Exotic and excited-state meson spectroscopy and radiative transitions from lattice QCD

Christopher Thomas, Jefferson Lab

In collaboration with: Jo Dudek, Robert Edwards, David Richardsand the Hadron Spectrum Collaboration

QNP 2009, Beijing, China

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Outline

• Introduction

• Light meson spectrum

• Charmonium radiative transitions

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Overview

Photoproduction at GlueX (JLab 12 GeV upgrade)

Light mesons

GlueX (JLab), BESIII, PANDA

Exotics (1-+, ...)?

Spectrum and photocouplings

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Spectroscopy on the lattice

Calculate energies and matrix elements (Z) from correlation functions of meson interpolating fields

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Variational Method

Consider a large basis of operators matrix of correlators Cij(t)

Generalised eigenvector problem:

Eigenvalues energies

Eigenvectors optimal linear combination of operators to overlap on to a state

Z(n) related to eigenvectors

(t >> t0)

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Spin and operator construction

On a lattice, 3D rotation group is broken to Octahedral Group

2D Example

Eigenstates of angular momentum are

On a lattice, the allowed rotations are + /2

Can’t distinguish e.g. J = 0 and J = 4

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Spin and operator construction

Construct operators which only overlap on to one spin in the continuum limit

On a lattice, 3D rotation group is broken to Octahedral Group

Irrep A1 A2 T1 T2 E

dim 1 1 3 3 2

cont. spins 0,4,6, ... 3,6,7, ... 1,3,4, ... 2,3,4, ... 2,4,5, ...

‘Subduce’ operators on to lattice irreps:

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Light Meson Spectroscopy

Dudek et al, arXiv:0909.0200

• Results here are with three degenerate ‘light’ quarks:

Exact SU(3) symmetry – all mesons in octet (, K, 8) are degenerate (singlet, 1, has different mass)

• Only connected diagrams – Isovectors (I=1) only

• Mp = 833 MeV

• Unquenched calculation (dynamical fermions)

• Show here mostly results with volume = 163 (Ls 1.96 fm)

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1-+

4--

4-+

J-+ J--

Exotic

First J = 4 on lattice!

mπ

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4++

0+- 2+-

J+- J++

More exotics

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J--

13S1

23S1

13D1

Vector Hybrid??

Z valuesMas

s /

MeV

This operator commutator of two covariant derivatives

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Z values – spin 2

J--

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What about multiparticle states?

Expect two-meson states above 2m

Where are they?

2m 1.7 GeV

()L=1 0-- 2.4 GeV (with min mom allowed on lattice)

Momentum constrained to discrete values on a lattice – discrete spectrum of multiparticle states

J--

Mas

s /

MeV

Preli

minary

163 203

T1--

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Charmonium radiative transitions

Meson – Photon coupling

BABAR, Belle, BES, CLEO-c

Exotic 1-

+ ?

Dudek, Edwards & CT, PR D79 094504 (2009)

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Same scale as many measured conventional charmonium transitions BUT very large for an M1 transition

Exotic 1-+ – Vector 1--

• Usually M1 spin flip 1/mc suppression

• Spin-triplet hybrid M1 transition without spin flip not suppressed

M1 multipole dominates

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More charmonium results

Vector (1--) hybrid candidate:

Vector – Psuedoscalar

Scalar – Vector

Axial – Vector

Dudek, Edwards & CT, PR D79 094504 (2009)

Tensor – Vector transitions

Identify 13P2, 13F2, 23P2 tensors from hierarchy of multipoles E1, M2, E3

Quenched, only disconnected diagrams, one volume and one lattice spacing

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Summary and Outlook

• Method successful – first calc. of excited state rad. trans. on lattice• Hybrid photocoupling is large: • Non-exotic vector hybrid candidate• Comparison with quark models

• Our first results on light mesons – technology and method work• Spin identification is possible• First spin 4 state extracted and confidentially identified on lattice• Exotics (and non-exotic hybrids?)• Ongoing work: different masses and volumes,

multi-meson operators, photocouplings ...

Charmonium

Lighter mesons

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Extra Slides

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Photocouplings on the Lattice

Calculate from 3-point correlators:

These couplings are what we want.Parameterize in terms of multipoles (like form factors)

nm

Known from 2-point analysis

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More on 3-points

Source (ti): Only (smeared) local operators ( = 5, i, 1) Momentum selected automatically from momentum cons.

