Lattice Hadron Physics · Lattice Hadron Physics James Zanotti The University of Adelaide CSSM...
Transcript of Lattice Hadron Physics · Lattice Hadron Physics James Zanotti The University of Adelaide CSSM...
Lattice Hadron PhysicsJames ZanottiThe University of Adelaide
CSSM Summer School, February 11  15, 2013, CSSM, Adelaide, Australia
QCD Hadron Spectrum
Excellent agreement between different collaborations/lattice formulations
Plot from A. Kronfeld [1203.1204]
ρ K K∗ η φ N Λ Σ Ξ Δ Σ∗
Ξ∗ Ωπ ηʹ′ ω0
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
(MeV)
H H* Hs Hs* Bc Bc
*
© 2012 Andreas Kronfeld/Fermi Natl Accelerator Lab.
bflavoured masses –4000 MeV
numerousquarkonium
omitted
π…Ω: BMW, MILC, PACSCS, QCDSF;ηηʹ′: RBC, UKQCD, Hadron Spectrum (ω);D, B: Fermilab, HPQCD, MohlerWoloshyn
Spectroscopy• Rich spectrum of particles observed in experiment
• Some deviate from standard quark model predictions
• Other “missing states” are yet to be observed
• Can we understand the full hadron spectrum from QCD?
Quark
AntiQuarkMesons
Baryons
Molecules/Multiquarks
HybridsGlueballs
Glue
Lattice 2012  spectroscopy overview
the light meson spectrum  experiments
3
GlueX CLAS12
p N
π,η,K...γ
ψ,χc π,η,K...
ψ,χcπ,η,K...
γ
BES III
p N
π,η,K...π
p p
π,η,K...
p p
Experiments• Many experiments dedicated to study meson and baryon spectroscopy
• GlueX and CLAS12 [Jefferson Lab]
• Compass [CERN]
• BES III [Beijing]
Lattice 2012  spectroscopy overview
L=2
L=1
L=3
L=0
the light meson spectrum
4
relatively simple models of hadrons:
bound states of constituent quarks and antiquarks “the quark model”
I=0, S=0 : η,φ,ω,fJ ...
I=1, S=0 : π,ρ,b1,aJ ...
I=½,S=±1 : K,K* ...
I1, S1
empirical meson flavour systematics
0
0.5
1.0
1.5
2.0
empirical JPC distribution
PDG summary(take with a pinch of salt)Jo Dudek [Lattice 2012]
Spectroscopy• Recall: Masses (energies) are extracted from (Euclidean) timedependence of
lattice correlators
• In general, works well for ground states (for large enough t, fit a single exponential)
• Extracting excited states from multiexponential fits difficult
• Try to optimise to isolate state of interest. i.e. make
• large for state, n, of interest as small for other states
G(~p, t) =NX
n=1
eEnt
2En
Of (0)n, pihn, pO†
i (0)↵
Zn(~p) hOni22En(~p)
O
Operators• Hadrons are extended objects ~1fm
• Using propagators computed from a point source perhaps only have small overlap with states of interest
• Gaugeinvariant Gaussian smearing starts with a point source
• and proceeds by the iterative scheme
• Repeating N times gives the resulting fermion source
~x
~x+ µ
a
0↵
= ac↵
~x,~x0t,t0
i
(~x, t) =X
~x
0
F (~x, ~x 0) i1(~x
0, t)
F (~x, ~x 0) =1
(1 + ↵)
~x,~x
0 +↵
6
3X
µ=1
U
µ
(~x, t)~x
0,~x+µ
+ U
†µ
(~x µ, t)~x
0,~xµ
!
N
(~x, t) =X
~x
0
F
N (~x, ~x 0) 0(~x0, t)
Example Effective Masses
Improved overlap with ground states
Extracting Excited States• Different operators/smearing have differing overlap strengths with a variety of
states of interest
• Set up a Generalised Eigenvalue Problem to cleanly isolate individual states
• E.g., express (p=0) baryon 2pt function as a sum over states
• Look for a linear combination that cleanly isolates state with overlap at time t0
• which leads to
Parity projectory
↵
↵ z↵
G
±ij
(t) =X
~x
Trn
±hOi
(x)Oj
(0)io
=X
↵
↵
i
↵
j
e
m↵t
Gij(t0)u↵j = ↵
i z↵em↵t0
Gij(t0 +t)u↵j = em↵tGij(t0)u
↵j
Oju↵j
Extracting Excited States• Multiplying on the left by [Gij(t0)]1 gives the generalised eigenvalue equation
• with eigenvalues
• and similarly
• We can diagonalise the correlation matrix
• allowing fits at earlier times
0.28
0.3
0.32
0.34
0.36
0.38
0.4
0.42
5 10 15 20
aE
1st
at
t/a
GEVP (4x4)GEVP (3x3)
smeared (R=90)smeared (R=22)
Plateau (from GEVP)
Figure 1: Plot of aEstat1 (t, t0 = 4a) against t/a as obtained from the GEVP for a 3×3 and
a 4×4 system, and from the effective mass plot of two correlators with different amountsof Gaussian smearing (22 and 90 iterations, respectively). The improved convergenceof the GEVP solutions can be seen clearly.
