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