David Richards Jefferson Laboratory/Hadron Spectrum
Collaboration
"Excited-state spectrum, and the low-energy degrees of freedom of QCD
Cairns, 23 July, 2015
Hadron Spectrum Collaboration
Outline• Spectroscopy: theory and experiment • Lattice QCD Spectroscopy Recipe Book • Results and insight
– Light-meson spectroscopy and isoscalar – Decay constants – Charmonium – Baryons, and the search for gluons
• Strong decays • Summary
Spectroscopy
Spectroscopy and QCD – What are the key degrees of freedom describing the
bound states - protons, neutrons, pions,???? • How do they change as we vary the quark mass -
Charmonium? – What is the origin of confinement, describing 99% of
observed matter? – If QCD is correct and we understand it, expt. data must
confront ab initio calculations – What is the role of the gluon in the spectrum – search for
exotics
Classic means of determining underlying degrees of freedom in a theory. Probe the strong interaction and its underlying field theory Quantum Chromodynamics (QCD)
P = (�1)l+1
C = (�1)l+s
Meson Spectrum
5
• Exotic Mesons are those whose values of JPC are in accessible to quark model: 0+-, 1-+, 2+-
– Multi-quark states: – Hybrids with excitations of the flux-tube • Study of hybrids: revealing gluonic degrees of freedom of QCD. • Glueballs: purely, or predominantly, gluonic states
LS1
S2
Simple quark model (for neutral mesons) admits only certain values of JPC
Baryon Spectroscopy• No baryon “exotics”, ie quantum numbers not accessible with
simple quark model; but may be hybrids! • Nucleon Spectroscopy: Quark model masses and amplitudes –
states classified by isospin, parity and spin.
• Missing, because our pictures do not capture correct degrees of freedom?• Do they just not couple to probes?
|q3>
|q2q>
Real%parts%of%N*%pole%values%%�
PDG� Ours�
N*%with%3*,%4*� 18�16�
N*%with%1*,%2*� 5�
PDG%4*�
PDG%3*�
Ours�
EBAC: Kamano, Nakamura, Lee, Sato - 2012
Quantum Chromodynamics (QCD)QED QCD
Photon, γ Gluons, GCharged particles, e, µ, u, d,… Quarks: u, d, s, c, b, t
Photon is neutral Gluons carry color chargeTheory is non-Abelian
αe =1/137 αs ~O(1)
Asymptotic Freedom Pert. Theory
Infrared Slavery Lattice QCD
Gluons in three-jet event
Gluons in spectrum!
Construct matrix of correlators with judicious choice of operators
Variational Method
Delineate contributions using variational method: solve
Eigenvectors, with metric C(t0), are orthonormal and project onto the respective states
Subleading terms ➙ Excited states
C
ij
(t, 0) =1
V3
X
~x,~y
hOi
(~x, t)O†j
(~y, 0)i =X
N
Z
N⇤i
Z
N
j
2EN
e
�EN t
ZNi ⌘ hN | O†
i (0) | 0i
C(t)v(N)(t, t0) = �N (t, t0)C(t0)v(N)(t, t0).
�N (t, t0) �! e�EN (t�t0),
v(N0)†C(t0)v
(N) = �N,N 0
ZNi =
p2mNemN t0/2v(N)⇤
j Cji(t0).
C(t) = ⇤0 | O(t)O(0)† | 0⌅ �⇥ e�Et
�2(t) ⇤�⌅0 | |O(t)O(0)†|2 | 0⇧ � C(t)2
��⇥ e�2m⇡t
ChallengesTo appreciate difficulty of extracting excited states, need to understand signal-to-noise ratio in two-point functions. Consider correlation function:
Then the fluctuations behave as
Signal-to-noise ratio degrades with increasing E - Solution: anisotropic lattice with lattice spacing at < as
DeGrand, Hecht, PRD46 (1992)
Cubic symmetry of lattices
a2
MEMT2
M2
For heavy quarks m at << 1
Glueball Spectroscopy - I
Observe emergence of degeneracies
Morningstar, Peardon 97,99Triple-gluon vertex - Pure Yang-Mills spectrum. Predicts existence of bound states.