2 The Generalized Eigenvalue Problem
2.1 The basic idea
We start from a matrix of Euclidean space correlation functions
Cij(t) = 〈Oi(t)O∗j (0)〉 =
∞∑
n=1
e−Entψniψ∗nj , i, j = 1, . . . , N (2.1)
ψni ≡ (ψn)i = 〈0Oin〉 En < En+1 .
Note that we have assumed nondegenerate energy levels. Spacetime could be continuous, but in practice applications will be for discretized field theories. In that case,we assume that either the theory has a hermitian, positive transfer matrix as for thestandard Wilson gauge theory [21,22], or we consider t, t0 large enough and the latticespacing, a, small enough such that the correlation functions are well represented by aspectral representation and complex energy contributions are irrelevant [23,24]. Statesn〉 with 〈mn〉 = δmn are eigenstates of the Hamiltonian (logarithm of the transfermatrix) and all energies, En, have the vacuum energy subtracted. Oj(t) are fields on
2
Alpha: 0902.1265
[(G(t0))1G(t0 +t)]iju
↵j = c↵u↵
i
c↵ = em↵t
v↵i [G(t0 +t)(G(t0))1]ij = c↵v↵j
v↵i Gij(t)uj / ↵
Extracting Excited States• Multiplying on the left by [Gij(t0)]1 gives the generalised eigenvalue equation
• with eigenvalues
• and similarly
• We can diagonalise the correlation matrix
• allowing fits at earlier times
• and cleanly isolate excited states
S. Mahbub:1011.0480
D. B. Leinweber
is interesting to examine the stability of the masses to different choices of bases to ascertain whetherone has reliably isolated single eigenstates of QCD. The relevant issues are: (i) whether or not theoperators are sufficiently far from collinear that numerical errors do not prevent diagonalisation ofthe correlation matrix and, (ii) whether or not the states of interest have significant overlap withthe subspace spanned by our chosen sets of operators. Since our correlation matrix diagonalisationsucceeded, except at large Euclidean times where statistical errors dominate, we conclude that ouroperators are sufficiently far from collinear.
Basis numbers 4, 7, 9 and 10 contain higher smearingsweep counts of 400 and 800, whichresults in a significant enhancement of errors for the second and third excited states. It is noted thatthe sources with sweep counts of 400, 800 and 1600 are very challenging as the smearing radii forthese sources are close to the wall source. The poor signaltonoise ratio for these sources makethe correlation matrix analysis more challenging and the eigenvalue analysis becomes unsuccessfulfor a large number of variational parameters (t0,!t). Therefore, the sources 400, 800 and 1600 areundesirable to work with.
Figure 2: (Color online). Masses of the nucleon, N 12+ states, from the projected correlation functions as
shown in Eq. (2.7). Each set of ground (g.s) and excited (e.s) states masses correspond to the diagonalizationof the correlation matrix for each set of variational parameters t0 (shown in major tick marks) and!t (shownin minor tick marks). Figure corresponds to kud = 0.13754 and for the 3rd basis.
The agreement among the three lowest lying eigenstates is remarkable and verifies that ourapproach successfully isolates true eigenstates [6, 14].
Basis number 3 has good diversity including both lower and higher smearings which is necessary for the extraction of masses over the entire heavy to light quark mass range. As a result, basis
5
[(G(t0))1G(t0 +t)]iju
↵j = c↵u↵
i
c↵ = em↵t
v↵i [G(t0 +t)(G(t0))1]ij = c↵v↵j
v↵i Gij(t)uj / ↵
t
4
FIG. 4: (Color online). Masses of the lowlying positiveparitystates of the nucleon. Physical values are plotted at the farleft. Lattice results for the Roper (filled triangles) reveal significant chiral curvature towards the physical mass.
FIG. 5: (Color online). A comparison of the lowlyingpositiveparity spectrum of dynamical QCD (full symbols)and quenched QCD results (open symbols) from Ref. [26].
ment, significant differences are observed for the Roperin the light quark mass regime. Once again, this emphasizes the role of dynamical fermion loops in creating themesonic dressings of the Roper.This investigation is the first to illustrate the manner
in which the Roper resonance of Nature manifests itselfin today’s best numerical simulations of QCD. The quarkmass dependence of the state revealed herein substantiates the essential role of dynamical fermions and theirassociated nontrivial lightmesonic dressings of baryons,which give rise to significant chiral nonanalytic curvature in the Roper mass in the chiral regime.This research was undertaken on the NCI National
Facility in Canberra, Australia, which is supported bythe Australian Commonwealth Government. We also acknowledge eResearch SA for generous grants of supercomputing time. This research is supported by the Australian Research Council.
