Hadron spectroscopy from lattice QCDth- · 2015. 6. 25. · Hadron spectroscopy from lattice QCD...
Transcript of Hadron spectroscopy from lattice QCDth- · 2015. 6. 25. · Hadron spectroscopy from lattice QCD...
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Hadron spectroscopy from lattice QCD
Christopher Thomas, University of Cambridge
Cracow School of Theoretical Physics, Zakopane, 20 – 28 June 2015
Partial support from the Science & Technology Facilities Council (UK)
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Outline
• Lecture 1 – Lattice QCD and some applications
• Lecture 2 – Hadron spectroscopy
• Lecture 3 – Resonances, scattering, etc
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Lecture 1
• Why lattice quantum chromodynamics?
• Introduction to lattice QCD
• Some applications
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General references
• “Lattice Gauge Theories, An Introduction”, Heinz Rothe (World Scientific, Lecture Notes in Physics, 4th edn. 2012)
• “Lattice Methods for Quantum Chromodynamics”, Thomas Degrand and Carleton DeTar (World Scientific, 2006)
• “Quantum Chromodynamics on the Lattice: An Introductory Presentation”, Christof Gattringer and Christian Lang (Springer, Lecture Notes in Physics, 2009, also available as an e-book)
• “Quantum fields on the lattice”, I. Monvay and G. Münster (CUP, 1994)
• Reviews from the annual International Symposium on Lattice Field Theory, http://www.bnl.gov/lattice2014/ and proceedings, http://pos.sissa.it/cgi-bin/reader/family.cgi?code=lattice
• INT Summer School on Lattice QCD for Nuclear Physics (2012) http://www.int.washington.edu/PROGRAMS/12-2c/
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The strong interaction
• Binds quarks hadrons: mesons and baryons (protons, neutrons, ...)
• Binds protons and neutrons nuclei
• Responsible for most of mass of conventional matter (99% of proton mass)
www.nndc.bnl.gov
Z
N
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The strong interaction
• Binds quarks hadrons: mesons and baryons (protons, neutrons, ...)
• Binds protons and neutrons nuclei
• Responsible for most of mass of conventional matter (99% of proton mass)
www.nndc.bnl.gov
Z
N
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Quantum Chromodynamics
SU(3) gauge field theory; quarks and gluons
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Quantum Chromodynamics
SU(3) gauge field theory; quarks and gluons
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quarks
gluons
Quantum Chromodynamics
g
SU(3) gauge field theory; quarks and gluons
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quarks
gluons
Quantum Chromodynamics
g
SU(3) gauge field theory; quarks and gluons
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Quantum Chromodynamics
[PDG 2014]
Large at low energies – can’t make perturbative expansion Asymptotic freedom
Running coupling constant
g
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Quantum Chromodynamics
[PDG 2014]
Large at low energies – can’t make perturbative expansion Asymptotic freedom
Running coupling constant
g
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• Non-perturbative regime:
• Confinement of quarks into hadrons
• Masses of hadrons (spectra), widths, transitions, ...
• Nuclei
• ...
• Models, effective field theories (EFTs), ...
• Based on some symmetry properties, (expected) physics of QCD, approximation in some regime.
• In general not derived from QCD
• May be only approach (currently) applicable to some problems
• Can be useful for getting insight into physics (complementary)
• Lattice QCD – numerical non-perturbative calculations in QCD
Why lattice QCD?
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Discretise theory on a 4d grid (spacing = a) – UV regulator Finite volume (L3 x T) finite no. of d.o.f. Quantised momenta for spatial periodic BCs
a
L QCD on a lattice
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Discretise theory on a 4d grid (spacing = a) – UV regulator Finite volume (L3 x T) finite no. of d.o.f. Quantised momenta for spatial periodic BCs
a
L QCD on a lattice
Quark fields on lattice sites
Gauge fields on links; U is an element of SU(3)
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Path integral formulation (continuum) – Integrate over all field configurations (infinite number) Observable:
QCD on a lattice
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Finite lattice – finite num. of quark and gluon fields to integrate over
Euclidean time: t - i t oscillating phase decaying exponential – amenable to numerical computation
c.f. statistical physics
Path integral formulation (continuum) – Integrate over all field configurations (infinite number) Observable:
QCD on a lattice
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Many possible discretisations which all continuum QCD as a 0. ‘Improved’ actions reduce discretisation effects, e.g. O(a).
Generic Euclidean action (gauge invariant):
QCD on a lattice
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Many possible discretisations which all continuum QCD as a 0. ‘Improved’ actions reduce discretisation effects, e.g. O(a).
Generic Euclidean action (gauge invariant):
Gauge fields, e.g.
(Suppressed spin and colour labels)
QCD on a lattice
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Many possible discretisations which all continuum QCD as a 0. ‘Improved’ actions reduce discretisation effects, e.g. O(a).
Generic Euclidean action (gauge invariant):
‘Naive’ fermions:
Technical problems with this... Various solutions each with advantages and disadvantages
Gauge fields, e.g.
QCD on a lattice
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QCD on a lattice
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QCD on a lattice
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Gauge fields (bosons) – complex matrices
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QCD on a lattice
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Fermion fields – anticommuting ‘Grassmann’ numbers
Gauge fields (bosons) – complex matrices
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Bilinear in fermion fields
QCD on a lattice
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Fermion fields – anticommuting ‘Grassmann’ numbers
Gauge fields (bosons) – complex matrices
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QCD on a lattice
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QCD on a lattice
Q has dim. ( (L/a)3 x (T/a) x 4 x 3)2 , e.g. (203 x 128 x 4 x 3)2 (107) 2 – huge!
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Fermion det. – nonlocal function of U very computationally expensive
Q-1 and det(Q) more expensive for small m
Historically, quenched approx: set det[Q] = 1 – don’t include quark loops
Now most calculations are dynamical (‘unquenched’) – include det[Q]
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QCD on a lattice
Q has dim. ( (L/a)3 x (T/a) x 4 x 3)2 , e.g. (203 x 128 x 4 x 3)2 (107) 2 – huge!