S�G[U ] =
�
Nc⇥g
8<
:X
x,s>s0
5
3u4s
Pss0 �1
12u6s
Rss0
�+
X
x,s
4
3u2su
2t
Pst �1
12u4su
2t
Rst
�9=
;
S�F [U,⇧,⇧] =
Xx⇧(x)
1
ut
(utm0 + Wt +
1
⇥f
X
s
Ws�
1
2
"1
2
✓⇥g⇥f
+1
⇤
◆1
utu2s
X
s
⌅tsFts +1
⇥f
1
u3s
X
s<s0
⌅ss0 Fss0
#)⇧(x).
�g = ⇤0
�f = ⇤0/⇥� = 3.5
Spectroscopy with Quarks• Anisotropic lattices - to precisely resolve energies • Variational method - with sufficient operator basis to delineate states • Identification of spin - Many Values of Lattice Spacing?
Anisotropic fermion action Edwards, Joo, Lin, PRD78 (2008)
Two anisotropy parameters to tune, in gauge and fermion sectors
Dispersion Relation as ' 0.12 fm
at ' 0.035 fm
Anisotropic Clover Generation - I
H-W Lin et al (Hadron Spectrum Collaboration), PRD79, 034502 (2009 )
Challenge: setting scale and strange-quark mass
Express physics in (dimensionless) (l,s) coordinates
Omega
Lattice coupling fixed
Tuning performed for three-flavor theory
Proportional to ms to LO ChPT
Proportional to ml to LO ChPT
Anisotropic Clover – II
Low-lying spectrum: agrees with experiment to 10%
No chiral extrapolations - resonances
m⇡ � 400 MeV
�(⇥x, t)�DiDj . . .�(⇥x, t) !D
m=�1 = ip2
⇣ !D
x
� i !D
y
⌘
!D
m=0 = i !D
z
!D
m=+1 = � ip2
⇣ !D
x
+ i !D
y
⌘.
Variational Method: Meson OperatorsAim: interpolating operators of definite (continuum) JM: OJM
h0 | OJM | J 0,M 0i = ZJ�J,J 0�M,M 0
14
Starting point
Introduce circular basis:
(�⇥D[1]J=1)
J,M =X
m1,m2
⌦1,m1; 1,m2
��J,M↵��m1
!Dm2�.
Use projection formula to find subduction under irrep. of cubic group - operators are closed under rotation!
Straighforward to project to definite spin - for example J = 0, 1, 2
Action of RIrrep, Row Irrep of R in Λ
Correlation functions: Distillation• Use the new “distillation” method. • Observe
• Truncate sum at sufficient i to capture relevant physics modes – we use 64: set “weights” f to be unity
• Meson correlation function
• Decompose using “distillation” operator as
Eigenvectors of Laplacian
Includes displacements
Perambulators
M. Peardon et al., PRD80,054506 (2009)
Momentum Projection at Source and Sink!
Distillation - II
• Meson correlation functions N3
• Baryon correlation functions N4
• Stochastic sampling of eigenvectors - stochastic LaPHMorningstar et al, Phys.Rev.D83:114505,2011
• Alternative idea: simpler orthonormal basis for the smearing function L
Colour-wave basis
X
i
�
⇤i
(x)�i
(y) = �(x� y),X
x
�
⇤i
(x)�j
(x) = �
ij
�
i
(x) = e
�ipx
�
s,s
0�
c,c
0
Z.Brown, K.Orginos, arXiv:1210.1953
Severely constrains baryon lattice sizes
Distillation - III
64 eigenvectors on Isotropic Lattice 16 eigenvectors on Isotropic Lattice
Non-local Pion Correlation Function Effective Mass
C�ij =
1
dim(�)
X
�
�0 | O[J]i(�)�O
[J]j(�)� | 0⇥
Hadspec collab. (dudek et al), 1004.4930, PRD82, 034508
Identification of Spin
0.2 0.4 0.6 0.8 1.0
Operators know their parentage
Exploit to determine spins
{
163 203
Isovector Meson Spectrum - I
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Exotic
Isovector spectrum with quantum numbers reliably identified
Dudek et al, PRL 103:262001 (2009)
Nf = 3 theory - three mass-degenerate “strange” quarks
Interpretation of Meson Spectrum
In each Lattice Irrep, state dominated by operators of particular J
D[2]J=1
Vanishes for unit gauge field
ZNi =
p2mNemN t0/2v(N)⇤
j Cji(t0).