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arXiv:0906.5433 [heplat] .[27] M. S. Mahbub, A. O. Cais, W. Kamleh, D. B. Leinweber,
and A. G. Williams, Phys. Rev., D82, 094504 (2010),arXiv:1004.5455 [heplat] .
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S. Mahbub:1011.5724
Excited States
Spectroscopy• Meson states allowed by the quark model
• States with P=(1)J but CP=1 forbidden
• The are exotic states: not just a qq pair
• Need to consider a large basis of operators to isolate higher spin states and access exotic quantum numbers
• E.g.
• Solve Generalised Eigenvalue Problem to isolate individual states

JPC = 0+, 0++, 1, 1+, 2, 2+, . . .
JPC = 0+, 0, 1+, 2+, 3+, . . .
!D = 1
2 (!D D)
!D
!D !D Fµ
The Excited Hadron Spectrum
• Includes high spin and light exotic states
• Most states identically flavour mixed strangelight mixing
4
0.5
1.0
1.5
2.0
2.5
exotics
isoscalar
isovector
YM glueball
negative parity positive parity
FIG. 4: Isoscalar meson spectrum labeled by JPC . The box height indicates the one sigma statistical uncertainty above andbelow the central value. The lightstrange content of each state (cos2 α, sin2 α) is given by the fraction of (black, green) andthe mixing angle for identified pairs is also shown. Horizontal square braces with ellipses indicate that additional states wereextracted in this JPC but were not robust. Grey boxes indicate the positions of isovector meson states extracted on the samelattice (taken from [9]). The mass scale is set using the procedure outlined in [9, 12]. Pink boxes indicate the position ofglueballs in the quarkless YangMills theory [6].
Acknowledgments: We thank our colleagues withinthe Hadron Spectrum Collaboration. Chroma [21] andQUDA [14, 15] were used to perform this work on clustersat Jefferson Laboratory under the USQCD Initiative andthe LQCD ARRA project. Support is from Jefferson Science Associates, LLC under U.S. DOE Contract No. DEAC0506OR23177 and Science Foundation Ireland underresearch grant 07/RFP/PHYF168.
∗ Electronic address: [email protected][1] K. Nakamura et al. (Particle Data Group), J.Phys.G
G37, 075021 (2010).[2] E. Klempt and A. Zaitsev, Phys. Rept. 454, 1 (2007).[3] N. H. Christ et al., Phys. Rev. Lett. 105, 241601 (2010).[4] C. McNeile, C. Michael, and C. Urbach (ETM), Phys.
Lett. B674, 286 (2009).[5] C. McNeile, C. Michael, and K. J. Sharkey (UKQCD),
Phys. Rev. D65, 014508 (2002).[6] C. J. Morningstar and M. J. Peardon, Phys. Rev. D60,
034509 (1999).[7] C. M. Richards, A. C. Irving, E. B. Gregory, and C. Mc
Neile (UKQCD), Phys. Rev. D82, 034501 (2010).[8] J. J. Dudek et al., Phys. Rev. Lett. 103, 262001 (2009).[9] J. J. Dudek et al., Phys. Rev. D82, 034508 (2010).
[10] M. Peardon et al., Phys. Rev. D80, 054506 (2009).[11] R. G. Edwards, B. Joo, and H.W. Lin, Phys. Rev. D78,
054501 (2008).[12] H.W. Lin et al., Phys. Rev. D79, 034502 (2009).[13] J. Bulava et al., Phys. Rev. D82, 014507 (2010).[14] M. A. Clark et al., Comput. Phys. Commun. 181, 1517
(2010).[15] R. Babich, M. A. Clark, and B. Joo (2010), 1011.0024.[16] V. Mathieu and V. Vento, Phys. Rev. D81, 034004
(2010).[17] C. E. Thomas, JHEP 10, 026 (2007).[18] R. Escribano and J. Nadal, JHEP 05, 006 (2007).[19] J. J. Dudek et al. (2010), 1011.6352.[20] J. Foley et al. (2010), 1011.0481.[21] R. G. Edwards and B. Joo, Nucl. Phys. B. Proc. Suppl.
140, 832 (2005).[22] F. E. Close and A. Kirk, Z.Phys. C76, 469 (1997).[23] Autocorrelations are found to be small  binning mea
surements by 10 produces no significant change in thefinal spectra.
[24] Since each state can be arbitrarily rephased withoutchanging the correlator matrix, the sign of α cannot bedetermined in this way. The angle α is related to theoctetsinglet mixing angle[1], θ by θ = α− 54.74.