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Use Importance Sampling Monte Carlo to evaluate numerically
Dominated by field cfgs of U where this is large
Sample integral with prob.
det(Q[U]) must be re-calculated for each U – expensive
Sample integral a finite number of times (num. of cfgs.) mean and statistical uncertainty
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QCD on a lattice
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QCD on a lattice
JLab
Oak Ridge National Lab, US
www.hpc.cam.ac.uk
NVIDIA
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• Continuum limit: lattice spacing, a 0 L = const, so N = L/a
• Volume, L >> physical size of problem e.g. L m >> 1
• Pion mass, m physical m
a
L
Lattice QCD
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• Continuum limit: lattice spacing, a 0 L = const, so N = L/a
• Volume, L >> physical size of problem e.g. L m >> 1
• Pion mass, m physical m
Setting the scale (determine a in physical units)
• Every dimensional quantity measured in terms of a
• ‘Set the scale’ by comparing with a physical observable calculated on the lattice to experimental value
• E.g. static quark potential, baryon mass, ... Set bare quark masses (mq) in action by comparing lattice computations of hadron masses with experimental masses
a
L
Lattice QCD
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Some applications
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Potential between two infinitely heavy quarks (static colour sources)
Static potential from lattice QCD
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Potential between two infinitely heavy quarks (static colour sources)
Compare length scale with experimental charmonium and bottomonium spectra
PR D79, 034502 (2009)
Static potential from lattice QCD
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Spectroscopy on the lattice
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Spectroscopy on the lattice
Calculate energies and matrix elements (“overlaps”, Z) from 2-point correlation functions of hadron interpolating fields “operators”
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Spectroscopy on the lattice
Calculate energies and matrix elements (“overlaps”, Z) from 2-point correlation functions of hadron interpolating fields “operators”
• is local but hadrons are extended objects 1 fm.
• Improve overlap onto states of interest (reduce overlap with UV modes) by spatially smearing quark fields.
q
fm
q
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Diagrams
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Wick’s theorem: contract in all possible ways
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Diagrams
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Wick’s theorem: contract in all possible ways
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Diagrams
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q
q
Wick’s theorem: contract in all possible ways
Diagrammatically: q
q
q
q x0 x1 x0 x1
‘Connected’ ‘Disconnected’
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Diagrams
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q
q
Wick’s theorem: contract in all possible ways
Diagrammatically:
N.B. these are not perturbation theory diagrams
q
q
q
q x0 x1 x0 x1
‘Connected’ ‘Disconnected’
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Diagrams
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Spectroscopy on the lattice
Calculate energies and matrix elements (“overlaps”, Z) from 2-point correlation functions of meson interpolating fields “operators”
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Spectroscopy on the lattice
Calculate energies and matrix elements (“overlaps”, Z) from 2-point correlation functions of meson interpolating fields “operators”
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Spectroscopy on the lattice
Calculate energies and matrix elements (“overlaps”, Z) from 2-point correlation functions of meson interpolating fields “operators”
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Spectroscopy on the lattice
Calculate energies and matrix elements (“overlaps”, Z) from 2-point correlation functions of meson interpolating fields “operators”
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t/a
C(t)
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Correlators
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t/a t/a
C(t) Meff
0.216
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Correlators
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Use only smeared local operators (e.g. i ). Set scale using M Nucleons & isovector mesons – only connected diagrams
BMW Collaboration, Durr et al, Science 322, 1224 (2008)
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Low-lying spectrum of hadrons
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Low-lying spectrum of hadrons BMW Collaboration, Durr et al, Science 322, 1224 (2008)
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Low-lying spectrum of hadrons BMW Collaboration, Durr et al, Science 322, 1224 (2008)
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Light baryons – comparison
Agreement between results from different lattice actions (extrapolated to continuum limit and physical m)
Alexandrou et al, PR D90, 074501 (2014)
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Dowdall et al (HPQCD) [PR D86, 094510 (2012)] Quarkonia and heavy-light mesons
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Dowdall et al (HPQCD) [PR D86, 094510 (2012)] Quarkonia and heavy-light mesons
Bs
Bc
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• Hadron form factors, radiative transitions
• Hadron structure, TMDs, etc
• Decay constants
• Weak matrix elements, flavour physics
SM tests and BSM constraints
• Nuclear physics / nuclei
• QCD at finite temperature and density
• Other field theories, BSM physics
• ...
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More applications
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Summary of lecture 1
Next time
• Why lattice quantum chromodynamics?
• A (very brief) introduction to lattice QCD
• Applications: static potential and some ‘simple’ hadron spectroscopy
• Excited hadron spectroscopy
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Lecture 2
• Background on hadron spectroscopy
• Excited hadron spectroscopy in lattice QCD
• Mesons
• Baryons
(I won’t review all lattice calculations of hadron spectra)
Hadron spectroscopy from lattice QCD
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Hadron spectroscopy
Masses and other properties of hadrons probe the non-perturbative regime of QCD.
• Relevant degrees of freedom?
• Confinement?
• Role of gluons?
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Hadron spectroscopy
Masses and other properties of hadrons probe the non-perturbative regime of QCD.
• Relevant degrees of freedom?
• Confinement?
• Role of gluons?
KLOE2 CLAS12
Experiments
LHCb
ATLAS CMS
BESIII
ELSA MAMI J-PARC Spring-8
+ others at 12 GeV JLab
+ others at GSI
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Hadron spectroscopy – mesons
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Hadron spectroscopy – mesons
Quark-antiquark pair: n 2S+1LJ
L
L ½
½
J = L S
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Hadron spectroscopy – mesons
Quark-antiquark pair: n 2S+1LJ
L
L ½
½
J = L S
Parity: P = (-1)(L+1)
Charge Conj Sym: C = (-1)(L+S)
JPC = 0- + , 0+ + , 1- - , 1+ + , 1+ - , 2- - , 2+ + , 2- +, …
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Hadron spectroscopy – mesons
Quark-antiquark pair: n 2S+1LJ
L
L ½
½
J = L S
Exotic JPC (0- -, 0+ -, 1 - +, 2
+ -, ...) or flavour quantum numbers – can’t just be a pair
Parity: P = (-1)(L+1)
Charge Conj Sym: C = (-1)(L+S)
JPC = 0- + , 0+ + , 1- - , 1+ + , 1+ - , 2- - , 2+ + , 2- +, …
E.g. multiquark systems (tetraquarks, molecular mesons)
Hybrid mesons (gluonic field excited)
Glueballs
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Isoscalars (I = 0) e.g. , ’, ,
Isovectors (I = 1) e.g. , , a1
Flavours of mesons
Light mesons u, d, s (anti)quarks
Hadron spectroscopy – mesons
Kaons (I = 1/2) e.g. K, K*
mu = md – isospin sym.
