyprus, page 51
Neutron magnetic moment
Hadron05, Cyprus, page 52
Octet Sigma magnetic moments
Hadron05, Cyprus, page 53
Delta magnetic moments
Hadron05, Cyprus, page 54
Proton and + magnetic moments
Hadron05, Cyprus, page 55
Baryon M
agnetic M
oments
Phys. L
ett. B, F
.X. L
ee et al
Hadron05, Cyprus, page 56
PolarizabilitiesPolarizabilities
New Topic
Hadron05, Cyprus, page 57
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
Hadron05, Cyprus, page 58
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:
Hadron05, Cyprus, page 59
Effective-mass plot for neutron mass shiftsEffective-mass plot for neutron mass shifts
Hadron05, Cyprus, page 60
Neutron Mass Shifts in Magnetic FieldsNeutron Mass Shifts in Magnetic Fields 2
2)( BcBm
Hadron05, Cyprus, page 61
Proton and NeutronProton and Neutron
Hadron05, Cyprus, page 62
Octet SigmasOctet Sigmas
Hadron05, Cyprus, page 63
DeltasDeltas
-
Hadron05, Cyprus, page 64
PionsPions
Hadron05, Cyprus, page 65
KaonsKaons
Hadron05, Cyprus, page 66
RhosRhos
Hadron05, Cyprus, page 67
Baryon Octet Magnetic PolarizabilitiesBaryon Octet Magnetic Polarizabilities
Hadron05, Cyprus, page 68
Baryon Decuplet Magnetic PolarizabilitiesBaryon Decuplet Magnetic Polarizabilities
Hadron05, Cyprus, page 69
Meson Magnetic PolarizabilitiesMeson Magnetic Polarizabilities
Hadron05, Cyprus, page 70
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
Hadron05, Cyprus, page
71
Reserve SlidesReserve Slides
New Topic
Hadron05, Cyprus, page
72
Electric Polarizabilities of Neutral ParticlesElectric Polarizabilities of Neutral Particles
Hadron05, Cyprus, page 73
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.
Hadron05, Cyprus, page 74
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.
Hadron05, Cyprus, page 75
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.
Hadron05, Cyprus, page 76
Comparison of different operators
At large time, they all project to the same state (KN scattering state).
preliminary