[25] Using the radiative transitions methodology of [22] applied to current PDG [1] branching fraction values.
Isoscalar Mesons [Dudek et al. 1102.4299]
m = 396 MeV
↵! = 1.7(2)o↵0 = 42(1)o
The Excited Hadron SpectrumBaryons [Edwards et al. 1212.5236]
11
FIG. 7: The lowest negativeparity states that are flavoroctetare shown for m = 391 MeV.
IV. SUMMARY
This work presents results for baryons based on latticeQCD using the 163128 anisotropic lattices that weredeveloped in Ref. [6]. Excited state spectra are calculatedfor baryons that can be formed from u, d and s quarks,namely the N , , , , and families of baryons, fortwo pion masses, 391 MeV, 524 MeV, and at the SU(3)F symmetric point corresponding to a pion mass of 702MeV.
The interpolating operators used incorporate covariantderivatives in combinations that correspond to angularmomentum quantum numbers L = 0, 1 and 2. The angular momenta are combined with quark spins to buildoperators that transform according to good total angular momentum, J , in the continuum. As noted in earlierworks, approximate rotational symmetry is realized atthe scale of hadrons, enabling us to identify reliably thespins in the spectrum up to J = 7
2 from calculations ata single lattice spacing.
The operators we have employed are classified according to the irreducible representations of SU(3)F flavor.At the pion masses used, the SU(3)F symmetry is brokenonly weakly and states in the spectra can be identifiedas being created predominantly by operators of definiteflavor symmetry 8F , 10F or 1F.
We find bands of states with alternating parities andincreasing energies. Each state has a welldefined spinand generally a dominant flavor content can be identified. The number of nonhybrid states of each spin andflavor in the lowestenergy bands is in agreement with the
FIG. 8: The lowest negativeparity states that are flavorsinglet (beige) and decuplet (yellow) are shown for m = 391MeV.
expectations based on weakly broken SU(6)O(3) symmetry. These states correspond to the quantum numbersof the quark model.Chromomagnetic operators are used to identify states
that have strong hybrid content. Usually these statesare at higher masses, about 0.7m, or more, above thelowest nonhybrid states. There is reasonable agreementbetween the number of positiveparity states with stronghybrid content and the expectations of Table IV that arebased on nonrelativistic quark spins, although a few ofthe expected states are not found at the lowest pion mass.With the inclusion into our basis of multihadron oper
ators, which couple eciently onto multihadron scattering states, we expect to find an increased number of levelsin the spectrum. As demonstrated in Ref. [16], using thetechnique of moving frames where the total momentum ofthe system is nonzero, the increased number of levels allows for the extensive mapping of the energy dependenceof scattering amplitudes, and hence, the determination ofresonances. The prospect of determining the propertiesof resonances provides a strong motivation for continuedwork on the spectra of baryons.
Acknowledgments
We thank our colleagues within the Hadron SpectrumCollaboration. Particular thanks to C. Shultz for his updates to our variational fitting code. Chroma [17] andQUDA [18, 19] were used to perform this work on clusters
m 390 MeV
+
0−
0
Y
+1−1
Ξ
Σ Σ
p(uud)n(udd)
Ξ (uss)(dss)
(uds) (uus)
I3Λ0(uds)
Σ −(dds)
Σ
Y
I3
Ω
Ξ (uss)0∗Ξ (dss)
Σ∗−(dds) ∗0(uds) Σ∗+(uus)
−1 +1
−2
∗−
Δ0(udd)0
−(sss)
Δ+(uud) Δ++(uuu)
−1
Δ−(ddd)
Figure 3: The lowest octet and decuplets for the spin 12 and for the 3
2 baryons (plottedin the I3–Y plane.
must be flat at the symmetric point, and furthermore that the combinations(M∆++ + M∆− + MΩ) and (M∆+ + M∆0 + MΣ∗+ + MΣ∗− + MΞ∗0 + MΞ∗−) willalso be flat. Technically these symmetrical combinations are in the A1 singletrepresentation of the permutation group S3. This is the symmetry group of anequilateral triangle, C3v. This group has 3 irreducible representations, [6], twodifferent singlets, A1 and A2 and a doublet E, with elements E+, E−. Some details of this group and its representations are given in Appendix A, while Table 1gives a summary of the transformations.
A1 E A2
Op E+ E−
Identity + + + +u ↔ d + + − −u ↔ s + mix −d ↔ s + mix −
u → d → s → u + mix +u → s → d → u + mix +
Table 1: A simplified table showing how the group operations of S3 act in the differentrepresentations: + refers to unchanged; − to reflection.