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Isoscalars (I = 0) e.g. , ’, ,
Isovectors (I = 1) e.g. , , a1
Flavours of mesons
Light mesons
Charmonium e.g. J/
Bottomonium e.g. ϒ
D, Ds, B, ...
u, d, s (anti)quarks
Hadron spectroscopy – mesons
Kaons (I = 1/2) e.g. K, K*
mu = md – isospin sym.
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Charmonium
PDG 2013
Quark Model (G&I)
M /
MeV
Y(4260)
X(3872)
Z+(4430)
c
J/
χc0
χc1 χc2 hc
Z+(3900)
Hadron spectroscopy – mesons
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Charmonium
PDG 2013
Quark Model (G&I)
M /
MeV
Y(4260)
X(3872)
Z+(4430)
c
J/
χc0
χc1 χc2 hc
Z+(3900)
Hadron spectroscopy – mesons
Puzzles (don’t fit expected pattern): X(3872), Y(4260), Z+(4430), Zc
+(3900),
Also: Zb+, Ds(2317), light scalars, ...
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Charmonium
PDG 2013
Quark Model (G&I)
M /
MeV
Y(4260)
X(3872)
Z+(4430)
c
J/
χc0
χc1 χc2 hc
Z+(3900)
Hadron spectroscopy – mesons
Puzzles (don’t fit expected pattern): X(3872), Y(4260), Z+(4430), Zc
+(3900),
Also: Zb+, Ds(2317), light scalars, ...
Exotic JPC ? • No sign in charmonium or bottomonium • Light sector: 1(1600) (1
- +) needs conf.
Flavour exotics
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• Missing states? • ‘Freezing’ of degrees of freedom? • Gluonic excitations? • Flavour structure
Hadron spectroscopy – baryons
1/2+
3/2+
5/2+
7/2+
9/2+ 1/2
-3/2
-5/2
-7/2
-9/2
-800
1000
1200
1400
1600
1800
2000
2200
2400
Mas
s (M
eV)
Nucleon (Exp): 4*, 3*, some 2*
2 2 1 4 5 3 1
N*
1/2+
3/2+
5/2+
7/2+ 1/2
-3/2
-5/2
-7/2
-800
1000
1200
1400
1600
1800
2000
2200
2400
Mas
s (M
eV)
Delta (Exp): 4*, 3*, some 2*
1 1 0
Δ*
2 3 2 1
Plots from Robert Edwards, data from PDG
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• Missing states? • ‘Freezing’ of degrees of freedom? • Gluonic excitations? • Flavour structure
Hadron spectroscopy – baryons
1/2+
3/2+
5/2+
7/2+
9/2+ 1/2
-3/2
-5/2
-7/2
-9/2
-800
1000
1200
1400
1600
1800
2000
2200
2400
Mas
s (M
eV)
Nucleon (Exp): 4*, 3*, some 2*
2 2 1 4 5 3 1
N*
1/2+
3/2+
5/2+
7/2+ 1/2
-3/2
-5/2
-7/2
-800
1000
1200
1400
1600
1800
2000
2200
2400
Mas
s (M
eV)
Delta (Exp): 4*, 3*, some 2*
1 1 0
Δ*
2 3 2 1
Plots from Robert Edwards, data from PDG
‘Roper’ resonance
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Can we compute spectra of hadrons (including unconventional hadrons), understand these observations and address puzzles with first-principles calculations in QCD? lattice QCD
Hadron spectroscopy
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Excited meson spectroscopy in LQCD – our approach
Energy eigenstates from:
[Hadron Spectrum Collaboration, PR D80 054506, PRL 103 262001, PR D82 034508, D84 074508, D85 014507]
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Excited meson spectroscopy in LQCD – our approach
Energy eigenstates from:
Interpolating operators
Ops have definite JP(C) in continuum (when p = 0) Here p = 0 and up to 3 derivatives included: • many ops in each channel (up to 26) • different spin and angular structures, include [Di, Dj]
[Hadron Spectrum Collaboration, PR D80 054506, PRL 103 262001, PR D82 034508, D84 074508, D85 014507]
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Large basis of ops matrix of correlators Generalised eigenvalue problem:
Variational method
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Large basis of ops matrix of correlators Generalised eigenvalue problem:
Eigenvalues energies (t >> t0)
Variational method
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Large basis of ops matrix of correlators Generalised eigenvalue problem:
Eigenvalues energies
Also optimal linear combination of operators to overlap on to a state
Z(n) related to eigenvectors
(t >> t0)
Variational method
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Large basis of ops matrix of correlators Generalised eigenvalue problem:
Eigenvalues energies
Also optimal linear combination of operators to overlap on to a state
Z(n) related to eigenvectors
(t >> t0)
Var. method uses orthog of eigenvectors; don’t just rely on separating energies
Variational method
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Continuum
Infinite number of irreps: J = 0, 1, 2, 3, 4, ...
Reduced symmetry and spin
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Continuum
Irrep A1 A2 T1 T2 E
Dim 1 1 3 3 2
Infinite number of irreps: J = 0, 1, 2, 3, 4, ...
Finite cubic lattice
Cont. Spin 0 1 2 3 4 ...
Irrep(s) A1 T1 T2 + E T1 + T2 + A2 A1 + T1 + T2 + E ...
Broken sym: 3D rotation group cubic group
Finite number of irreps Λ: A1, A2, T1, T2, E (+ others for half-integer spin)
Reduced symmetry and spin
‘Subduce’ operators into lattice irreps (J ):
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Continuum
Irrep A1 A2 T1 T2 E
Dim 1 1 3 3 2
Infinite number of irreps: J = 0, 1, 2, 3, 4, ...
Finite cubic lattice
Cont. Spin 0 1 2 3 4 ...
Irrep(s) A1 T1 T2 + E T1 + T2 + A2 A1 + T1 + T2 + E ...