look at the ‘overlaps’
hybrid?
x x x x
ground state is dominantly
x x x x
1st excited state is dominantly
with some
x x x x
2nd excited state is dominantly
with some
x x x x
3rd excited state is dominantly
hybrid?
with some
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Anti-commutator of covariant derivative: vanishes for unit gauge!
Dudek, arXiv:1106.5515
1--
Use lattice QCD to build phenomenology of bound states
• Spin-identified single-particle spectrum: states of spin as high as four • Hidden flavor mixing angles extracted - except 0-+, 1++ near ideal mixing • First determination of exotic isoscalar states: comparable in mass to isovector
Isoscalar Meson Spectrum
22
0.5
1.0
1.5
2.0
2.5
exotics
isoscalar
isovectorYM glueball
negative parity positive parity
J. Dudek et al., PRD73, 11502
Diagonalize in 2x2 flavor space
C =
✓�C�� + 2D��
⇥2D�s
⇥2Ds� �Css +Dss
◆.
Dudek et al, arXiv:1309.2608, arXiv:0909.0200
23
Charmonium
163
243
0-+0-+ 1--1-- 2-+2-+ 2--2-- 3--3-- 4-+4-+ 4--4-- 0++0++ 1+-1+- 1++1++ 2++2++ 3+-3+- 3++3++ 4++4++ 1-+1-+ 0+-0+- 2+-2+- 1-+1-+
DDDD
DsDsDsDs
0.50
0.55
0.60
0.65
0.70
0.75
0.80
a tm
Exotics
Volume-dependence small ➨ quote results at larger volume
Operator construction follows light-quark Ignore annihilation contributions Charm quark mass set from ηc with scale set using Ω
Liuming Liu et al, arXiv: 1204.5425
24
Charmonium - II
DDDD
DsDsDsDs
0-+0-+ 1--1-- 2-+2-+ 2--2-- 3--3-- 4-+4-+ 4--4-- 0++0++ 1+-1+- 1++1++ 2++2++ 3+-3+- 3++3++ 4++4++ 1-+1-+ 0+-0+- 2+-2+-0
500
1000
1500
M-Mh cHMe
VL
Appearance of multiplets from n 2s+1LJ quark potential model
S-wave
D-wave
P-wave
DDDD
DsDsDsDs
0-+0-+ 1--1-- 2-+2-+ 1-+1-+ 0++0++ 1+-1+- 1++1++ 2++2++ 3+-3+- 0+-0+- 2+-2+-0
500
1000
1500M-Mh cHMe
VL
1--1-- 0-+0-+ 1-+1-+ 2-+HT2L2-+HT2L 2-+HEL2-+HEL
1,3HybJ1,3HybJ 8pNR, rNR< ¥ DJ=1
@2D8pNR, rNR< ¥ DJ=1@2D
0
2
4
6
8
Z
Hybrid supermultiplet
Pseudoscalar Decay Constants
• Expectation from WT identity
25
f⇡N ⌘ 0, N � 0
e.g. Chang, Roberts, Tandy, arXiv:1107.4003
• Compute in LQCD
McNeile and Michael, hep-lat/0607032
fX =1
m2X
mqh0 | ⇡ | Xi
Pseudoscalar Decay - II
26
• Axial-vector current mixes on lattice E. Mastropas, DGR, PRD(2014)
AI4 = (1 +mat⌦m)
AU
4 � 1
4(⇠ � 1)at@4P
�
C
A4,N (t) =1
V3
X
~x,~y
h0 | A4(~x, t)⌦†N
(~y, 0) | 0i �! e
�mN t
m
N
f
⇡N
where ⌦N =p2mNe�mN t0/2v(N)
i Oi
27