We list some of these invariant mass combinations in Table 2. The permutation group S3 yields a lot of useful relationships, but cannot capture theentire structure. For example, there is no way to make a connection between the
8
+
0−
0
Y
+1−1
Ξ
Σ Σ
p(uud)n(udd)
Ξ (uss)(dss)
(uds) (uus)
I3Λ0(uds)
Σ −(dds)
Σ
Y
I3
Ω
Ξ (uss)0∗Ξ (dss)
Σ∗−(dds) ∗0(uds) Σ∗+(uus)
−1 +1
−2
∗−
Δ0(udd)0
−(sss)
Δ+(uud) Δ++(uuu)
−1
Δ−(ddd)
Figure 3: The lowest octet and decuplets for the spin 12 and for the 3
2 baryons (plottedin the I3–Y plane.
must be flat at the symmetric point, and furthermore that the combinations(M∆++ + M∆− + MΩ) and (M∆+ + M∆0 + MΣ∗+ + MΣ∗− + MΞ∗0 + MΞ∗−) willalso be flat. Technically these symmetrical combinations are in the A1 singletrepresentation of the permutation group S3. This is the symmetry group of anequilateral triangle, C3v. This group has 3 irreducible representations, [6], twodifferent singlets, A1 and A2 and a doublet E, with elements E+, E−. Some details of this group and its representations are given in Appendix A, while Table 1gives a summary of the transformations.
A1 E A2
Op E+ E−
Identity + + + +u ↔ d + + − −u ↔ s + mix −d ↔ s + mix −
u → d → s → u + mix +u → s → d → u + mix +
Table 1: A simplified table showing how the group operations of S3 act in the differentrepresentations: + refers to unchanged; − to reflection.
We list some of these invariant mass combinations in Table 2. The permutation group S3 yields a lot of useful relationships, but cannot capture theentire structure. For example, there is no way to make a connection between the
8
Resonances
• Many states are actually resonances
• If quarks are light enough, state can decay strongly
• e.g.
• How can we be sure that the state we measure is the true resonance state and not some superposition of states?
• [see Ross’ talk yesterday]
• Include two particle operators into your simulations
! ++ ! p+Lüscher
Map out volumedependence of energy levels
VolumeEner
gy
Phas
esh
ift
Energy
Wiese [Lat88] “free” 2particle
discrete spectrum
Lüscher NPB(1991)
Two Particle States
• Example
• If you are below threshold (particle can decay), need to consider two particle system
• Not only need twopoint function
• But also
!
!
• Recall meson 2pt function
• For twoparticle system
Two Particle States
hO(x)O†(0)i = h a(x) b(x) b(0)† a(0)i
+ ! +
hd(x)u(x)u(x)d(x) u(0)†d(0)d(0)†
u(0)i
• Recall meson 2pt function
• For twoparticle system
Two Particle States
hO(x)O†(0)i = h a(x) b(x) b(0)† a(0)i
+ ! +
hd(x)u(x)u(x)d(x) u(0)†d(0)d(0)†
u(0)i
• Recall meson 2pt function
• For twoparticle system
Two Particle States
hO(x)O†(0)i = h a(x) b(x) b(0)† a(0)i
+ ! +
hd(x)u(x)u(x)d(x) u(0)†d(0)d(0)†
u(0)i
• Recall meson 2pt function
• For twoparticle system
Two Particle States
hO(x)O†(0)i = h a(x) b(x) b(0)† a(0)i
+ ! +
hd(x)u(x)u(x)d(x) u(0)†d(0)d(0)†
u(0)i
Requires “alltoall” propagatorS(x, x)
Extracting Resonance• Construct correlation matrix
• Diagonalise to extract two energy levels
G! G!
G! G!
G1 00 G2
E1, E2
Lüscher
Map out volumedependence of energy levels
Volume
Ener
gy
Phas
esh
ift
Energy
Wiese [Lat88] “free” 2particle
discrete spectrum
Lüscher NPB(1991)
x
x
Extracting Resonance14
0
20
40
60
80
100
120
140
160
180
0.14 0.15 0.16 0.17 0.18
FIG. 10. P wave elastic scattering phaseshift, 1(Ecm), determined from solution of Eq. 7 applied to the finitevolumespectra shown in Fig. 9 under the assumption that `>1 = 0. Energy region plotted is from threshold to KK threshold.
atmR = 0.15226(34)(11)2
41 0.14 0.09
1 0.321
3
5g = 5.06(15)(2)
R/at = 16.6(52)(17)
2/Ndof =
43.6293 = 1.68,
which shows a slightly improved quality of fit, althoughthere is clearly some correlation between the coupling g
and the range R. The range expressed in physical unitsR = R
at
atm
mphys
0.6 ± 0.2 fm would seem to be reason
able on the usual hadronic scale. The resulting energydependence is shown by the red curve in Fig. 11 whereit is seen to approach 180 more rapidly than the simpleBreitWigner.