Broken sym: 3D rotation group cubic group
Finite number of irreps Λ: A1, A2, T1, T2, E (+ others for half-integer spin)
Reduced symmetry and spin
Relevant symmetry reduced further for hadrons at non-zero momentum
‘Subduce’ operators into lattice irreps (J ):
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JHEP 07 (2012) 126 – Liu, Moir, Peardon, Ryan, CT, Vilaseca, Dudek, Edwards, Joó, Richards
Hadron Spectrum Collaboration Excited charmonia
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• Dynamical ‘clover’ u, d and s quarks, mu = md < ms [Nf = 2+1]
• Relativistic c quark
• Anisotropic – finer in temporal dir (as/at 3.5),
• m 400 MeV (not physical m)
• One lattice spacing, as 0.12 fm (no extrap. to contin. limit)
• Two volumes: 163, 243 (Ls 1.9, 2.9 fm)
• Only compute connected contributions
c
c
c
c
c
c
JHEP 07 (2012) 126 – Liu, Moir, Peardon, Ryan, CT, Vilaseca, Dudek, Edwards, Joó, Richards
Hadron Spectrum Collaboration Excited charmonia
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J/ ’
m at = 0.53726(4) m at = 0.6713(5)
Excited charmonia
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JPC
c J/ χc0 χc1 χc2 hc
Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
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Exotics
JPC
c J/ χc0 χc1 χc2 hc
Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
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S-wave
Exotics
JPC
c J/ χc0 χc1 χc2 hc
L ½
½
J = L S
Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
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S-wave
P-wave
Exotics
JPC
c J/ χc0 χc1 χc2 hc
L ½
½
J = L S
Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
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S-wave
P-wave
D-wave
Exotics
JPC
c J/ χc0 χc1 χc2 hc
L ½
½
J = L S
Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
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S-wave
P-wave
D-wave
F-wave
Exotics
JPC
c J/ χc0 χc1 χc2 hc
L ½
½
J = L S
Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
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S-wave
P-wave
D-wave
F-wave
Exotics
JPC
c J/ χc0 χc1 χc2 hc
L ½
½
J = L S
Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
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S-wave
P-wave
D-wave
F-wave
part of G-wave Exotics
JPC
c J/ χc0 χc1 χc2 hc
L ½
½
J = L S
Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
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S-wave
P-wave
D-wave
F-wave
part of G-wave Exotics
JPC
c J/ χc0 χc1 χc2 hc
L ½
½
J = L S
Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
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c J/ χc0 χc1 χc2 hc
JPC
Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
in L=0, with gluonic 1+-, scale 1.2 – 1.3 GeV
Large overlap with op
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c J/ χc0 χc1 χc2 hc
JPC
in L=1, with gluonic 1+- Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
in L=0, with gluonic 1+-, scale 1.2 – 1.3 GeV
Large overlap with op
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Y(4260)?
c J/ χc0 χc1 χc2 hc
JPC
in L=1, with gluonic 1+- Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
in L=0, with gluonic 1+-, scale 1.2 – 1.3 GeV
Large overlap with op
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Y(4260)?
c J/ χc0 χc1 χc2 hc
X(3872)?
JPC
in L=1, with gluonic 1+- Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
in L=0, with gluonic 1+-, scale 1.2 – 1.3 GeV
Large overlap with op
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Y(4260)?
c J/ χc0 χc1 χc2 hc
X(3872)?
JPC
in L=1, with gluonic 1+- Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
in L=0, with gluonic 1+-, scale 1.2 – 1.3 GeV
Large overlap with op
Compare pattern of lightest hybrids with models [Dudek, PR D84, 074023 (2011)]
Bag model hybrids: 0- +, 1- +, 2- +, 1- -
Constituent gluon in L=1 (1+ -): 0- +, 1- +, 2- +, 1- -
Flux tube: 1+ +, 1- -, 0- +, 0+ -, 1- +, 1+ -, 2- +, 2+ -
Constituent gluon in L=0 (1- -): 0- - , 1- +, non-exotics
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Y(4260)?
c J/ χc0 χc1 χc2 hc
X(3872)?
JPC
in L=1, with gluonic 1+- Excited charmonia
M 400 MeV [HadSpec, JHEP 07 (2012) 126]
in L=0, with gluonic 1+-, scale 1.2 – 1.3 GeV
Large overlap with op
Compare pattern of lightest hybrids with models [Dudek, PR D84, 074023 (2011)]
Bag model hybrids: 0- +, 1- +, 2- +, 1- -
Constituent gluon in L=1 (1+ -): 0- +, 1- +, 2- +, 1- -
Flux tube: 1+ +, 1- -, 0- +, 0+ -, 1- +, 1+ -, 2- +, 2+ -
Constituent gluon in L=0 (1- -): 0- - , 1- +, non-exotics
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Light mesons
u and d quarks are degenerate – isospin symmetry
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Isoscalars (I = 0) e.g. , ’, ,
Isovectors (I = 1) e.g. , , a1 – only connected contributions
Light mesons
q
q'
u and d quarks are degenerate – isospin symmetry
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Isoscalars (I = 0) e.g. , ’, , QCD annihilation dynamics
Light isoscalar mesons
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ms = mu = md [SU(3) sym] – eigenstates are octet, singlet
Isoscalars (I = 0) e.g. , ’, , QCD annihilation dynamics
Light isoscalar mesons
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ms = mu = md [SU(3) sym] – eigenstates are octet, singlet
ms ≠ mu = md mixing
‘Ideal mixing’
Isoscalars (I = 0) e.g. , ’, , QCD annihilation dynamics
Light isoscalar mesons
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ms = mu = md [SU(3) sym] – eigenstates are octet, singlet
ms ≠ mu = md mixing
‘Ideal mixing’
In general
Isoscalars (I = 0) e.g. , ’, , QCD annihilation dynamics
Light isoscalar mesons
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ms = mu = md [SU(3) sym] – eigenstates are octet, singlet
ms ≠ mu = md mixing
‘Ideal mixing’
Experimentally , (1--) and f2(1270), f2’(1525) (2++) – close to ‘ideal’
, ’ (0-+) – closer to octet-singlet
In general
Isoscalars (I = 0) e.g. , ’, , QCD annihilation dynamics
Light isoscalar mesons
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ms = mu = md [SU(3) sym] – eigenstates are octet, singlet
ms ≠ mu = md mixing
‘Ideal mixing’
Experimentally , (1--) and f2(1270), f2’(1525) (2++) – close to ‘ideal’
, ’ (0-+) – closer to octet-singlet
In general
Can also mix with glueballs
Isoscalars (I = 0) e.g. , ’, , QCD annihilation dynamics
Light isoscalar mesons
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Light isoscalar mesons
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Operator basis doubled in size c.f. isovectors:
No glueball ops for now
Light isoscalar mesons
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Connected and disconnected contrib. required
q
q
q
q
q
q
Operator basis doubled in size c.f. isovectors:
No glueball ops for now
Same lattice setup as before
Light isoscalar mesons
NVIDIA
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Light mesons (isospin = 0 and 1)
Dudek, Edwards, Guo, CT (HadSpec), PR D88, 094505 (2013)
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S-wave
P-wave D-wave
Light mesons (isospin = 0 and 1)
Dudek, Edwards, Guo, CT (HadSpec), PR D88, 094505 (2013)
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Lightest hybrid mesons
Light mesons (isospin = 0 and 1)
Dudek, Edwards, Guo, CT (HadSpec), PR D88, 094505 (2013)
in L=0, with gluonic 1+-, scale 1.2 – 1.3 GeV
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Hidden-flavour mixing: - ’ = 46(1), f1 – f1’ = 27(2), 1-+ exotics 21(5)
Light mesons (isospin = 0 and 1)
Dudek, Edwards, Guo, CT (HadSpec), PR D88, 094505 (2013)
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Light mesons (isospin = 0 and 1)
Dudek, Edwards, Guo, CT (HadSpec), PR D88, 094505 (2013)
Volume and m dependence
m /
MeV
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Light mesons (isospin = 0 and 1)
Dudek, Edwards, Guo, CT (HadSpec), PR D88, 094505 (2013)
Volume and m dependence
m /
MeV
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Light mesons (isospin = 0)
Michael, Ottnad, Urbach (ETM), PRL 111, 181602 (2013)
Twisted mass quarks [Nf = 2+1+1] Extrapolate in a and m: : 551 8 6 MeV, ’: 1006 54 38 + 61 MeV : 46 1 3
[c.f. HadSpec 46(1) @ m 400 MeV]
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Morningstar and Peardon, PR D60, 034509 (1999)
• No fermion fields – computationally much less expensive
• Operators are closed loops of links, with different spatial symmetries
Glueballs in pure gauge theory (SU(3) Yang-Mills)
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Baryons
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Excited baryon spectroscopy – our approach
Energy eigenstates from:
Interpolating operators
Up to 2 derivs:
• Again, many ops in each channel with different spin and angular structures • Same lattice setup as before but only one volume (163 , Ls 1.9 fm)
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N and baryons [HadSpec, PR D84 074508, PR D85 054016]
4 5 3 1 2 3 2 1
2 2 1 1 1
N (I = 1/2, |S| = 0) (I = 3/2, |S| = 0)
Counting in lowest bands as expected in non. rel. quark model, SU(6) x O(3) (flavour x spin x space), no ‘freezing of d.o.f.’
m ≈ 400 MeV
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N and baryons [HadSpec, PR D84 074508, PR D85 054016]
4 5 3 1 2 3 2 1
2 2 1 1 1
N (I = 1/2, |S| = 0) (I = 3/2, |S| = 0)
Counting in lowest bands as expected in non. rel. quark model, SU(6) x O(3) (flavour x spin x space), no ‘freezing of d.o.f.’
Hybrid baryons Gluonic 1+- (8c) with qqq (8c), scale 1.2 – 1.3 GeV
Blue: large overlap with ops [Di, Dj] Fij
m ≈ 400 MeV
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Flavour structure of excited baryons
ms = mu = md SU(3) flavour symmetry, M 700 MeV, qqq
• Again, multiplicities in lowest bands as expected in non. rel. quark model SU(6) x O(3)
• No ‘freezing’ of d.o.f.
[HadSpec, PR D87, 054506 (2013)]
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Flavour structure of excited baryons
ms = mu = md SU(3) flavour symmetry, M 700 MeV, qqq
• Again, multiplicities in lowest bands as expected in non. rel. quark model SU(6) x O(3)
• No ‘freezing’ of d.o.f.
Gluonic 1+- (8c) with qqq (8c), scale 1.2 – 1.3 GeV
Large overlap with ops [Di, Dj] Fij
[HadSpec, PR D87, 054506 (2013)]
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Flavour structure of excited baryons
ms = mu = md SU(3) flavour symmetry, M 700 MeV, qqq
• Again, multiplicities in lowest bands as expected in non. rel. quark model SU(6) x O(3)
• No ‘freezing’ of d.o.f.
Gluonic 1+- (8c) with qqq (8c), scale 1.2 – 1.3 GeV
Large overlap with ops [Di, Dj] Fij
[HadSpec, PR D87, 054506 (2013)]
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Flavour structure of excited baryons (I = 0, |S| = 1) (I = 1, |S| = 1)
(I = 1/2, |S| = 2) (I = 0, |S| = 3)
8F
10F
1F
ms > mu = md
Broken SU(3) flav. sym.
m 400 MeV
[HadSpec, PR D87, 054506 (2013)]
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Padmanath et al (HadSpec), PR D91, 094502 (2015) Excited charm (cc) baryons
• Again, pattern in lowest bands consistent with non. rel. quark model
• Spectra don’t support quark-diquark picture • Also triply-charmed (ccc) baryons
m 400 MeV
Large overlap with ops [Di, Dj]
cc cc
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Excited bottom (bbb) baryons Meinel, PR D85, 114510 (2012)
• NRQCD action for b quark • Dynamical domain-wall u,d and s quarks • Two different lattice spacings • A number of m – extrapolate to physical m
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Excited bottom (bbb) baryons Meinel, PR D85, 114510 (2012)
• NRQCD action for b quark • Dynamical domain-wall u,d and s quarks • Two different lattice spacings • A number of m – extrapolate to physical m
m2 / GeV2
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Excited bottom (bbb) baryons Meinel, PR D85, 114510 (2012)
• NRQCD action for b quark • Dynamical domain-wall u,d and s quarks • Two different lattice spacings • A number of m – extrapolate to physical m
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Summary of lecture 2
• Background on hadron spectroscopy
• Excited hadron spectroscopy in lattice QCD
• Mesons
• Baryons
But so far we’ve neglected the fact that many of these hadrons are above the threshold for strong decay…
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Summary of lecture 2
Next time
• Background on hadron spectroscopy
• Excited hadron spectroscopy in lattice QCD
• Mesons
• Baryons
But so far we’ve neglected the fact that many of these hadrons are above the threshold for strong decay…
• Scattering, resonances, etc
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• Scattering, resonances, etc in lattice QCD
• Some examples:
• The ρ resonance in elastic scattering
• Coupled-channel K , K η scattering
• Some Ds mesons and charmonia
(I won’t review all lattice calculations)
Lecture 3
Hadron spectroscopy from lattice QCD
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Based on Klempt & Zaitsev, Phys Rep 454, 1 (2007)
Experimental light isovector (I = 1) mesons
...