• Strong suppression for second HYBRID state
0
0.2
0.4
0.6
0.8
1
1.2
0.0694(2) 0.274(3) 0.367(3) 0.42(1) 0.499(5)
a0xD3_J131_J0__J0_A1a1xD3_J131_J1__J0_A1
b1xD1_J1__J0_A1b1xD3_J130_J1__J0_A1b1xD3_J132_J1__J0_A1b1xD3_J132_J3__J4_A1
pion_2xD0_J0__J0_A1pion_2xD2_J0__J0_A1
pionxD0_J0__J0_A1pionxD2_J0__J0_A1
rho_2xD2_J1__J0_A1rhoxD2_J1__J0_A1
Look at overlaps with different classes of operators
Matrix elements key to understanding structure of state
Construct basis of 3-quark interpolating operators in the continuum:
Subduce to lattice irreps:
163 ⇥ 128 lattices m⇡ = 524, 444 and 396 MeV
Hu
Excited Baryon Spectrum - I
28
O[J]n�,r =
X
M
SJ,Mn�,rO[J,M ] : � = G1g/u, Hg/u, G2g/u
⇣NM ⌦
�32
��1M⌦D[2]
L=2,S
⌘J=72
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
R.G.Edwards et al., arXiv:1104.5152
Observe remarkable realization of rotational symmetry at hadronic scale: reliably determine spins up to 7/2, for the first time in a lattice calculation
Continuum antecedents
“Flavor” x Spin x Orbital
Excited Baryon Spectrum - II
29
Broad features of SU(6)xO(3) symmetry. Counting of states consistent with NR quark model. Inconsistent with quark-diquark picture or parity doubling.
[70,1-] [70,1-]
[56,0+]
[56,0+]
N 1/2+ sector: need for complete basis to faithfully extract states
[70, 0+], [56, 2+], [70, 2+], [20, 1+]
Roper Resonance
30
Kamleh et al., arXiv:1212.2314
Look at Radial wave function ⇒2S state
D[2]l=1,M
Hybrid Baryon Spectrum
31
Original analysis ignore hybrid operators of form
1.0
1.5
2.0
2.5
3.0
Dudek, Edwards, arXiv:1201.2349
Letter of Intent to PAC43
32
Hybrid Roper?
33
Expectation for gluonic Roper
CLAS
Putting it Together
34
0
500
1000
1500
2000
Subtract ρ Subtract N
Common mechanism in meson and baryon hybrids: chromomagnetic field with Eg ∼ 1.2 - 1.3 GeV
Flavor Structure
35
One derivative
Two derivative
36
C�ij =
1
dim(�)
X
�
�0 | O[J]i(�)�O
[J]j(�)� | 0⇥
Block-diagonal in spin-flavor
Examine Flavor structure of baryons constructed from u, d s quarks.
• Can identify predominant flavor for each state: Yellow (10F), Blue (8F), Beige (1F). • SU(6) x O(3) Counting • Presence of “hybrids” characteristic across all +ve parity channels: BOLD Outline
R. Edwards et al., Phys. Rev. D87 (2013) 054506
38
Recent extension to doubly-charmed baryons
Padmanath et al., arXiv:1502.01845
The elephant in the room...