The particular form of the damping function is amodeldependent choice and we can explore the sensitivity by trying other parameterisations. For example agaussian form (previously considered in a quark modelstudy [35]),
gau.`=1 (Ecm) =
g
2
6
p
3cm
E
2cm
e
p2cm/62
e
p2R/62 . (11)
Fitting the same dataset we obtain
atmR = 0.15224(34)(14)2
41 0.18 0.16
1 0.471
3
5g = 5.08(17)(3)
at = 0.029(7)(3)
2/Ndof =
43.5293 = 1.67,
indicating that the particular functional form of thedamping appears to be relatively unimportant. In phys
ical units, = at · mphys
atm 160(40)MeV. The energy
dependence is shown by an orange curve in Fig. 11 thatlies almost exactly on the red curve already described.Another parameterisation that has been used to fit ex
perimental phaseshift data is provided by Pelaez andYndurain (see Ref. [36] and their subsequent papers),
cot 1(Ecm) =Ecm
2p3cm(m2
R E
2cm)
"
2m2
m
2REcm
+B0 +B1Ecm p
s0 E
2cm
Ecm +p
s0 E
2cm
#,
which, while it appears cosmetically to be very di↵erent to a BreitWigner, in fact has an energy dependence which is rather similar, with the three parametersmR, B0, B1 able to conspire to provide damping. Theadditional parameter, s0, is not allowed to float, and following the proposers’ suggestion is set to 2m +m, asdetermined on this lattice, at
ps0 = 0.29. Fitting yields
atmR = 0.15227(34)(12)2
41 0.06 0.05
1 0.991
3
5B0 = 2.71(77)(21)B1 = 6.0(33)(9)
2/Ndof =
43.7293 = 1.68,
a reasonable description of the data. The extremely highdegree of correlation between B0 and B1 suggests thatthey may not be the most natural way to parameterisethis amplitude. The energy dependence is plotted inFig. 11 using a green curve that lies almost exactly underthe orange and red curves already plotted.We have presented the data and fits in units of
the inverse temporal lattice spacing thus far to avoidambiguities with how one sets the lattice scale. If
Dudek et al.[1212.0830]
Spectroscopy• Light hadron spectrum is in good shape
• Significant advancement in the determination of excited states
• Now confronted with new challenges:
• Isospin breaking
• QED effects
• Resonances (See plenary talk by D. Mohler at Lattice 2012)
(See plenary talk by T. Izubuchi at Lattice 2012)
• To date, all Lattice simulations have been performed with degenerate up and down quark masses
• Progress by several collaborations in determining the effect on some observables
• QCDSF (1206.3156):
• Rome 123 (1110.6294):
Isospin Breaking (md mu)
Nf = 2, Nf = 2 + 1, Nf = 2 + 1 + 1
(md mu)
Mn Mp
Mn Mp = 3.13(55)
M M+ = 8.10(136)
M M0 = 4.98(85)
md mu(MS, 2GeV ) = 2.35(25)
FK+/F+
FK/F 1 = 0.0039(4)
Mn Mp = 2.8(7)Shanahan (’12)
QED Effects• Many Lattice QCD results are achieving high precision
• QED effects may not be negligible and should be included
• Although in some cases, QED can be treated perturbatively, this is not always the case
• Currently two main methods employed:
• Quenched QED
• Dynamical QED via reweighting
(See plenary talk by T. Izubuchi at Lattice 2012)
f, fK 1%, (f/fK) 0.5%
QCD+QED Lattice simulation
QED Effects(See plenary talk by T. Izubuchi at Lattice 2012)
• E.g. Quark masses from QCD+QED simulation [PRD82, 094508 (2010)]
• and for np:
• Combine with previous QCD result
c.f. experiment: 1.2933321(4) MeV
Quark mass from QCD+QED simulation
[PRD82 (2010) 094508 [47pages]]
mu = 2.24 ± 0.10 ± 0.34 MeV
md = 4.65 ± 0.15 ± 0.32 MeV
ms = 97.6 ± 2.9 ± 5.5 MeV
md mu = 2.411 ± 0.065 ± 0.476 MeV
mud = 3.44 ± 0.12 ± 0.22 MeV
mu/md = 0.4818 ± 0.0096 ± 0.0860
ms/mud = 28.31 ± 0.29 ± 1.77,
• MS at 2 GeV using NPR/SMOM scheme.
• Particular to QCD+QED, finite volume error is large: 14% and 2% for mu and md.
• This would be due to photon’s nonconfining feature (vs gluon).
• Volume, a2, chiral extrapolation errors are being removed.
• Applications for Hadronic contribution to (g 2)µ in progress.
Taku Izubuchi, Lattice 2012, Cairns, June 25, 2012 15
(Mn Mp)QED = 0.54(24)
(Mn Mp) = 2.14(42) MeV
H,H : , K p, n, . . .
O : Vµ, Aµ, . . .