Hadron Spectroscopy
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Based on Klempt & Zaitsev, Phys Rep 454, 1 (2007)
Experimental light isovector (I = 1) mesons
• Many states are resonances • Most of the puzzling states are
close to or above threshold(s)
...
Hadron Spectroscopy
– resonance
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Maiani & Testa (1990): scattering matrix elements cannot be extracted from infinite-volume Euclidean-space correlation functions (except at threshold).
Scattering in LQCD
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Continuous spectrum Infinite volume
Two hadrons: non-interacting
Scattering in LQCD
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Continuous spectrum
Discrete spectrum
Infinite volume
Finite volume
Two hadrons: non-interacting
L
periodic b.c.s (torus)
Scattering in LQCD
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Discrete spectrum
Finite volume
Two hadrons: interacting
L
c.f. 1-dim:
Continuous spectrum Infinite volume
periodic b.c.s (torus)
scattering phase shift
Scattering in LQCD
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Scattering amplitude, f
p′
p
Partial-wave expansion:
Scattering in an infinite volume – reminder
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Scattering amplitude, f
p′
p
Partial-wave expansion:
Scattering in an infinite volume – reminder
Elastic scattering and the phase shift, ℓ(E)
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Scattering – ‘Lüscher method’ Lüscher, NP B354, 531 (1991); extended by many others
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i, j label channels e.g. K, K
Scattering – ‘Lüscher method’ Lüscher, NP B354, 531 (1991); extended by many others
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i, j label channels e.g. K, K
Scattering – ‘Lüscher method’ Lüscher, NP B354, 531 (1991); extended by many others
Infinite-volume scattering t-matrix
Here
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effect of finite vol. i, j label channels e.g. K, K
Reduced symmetry ℓ mix
[all ℓ that subduce to a given lattice irrep () mix]
Scattering – ‘Lüscher method’ Lüscher, NP B354, 531 (1991); extended by many others
Infinite-volume scattering t-matrix
Here
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effect of finite vol. i, j label channels e.g. K, K
Reduced symmetry ℓ mix
[all ℓ that subduce to a given lattice irrep () mix]
Scattering – ‘Lüscher method’ Lüscher, NP B354, 531 (1991); extended by many others
Infinite-volume scattering t-matrix
Ignore
Integrals over momenta in loops sums over momenta Difference 1/L3 effects
effects
Here
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effect of finite vol. i, j label channels e.g. K, K
Reduced symmetry ℓ mix
[all ℓ that subduce to a given lattice irrep () mix]
Given t(E): solns finite-vol. spec. {E*}
We need: spectrum t-matrix
Scattering – ‘Lüscher method’ Lüscher, NP B354, 531 (1991); extended by many others
Infinite-volume scattering t-matrix
Here
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Elastic scattering
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E.g. Relativistic Breit-Wigner param. (mR, gR) for an isolated resonance
68
‘narrow’ ‘broad’
Ecm
Elastic scattering
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E.g. Relativistic Breit-Wigner param. (mR, gR) for an isolated resonance
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‘narrow’ ‘broad’
Ecm
ER E
Elastic scattering
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If assume only lowest ℓ relevant [near threshold t k2ℓ ]
can solve equ. for each energy level {E*} phase shift (E)
Alternatively parameterise t(E) and fit {E*lat} to {E*param}
Elastic scattering
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If assume only lowest ℓ relevant [near threshold t k2ℓ ]
can solve equ. for each energy level {E*} phase shift (E)
Alternatively parameterise t(E) and fit {E*lat} to {E*param}
Need many (multi-hadron) energy levels
• Single and multi-hadron ops
• Non-zero Pcm, different box sizes and shapes, twisted b.c.s, ...
Map out phase shift resonance parameters etc
Elastic scattering
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The resonance in scattering
P-wave (JPC = 1--, I = 1)
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Feng, Jansen, Renner (ETMC), [PR D83 094505 (2011)] Nf = 2, M ≈ 480, 420, 330, 290 MeV
Lang, Mohler, Prelovsek, Vidmar, [PR D84 054503 (2011)] Nf = 2, m ≈ 266 MeV
The resonance in scattering
P-wave (JPC = 1--, I = 1)
Aoki et al (PACS-CS), [PR D84 094505 (2011)] Nf = 2+1, m ≈ 410, 300 MeV
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‘single-meson’ ops.
and ops.
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Dudek, Edwards, CT (HadSpec), PR D87, 034505 (2013)
M 400 MeV, 3 volumes (L ≈ 2 – 3 fm, m L ≈ 4 - 6), as 0.12 fm, as/at 3.5
for various different P and
The resonance in scattering
Use many
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‘single-meson’ ops.
and ops.
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Dudek, Edwards, CT (HadSpec), PR D87, 034505 (2013)
M 400 MeV, 3 volumes (L ≈ 2 – 3 fm, m L ≈ 4 - 6), as 0.12 fm, as/at 3.5
for various different P and
The resonance in scattering
Use many
The l = 1 partial wave can mix with l = 3 and higher.
Find no significant signal for l = 3 and so assume l > 3 0 in this energy range.
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+ similar diagrams
[PR D87, 034505] I=1 – diagrams
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I=1 – spectra [PR D87, 034505]
L / as
at E
thresh.
(1)(-1)
thresh.
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I=1 – spectra [PR D87, 034505]
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M 400 MeV
[PR D87, 034505] The resonance in scattering
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[PR D87, 034505] The resonance in scattering
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M = 854.1 ± 1.1 MeV = 12.4 ± 0.6 MeV g = 5.80 ± 0.11
[M 400 MeV]
[PR D87, 034505] The resonance in scattering
c.f. experimentally M = 775.49 ± 0.34 MeV = 149.1 ± 0.8 MeV g ≈ 5.9
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The resonance – other calcs.
Pelissier, Alexandru, [PR D87 014503 (2013)] Nf = 2, M ≈ 300 MeV
M. Döring
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JP = 0+ , K0*(1430), ...
JP = 1- K*(892), ...
JP = 2+ K2*(1430), ...
Isospin = 1/2 Strangeness = 1
K, ηK (I=1/2) coupled-channel scattering
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Wilson, Dudek, Edwards, CT (HadSpec), PRL 113, 182001; PR D91, 054008
JP = 0+ , K0*(1430), ...
JP = 1- K*(892), ...
JP = 2+ K2*(1430), ...
‘single-meson’
+ K + K ops.