1
1 1 1 1
1
2 1
1 4 4 2 1
0.4
0.6
0.8
1.0
1.2
1.4
1.61
1 11 1 1 1
1 2 2 2 1
1
2 1
1 4 4 2 1
2 1 1 1
2 1
1
1 1
1 1
1
2 2
1
1 1
1 1
1
2 2
2 1 1 1
1 1
1 1 1
1 2 2 2 1
1 1 2 1
1 1 1 1
1 1 1 1
1 2 2 2 1
1 1 2 1
1 1 1 1
2 4 4 4 2
2 4 5 3 1
11
1 1
1 1 1
1 1
1
1 1
1 1 1
1 1
1 1
1 1 1 2 1
1
2 1
1 4 4 2 1
1
2 1
1 4 4 2 1
2 1 1 1
2 1
1 1
2 2 1 1
1 4 4 2 1
1 1
2 2 1 1
1 4 4 2 1
3 2 2 1
1 4 3 2 1
2 7 7 4 2
1 5 4 3 2
1 4 3 2 1
2 7 7 4 2
1 5 4 3 2
1 4 3 2 1
4 14 14 8 4
1
2 1
3 2 1 1
1
2 1
3 2 1 1
2 1 1 1
1
2 3 3 3 1
1
2 3 3 3 1
4 11 11 8 4
1
2 3 3 3 1
1
2 3 3 3 1
4 11 11 8 4
2 1 11
2 5 6 4 2
2 1 11
2 5 6 4 2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Meson spectrum on two volumes: dashed lines denote expected (non-interacting) multi-particle energies.
Calculation is incomplete.
Allowed two-particle contributions governed by cubic symmetry of volume
States unstable under strong interactions
Luescher: energy levels at finite volume ↔ phase shift at corresponding k
O�,�⇥⇥ (|�p|) =
X
m
S⇤,m�,�
X
p
Y m⇤ (p)O⇥(�p)O⇥(��p)
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Momentum-dependent I = 2 ππ Phase Shift
40
Dudek et al., Phys Rev D83, 071504 (2011)
Operator basis
Total momentum zero - pion momentum ±p
41
Resonant I = 1 ππ Phase Shift
0
20
40
60
80
100
120
140
160
180
800 850 900 950 1000 1050
Dudek, Edwards, Thomas, Phys. Rev. D 87, 034505 (2013)
Feng, Renner, Jansen, PRD83, 094505 PACS-CS, PRD84, 094505 Alexandru et al Lang et al., PRD84, 054503
42
First - and Successful - inelastic
Dudek, Edwards, Thomas, Wilson, PRL, PRD
deth�ij�JJ 0 + i⇢i t
(J)ij (Ecm)
⇣�JJ 0 + iM~P⇤
JJ 0(piL)⌘ i
= 0
tii =(⌘e2i�i�1)
2i⇢i, tij =
p1�⌘2 ei(�i+�j)
2p⇢i ⇢j
Parametrized as phase shift + inelasticity
Pole positions in complex plane
Inelastic in ππ KK channelWilson, Briceno, Dudek, Edwards, Thomas, arXiv:1507.02599
Inelastic Threshold
Decreasing Pion Mass
Summary• Spectroscopy of excited states affords an excellent theatre in which to study
QCD in low-energy regime. • Determining the quantum numbers and the study of the “single-hadron” states
a solved problem • Lattice calculations used to construct new “phenomenology” of QCD
– Quark-model like spectrum, common mechanism for gluonic excitations in mesons and baryons. LOW ENERGY GLUONIC DOF
• Prediction - there are exotics in a range accessible to the 12 GeV Upgrade of Jefferson Lab!
• Next step for lattice QCD: – Calculations at closer-to-physical pion masses - isotropic lattices. – Baryons a challenge…. …New improved methods in progress…
• Properties - radiative transitions, form factors. Essential both to predict production rates and to interpret structure
Variational Method + DistillationSingle “distilled” correlator
Fit to
C(t) = Ae�m0t+Be�m0t
and plot C(t)/e�m0t
Fit to �0(t, t0) = (1�A)e�m0(t�t0)+Ae�m0(t�t0)
and plot C(t)/e�m0(t�t0)
Reduced contribution of excited states24
3 ⇥ 64 isotropic lattice a2 ' 0.75 fm, SU(3)
…And for Rho