H OH
Calculating Matrix Elements
Calculating Matrix Elements
h(p0, s0)Jµ(~q)(p, s)i =
(0 · )PµG1(Q2) [(0 · q)µ ( · q)0µ]G2(Q
2) + ( · q)(0 · q) Pµ
(2m)2G3(Q
2)
h(p0, s0)Jµ(~q)(p, s)i =
u↵(p0, s0)
g↵
µa1(Q
2) +Pµ
2Ma2(Q
2) q↵q
(2M)2µc1(Q
2) + dPµ
2Mc2(Q
2)
u(p, s)
hN(p0, s0)Jµ(~q)N(p, s)i = u(p0, s0)hµF1(q
2) + iµ q2m
F2(q2)iu(p, s)
h(p0)Jµ(~q)(p)i = PµF(q2)
q2 = Q2 = (p0 p)2
Pµ = p0µ + pµSpin0
Spin1/2
Spin1
Spin3/2
G(t, , p, p0) =X
~x2,~x1
e
i~p
0·(~x2~x1)e
i~p·~x1↵
T
↵
(~x2, t)O(~x1, )
(0)
↵
Lattice 3pt Functions
• Create a state (with quantum numbers of the proton) at time t=0
G(t, , p, p0) =X
~x2,~x1
e
i~p
0·(~x2~x1)e
i~p·~x1↵
T
↵
(~x2, t)O(~x1, )
(0)
↵
Lattice 3pt Functions
• Create a state (with quantum numbers of the proton) at time t=0
• Insert an operator, , at some time O
G(t, , p, p0) =X
~x2,~x1
e
i~p
0·(~x2~x1)e
i~p·~x1↵
T
↵
(~x2, t)O(~x1, )
(0)
↵
Lattice 3pt Functions
• Create a state (with quantum numbers of the proton) at time t=0
• Insert an operator, , at some time
• Annihilate state at final time t
O
G(t, , p, p0) =X
~x2,~x1
e
i~p
0·(~x2~x1)e
i~p·~x1↵
T
↵
(~x2, t)O(~x1, )
(0)
↵
Lattice 3pt Functions
• Create a state (with quantum numbers of the proton) at time t=0
• Insert an operator, , at some time
• Annihilate state at final time t
O
• Insert complete set of states
• Make use of translational invariance
• Evolve to large Euclidean times to isolate ground state
Lattice 3pt Functions
I =X
B,p,s
B, p, sihB, p, s
(~x, t) = e
Hte
i ~P ·~x(0)ei
~P ·~xe
Ht
I =X
B0,p0,s0
B0, p0, s0ihB0, p0, s0
G(t, , ~p, ~p 0) =X
B,B0
X
s,s0
eEB0 (~p 0)(t)eEB(~p)↵
↵(0)
B0, p0, s0↵B0, p0, s0
O(~q)B, p, s
↵B, p, s
(0)
↵
G(t, , p, p0) =X
~x2,~x1
e
i~p
0·(~x2~x1)e
i~p·~x1↵
T
↵
(~x2, t)O(~x1, )
(0)
↵
0 t
G(t, , ~p, ~p 0) =X
s,s0
eE~p 0 (t)eE~p↵
↵(0)
N(p0, s0)↵N(p0, s0)
O(~q)N(p, s)
↵Np, s)
(0)
↵
• Consider a pion 3pt function
• With interpolating operator
• And insert the local operator (quark bilinear)
Lattice 3pt Functions pion
(x) = d(x)5u(x)
G(t, , p, p) =
x2,x1
eip·(x2x1)eip·x1
T(x2, t)O(x1, ) †(0)
q(x)Oq(x): Combination of
matrices and derivativesO
d(x2)5u(x2)u(x1)Ou(x1)u(0)5d(0) uquark
• Consider a pion 3pt function
• With interpolating operator
• And insert the local operator (quark bilinear)
Lattice 3pt Functions pion
(x) = d(x)5u(x)
G(t, , p, p) =
x2,x1
eip·(x2x1)eip·x1
T(x2, t)O(x1, ) †(0)
q(x)Oq(x): Combination of
matrices and derivativesO
d(x2)5u(x2)u(x1)Ou(x1)u(0)5d(0) uquark
Lattice 3pt Functions
• all possible Wick contractions
da(x2)5ua
(x2)ub(x1)u
b(x1)uc
(0)5dc(0)
pionuquark
Lattice 3pt Functions
• all possible Wick contractions
• connected
da(x2)5ua
(x2)ub(x1)u
b(x1)uc
(0)5dc(0)
pion
Scad(0, x2)5Sab
u(x2, x1)Sbcu(x1, 0)5
uquark
Lattice 3pt Functions
• all possible Wick contractions
• connected
• disconnected
da(x2)5ua
(x2)ub(x1)u
b(x1)uc
(0)5dc(0)
Scad(0, x2)5Sab
u(x2, x1)Sbcu(x1, 0)5
Scad(0, x2)5Sac
u(x2, 0)5Sbbu(x1, x1)
pionuquark
Lattice 3pt Functions
• all possible Wick contractions
• connected
• disconnected
da(x2)5ua
(x2)ub(x1)u
b(x1)uc
(0)5dc(0)