M = 391 MeV, MK = 549 MeV, Mη = 589 MeV; 3 volumes as before
Isospin = 1/2 Strangeness = 1
K, ηK (I=1/2) coupled-channel scattering
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K, ηK (I=1/2) – diagrams
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K, ηK (I=1/2) spectra P = [0,0,0] A1+
Neglect ℓ 4: only ℓ = 0 contributes
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Extension of Lüscher method to inelastic scattering: relate finite vol. energy levels to infinite vol. scattering t-matrix.
Underdetermined problem parameterize Ecm dependence of t-matrix and fit {E*lat} to {E*param}
K, ηK (I=1/2) coupled-channel scattering
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K-matrix param. – respects unitarity (conserve prob.) and flexible
Extension of Lüscher method to inelastic scattering: relate finite vol. energy levels to infinite vol. scattering t-matrix.
Underdetermined problem parameterize Ecm dependence of t-matrix and fit {E*lat} to {E*param}
K, ηK (I=1/2) coupled-channel scattering
Use various different params for K (see the paper for details)
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K, ηK (I=1/2) spectra P = [0,0,0] A1+
Neglect ℓ 4: only ℓ = 0 contributes
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Neglect ℓ 4: only ℓ = 0 contributes
K, ηK (I=1/2) spectra P = [0,0,0] A1+
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K, ηK (I=1/2) spectra
> 100 energy levels in total
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K (I=1/2): P-wave near threshold (well below ηK threshold)
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K (I=1/2): P-wave near threshold (well below ηK threshold)
Bound state just below K threshold
Breit-Wigner param. gR = 5.93(26) c.f. K* exp. = 5.52(16) MR = 933(1) MeV c.f. K* exp. ≈ 892 MeV
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K, ηK (I=1/2): S & P-waves
S-wave
(73 energy levels)
2/Ndof = 49.1/(61 - 6) = 0.89 2/Ndof = 15.0/(19 - 5) = 1.0
P-wave
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K, ηK (I=1/2): D-wave
Assume ℓ 3 negligible
Up to K threshold; neglect coupling to K
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K, ηK (I=1/2): t-matrix poles
Circle = on physical sheet Square = on unphysical sheet(s)
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Broad resonance c.f. K0
*(1430)
Narrow resonance c.f. K2
*(1430) (exp. B.R. to K 50%)
K, ηK (I=1/2): t-matrix poles
Circle = on physical sheet Square = on unphysical sheet(s)
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Broad resonance c.f. K0
*(1430)
Narrow resonance c.f. K2
*(1430) (exp. B.R. to K 50%)
Bound state just below threshold c.f. K*(892)
K, ηK (I=1/2): t-matrix poles
Circle = on physical sheet Square = on unphysical sheet(s)
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Broad resonance c.f. K0
*(1430)
Narrow resonance c.f. K2
*(1430) (exp. B.R. to K 50%)
Bound state just below threshold c.f. K*(892)
K, ηK (I=1/2): t-matrix poles
Circle = on physical sheet Square = on unphysical sheet(s)
Virtual bound state [pole below threshold on unphysical sheet(s)]
c.f. unitarised pt [Nebreda & Pelaez, PR D81, 034035 (2010)]
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Some other recent lattice calculations
First go at including multi-hadron operators in the open-charm sector: • Mohler et al [PR D87, 034501 (2012)] – 0+ D and 1+ D* resonances • Mohler et al [PRL 111, 222001 (2013)] – 0+ Ds(2317) below D K threshold • Lang et al [PRD 90, 034510 (2014)] – 0+ Ds(2317) and 1+ Ds1(2460), Ds1(2536)
... charmonium: • Ozaki, Sasaki [PR D87, 014506 (2013)] – no sign of Y(4140) in J/ • Prelovsek & Leskovec [PRL 111, 192001 (2013)] – 1+ + near/below DD* – X(3872)? • Prelovsek et al [PL B727, 172 ; PR D91, 014504 (2015)] – no sign of Z+(3900) in 1+-
• Chen et al [PR D89, 094506 (2014)] – find 1+ + I=1 is weakly repulsive
... light and strange meson: • Lang et al [PR D86, 054508 (2012), PR D88, 054508 (2013)]
– K in s-wave (0+) and p-wave (1-) including K* resonances • Lang et al [JHEP 04 (2014) 162] – channels relevant for a1(1260) & b1(1235)
... baryons: • Lang & Verduci [PR D87, 054502 (2013)] – N with JP=1/2- I=1/2 • Alexandrou et al [PR D88, 031501 (2013)] – different approach for (3/2+ I=3/2)
• Also see reviews e.g. from Lattice 2014 or 2013
(not complete list)
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Ds mesons Lang et al [PRD 90, 034510 (2014)]
(1) Clover [Nf = 2] (Hasenfratz et al), m = 266 MeV, m L 2.7, a 0.12 fm [small volume] (2) Clover [Nf = 2+1] (PACS-CS), m = 156 MeV, M L 2.3, a 0.09 fm [small volume]
JP = 0+ [relevant for Ds0(2317)]: 4 Ds + 3 DK ops
0+
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Ds mesons Lang et al [PRD 90, 034510 (2014)]
(1) Clover [Nf = 2] (Hasenfratz et al), m = 266 MeV, m L 2.7, a 0.12 fm [small volume] (2) Clover [Nf = 2+1] (PACS-CS), m = 156 MeV, M L 2.3, a 0.09 fm [small volume]
JP = 0+ [relevant for Ds0(2317)]: 4 Ds + 3 DK ops
0+
2
0
0 2
11)(cot pr
app
(1) a0 = -0.756(25) r0 = -0.056(31) m – (mK + mD) = -78.9(5.4)(0.8) MeV
(2) a0 = -1.33(20) r0 = 0.27(17) m – (mK + mD) = -36.6(16.6)(0.5) MeV
c.f. Ds0(2317) exp. ≈ -45.1 MeV
[Use two points per ensemble]
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Ds mesons Lang et al [PRD 90, 034510 (2014)]
(1) Clover [Nf = 2] (Hasenfratz et al), m = 266 MeV, m L 2.7, a 0.12 fm [small volume] (2) Clover [Nf = 2+1] (PACS-CS), m = 156 MeV, M L 2.3, a 0.09 fm [small volume]