TrSd(0, x2)5Su(x2, x1)Su(x1, 0)5
Tr Sd(0, x2)5Su(x2, 0)5
Tr
Su(x1, x1)
pionuquark
Lattice 3pt Functions
• all possible Wick contractions
• connected
• disconnected
• alltoall propagators
da(x2)5ua
(x2)ub(x1)u
b(x1)uc
(0)5dc(0)
S†(x, 0) = 5S(0, x)5
Tr S†
d(x2, 0)Su(x2, 0)Tr
Su(x1, x1)
TrS†
d(x2, 0)Su(x2, x1)Su(x1, 0)
5hermiticity
pion
• Use the following interpolating operator to create a proton
• And insert the local operator (quark bilinear)
• Perform all possible (connected) Wick contractions uquark (4 terms)
protonG(t, ; p, p) =
x2,x1
eip·(x2x1)eip·x1
T(t, x2)O(, x1) (0)
(x) = abcuTa(x) C5 db(x)
uc
(x)
abcabc uTa(x2) C5 db(x2)
uc
(x2)u(x1)Ou(x1)uc(0)
db
(0)C5uTa
(0)
Lattice 3pt Functions
q(x)Oq(x) : Combination of matrices and derivatives
O
• Use the following interpolating operator to create a proton
• And insert the local operator (quark bilinear)
• Perform all possible (connected) Wick contractions uquark (4 terms)
protonG(t, ; p, p) =
x2,x1
eip·(x2x1)eip·x1
T(t, x2)O(, x1) (0)
(x) = abcuTa(x) C5 db(x)
uc
(x)
abcabc uTa(x2) C5 db(x2)
uc
(x2)u(x1)Ou(x1)uc(0)
db
(0)C5uTa
(0)
Lattice 3pt Functions
q(x)Oq(x) : Combination of matrices and derivatives
O
• Use the following interpolating operator to create a proton
• And insert the local operator (quark bilinear)
• Perform all possible (connected) Wick contractions uquark (4 terms)
protonG(t, ; p, p) =
x2,x1
eip·(x2x1)eip·x1
T(t, x2)O(, x1) (0)
(x) = abcuTa(x) C5 db(x)
uc
(x)
abcabc uTa(x2) C5 db(x2)
uc
(x2)u(x1)Ou(x1)uc(0)
db
(0)C5uTa
(0)
Lattice 3pt Functions
q(x)Oq(x) : Combination of matrices and derivatives
O
• Use the following interpolating operator to create a proton
• And insert the local operator (quark bilinear)
• Perform all possible (connected) Wick contractions uquark (4 terms)
protonG(t, ; p, p) =
x2,x1
eip·(x2x1)eip·x1
T(t, x2)O(, x1) (0)
(x) = abcuTa(x) C5 db(x)
uc
(x)
abcabc uTa(x2) C5 db(x2)
uc
(x2)u(x1)Ou(x1)uc(0)
db
(0)C5uTa
(0)
Lattice 3pt Functions
q(x)Oq(x) : Combination of matrices and derivatives
O
• Use the following interpolating operator to create a proton
• And insert the local operator (quark bilinear)
• Perform all possible (connected) Wick contractions
abcabc uTa(x2) C5 db(x2)
uc
(x2)d(x1)Od(x1)uc(0)
db
(0)C5uTa
(0)
dquark (2 terms)
protonG(t, ; p, p) =
x2,x1
eip·(x2x1)eip·x1
T(t, x2)O(, x1) (0)
(x) = abcuTa(x) C5 db(x)
uc
(x)
Lattice 3pt Functions
q(x)Oq(x) : Combination of matrices and derivatives
O
• Use the following interpolating operator to create a proton
• And insert the local operator (quark bilinear)
• Perform all possible (connected) Wick contractions
abcabc uTa(x2) C5 db(x2)
uc
(x2)d(x1)Od(x1)uc(0)
db
(0)C5uTa
(0)
dquark (2 terms)
protonG(t, ; p, p) =
x2,x1
eip·(x2x1)eip·x1
T(t, x2)O(, x1) (0)
(x) = abcuTa(x) C5 db(x)
uc
(x)
Lattice 3pt Functions
q(x)Oq(x) : Combination of matrices and derivatives
O
• Pictorially:
• uquark
• dquark
• squark
• quarkline disconnected contributions drop out in isovector quantities (ud) if isospin is exact (mu=md)
protonLattice 3pt Functions