JP = 1+ [relevant for Ds1(2460), Ds1(2536)]: 8 Ds + 3 D*K ops
1+
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Ds mesons Lang et al [PRD 90, 034510 (2014)]
(1) Clover [Nf = 2] (Hasenfratz et al), m = 266 MeV, m L 2.7, a 0.12 fm [small volume] (2) Clover [Nf = 2+1] (PACS-CS), m = 156 MeV, M L 2.3, a 0.09 fm [small volume]
JP = 1+ [relevant for Ds1(2460), Ds1(2536)]: 8 Ds + 3 D*K ops
1+ (1) a0 = -0.665(25) r0 = -0.106(37) m – (mK + mD*) = -93.2(4.7)(1.0) MeV
(2) a0 = -1.15(19) r0 = 0.13(22) m – (mK + mD*) = -43.2(13.8)(0.6) MeV
c.f. Ds1(2460) exp. ≈ -44.7 MeV
[Use two points per ensemble]
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Ds mesons Lang et al [PRD 90, 034510 (2014)]
(1) Clover [Nf = 2] (Hasenfratz et al), m = 266 MeV, m L 2.7, a 0.12 fm [small volume] (2) Clover [Nf = 2+1] (PACS-CS), m = 156 MeV, M L 2.3, a 0.09 fm [small volume]
JP = 1+ [relevant for Ds1(2460), Ds1(2536)]: 8 Ds + 3 D*K ops
1+ (1) a0 = -0.665(25) r0 = -0.106(37) m – (mK + mD*) = -93.2(4.7)(1.0) MeV
(2) a0 = -1.15(19) r0 = 0.13(22) m – (mK + mD*) = -43.2(13.8)(0.6) MeV
c.f. Ds1(2460) exp. ≈ -44.7 MeV
[Use two points per ensemble]
Also “Ds1(2536)” [see no coupling to D*K]
(1) m – (mK + mD*) = -53(12) MeV
(2) m – (mK + mD) = 56(11) or 50(8) MeV
c.f. Ds1(2536) exp. m – (mK + mD) ≈ 31 MeV
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Clover [Nf = 2] (Hasenfratz et al), m = 266 MeV, m L 2.7, a 0.12 fm [small volume] 92
Charmonium Prelovsek, Leskovec [PRL 111, 192001 (2013)]
X(3872) [JPC = 1++] near/below D D* threshold Look in I=0 (one vol, one Pcm)
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Clover [Nf = 2] (Hasenfratz et al), m = 266 MeV, m L 2.7, a 0.12 fm [small volume] 92
Charmonium Prelovsek, Leskovec [PRL 111, 192001 (2013)]
X(3872) [JPC = 1++] near/below D D* threshold Look in I=0 (one vol, one Pcm)
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Clover [Nf = 2] (Hasenfratz et al), m = 266 MeV, m L 2.7, a 0.12 fm [small volume] 92
Charmonium Prelovsek, Leskovec [PRL 111, 192001 (2013)]
X(3872) [JPC = 1++] near/below D D* threshold
fm1.05.0
fm4.07.1
2
11)(cot
0
0
2
0
0
r
a
pra
pp
[2 points]
Look in I=0 (one vol, one Pcm)
X - (c + 3 J/)/4 = 815(7) MeV exp = 803.1(2) MeV
X - (D + D*) = -11(7) MeV exp = -8.2(3) MeV [D+ D*- ] = -0.2(3) MeV [D0 D*0]
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Clover [Nf = 2] (Hasenfratz et al), m = 266 MeV, m L 2.7, a 0.12 fm [small volume] 92
Charmonium Prelovsek, Leskovec [PRL 111, 192001 (2013)]
X(3872) [JPC = 1++] near/below D D* threshold
fm1.05.0
fm4.07.1
2
11)(cot
0
0
2
0
0
r
a
pra
pp
[2 points]
Also look in I=1 no evidence for bound state/resonance
Look in I=0 (one vol, one Pcm)
X - (c + 3 J/)/4 = 815(7) MeV exp = 803.1(2) MeV
X - (D + D*) = -11(7) MeV exp = -8.2(3) MeV [D+ D*- ] = -0.2(3) MeV [D0 D*0]
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Various Zc+ structures in exp.
e.g. Zc+(3900), Zc
+(4020), Zc+(4200),
JPC = ??-
Look in JPC = 1+- I=1. Many two-meson and some ‘4-quark’ ops
• Weak interaction
• No Z+ candidate
Prelovsek et al [PR D91, 014504 (2015)]
Clover [Nf = 2] (Hasenfratz et al), m = 266 MeV, m L 2.7, a 0.12 fm [small volume]
Charmonium
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Various Zc+ structures in exp.
e.g. Zc+(3900), Zc
+(4020), Zc+(4200),
JPC = ??-
Look in JPC = 1+- I=1. Many two-meson and some ‘4-quark’ ops
• Weak interaction
• No Z+ candidate
Prelovsek et al [PR D91, 014504 (2015)]
Clover [Nf = 2] (Hasenfratz et al), m = 266 MeV, m L 2.7, a 0.12 fm [small volume]
Charmonium
No sign of any Zc+ up to 4.2 GeV
Only see non-interacting energies.
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Scattering channels with 3 or more hadrons?
• Much more complicated than 2-hadron scattering.
• No straightforward analogue of the determinant equation.
• Theoretical work is ongoing, e.g. • Polejaeva, Rusetsky [EPJA 48, 67 (2012)] • Kreuzer, Griesshammer [EPJA 48, 93 (2012)] • Roca, Oset [PR D85, 054507 (2012) ] • Briceno, Davoudi [PR D87, 094507 (2013)] • Hansen, Sharpe [PR D90, 116003 (2014)] • Meissner, Rios, Rusetsky [PRL 114, 091602 (2015)] • Hansen, Sharpe [1504.04248]
• No real applications yet.
• Another reason why calculating at physical m is challenging (particularly for light mesons): more >2 hadron channels open
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• Scattering, resonances, etc in lattice QCD
• Some examples:
• The ρ resonance in elastic scattering
• Coupled-channel K , K η scattering
• Some Ds mesons and charmonia
Summary of lecture 3
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• Significant progress in computing spectra of (excited) hadrons using lattice QCD in last few years – improved algorithms, clever techniques, more powerful computers and novel use of technology (e.g. GPUs)
• I’ve aimed to give some idea of what goes into these lattice calculations, some highlights of results and some interpretation.
• Calculating properties of unstable hadrons is currently a very active area – only recently have we been able to do this in practice. There is still a lot to do here.
• Masses only get you so far. We can also compute other properties of hadrons that probe their structure using lattice QCD: e.g. form factors, transition amplitudes. Again, there is interesting work going on, but that’s another set of lectures...
Conclusions
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