Local vector current: j (ti < t < tf)

Sink (tf): Use ‘best’ operator, O(n), from 2- point analysisSpecify pf (usually pf = 0 0 0 )

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Charmonium radiative transitions

Electric (Ek), Magnetic (Mk)

Dipole E1, Quadrupole E2, Octupole E3

Multipoles

Ji = Jf k (k > 0)

No parity change: Ek for even k, Mk for odd kParity change: Ek for odd k, Mk for even k

Experimentally measure multipoles at Q2 = 0

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Spin on the lattice

Rotation group:

• infinite number of irreducible representations (irreps)• J = 0, 1, 2, 3, 4, ...

Lattice:

• broken to octahedral group (group of rotations of a cube)• finite number of irreps: A1, A2, E, T1, T2 (+ others for half-integer)

Irrep A1 A2 T1 T2 E

dim 1 1 3 3 2

cont. spins 0,4,6, ... 3,6,7, ... 1,3,4, ... 2,3,4, ... 2,4,5, ...

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Lattice systematics – charmonium

• Quenched anisotropic lattice (as/at = 3)

• Clover fermion action

• Vector current three-point functions from sequential source technology

• Only connected diagrams (OZI justification?)

• Fixed lattice spacing, at-1 = 6.05 GeV 0.033 fm

• Fixed volume (123 x 48) (Ls 1.2 fm)

• Extrapolation to Q2 = 0

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Lattice systematics – light mesons

• UNQUENCHED anisotropic lattice (as/at = 3.5)

• Two light clover quarks and one strange quark• in first results strange and light degenerate (other masses

underway)

• Only connected diagrams – isovector states

• Fixed lattice spacing, at-1 = 5.62(4) GeV 0.035 fm

• First volume = 163 x 128 (Ls 1.96 fm) (other volumes underway)

• First results are with Mp = 833 MeV (other masses underway)

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Charmonium radiative transitions

Lots more results and details in paper: Dudek, Edwards & CT, PR D79 094504 (2009)

Only a brief mention here...

• Caveats:• Quenched (no quark loops; no light quarks at all) • One lattice spacing and volume• Only connected diagrams

Also: Dudek et al PR D77 034501 (2008) ; Dudek & Rrapaj PR D78 094504 (2008)

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Radiative Transition Results

• Photon only couples to quark and not antiquark

• Don’t explicitly include the quark electric charge

• Actually compute

• Plot in terms of temporal lattice spacing (at-1 = 6.05 GeV, from static pot.)

• Constant term in t dependence; fit Q2 form (or similar):

•

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Tensor 2++ – Vector 1--

PDG08: 406(31) keV

• Same hierarchy as expected:

• Ratio |M2/E1| is considerably larger than experiment

|E1(0)| > |M2(0)| >> |E3(0)|

Quark models (13P2) 290 – 420 keV

E1, M2, E3

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Tensor 2++ – Vector 1--

Completely different hierarchy! |E3(0)| > |M2(0)| , |E1(0)|

E1, M2, E3

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Tensor 2++ – Vector 1--

Reverted to expected hierarchy: |E1(0)| > |M2(0)| >> |E3(0)|

Quark models (23P2) 50 – 80 keV

E1, M2, E3

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Tensor 2++ – Vector 1--

Interpretation: single quark transition model

In general: E1 , M2 , E3 (k = 1,2,3)

If only a single quark is involved (3P2 3S1):j = 1/2 j = 1/2 k = 1,2 only and E3 = 0|E1(0)| > |M2(0)| >> |E3(0)|

If instead tensor is 3F2 (3F2 3S1):j = 5/2 j = 1/2 k = 2,3 only and E1 = 0|E3(0)| > |M2(0)| >> |E1(0)|

Interpretation:

cc2 – 13P2

c’c2 – 13F2

c’’c2 – 23P2

Supported by spectrum analysis

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Vector 1-- – Pseudoscalar 0-+

Spectrum results (Dudek et al PR D77 034501 (2008) ):

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Vector 1-- – Pseudoscalar 0-+Only M1

Quark model: • spin flip ( 1/mc) gives suppression• ’ is 23S1 11S0 – further suppressed

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Vector 1-- – Pseudoscalar 0-+

Quark model: 13D1 11S0 has same leading Q2 behaviour as 23S1 11S0

Only M1

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Vector 1-- – Pseudoscalar 0-+

Much larger than other1-- 0-+ M1 trans

Analogous to 1-+ hybrid to vector trans:M1 with no spin flip

Spectrum analysis suggests a vector hybrid (spin-singlet)

c.f. flux tube model 30 – 60 keV

Only M1

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Vector 1-- – Pseudoscalar 0-+

Loops

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Scalar 0++ – Vector 1--Only E1

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Axial 1++ – Vector 1--

c.f. PDG08: 320(20) keVc.f. quark models (13P1) 215 – 314 keVExpected hierarchy: |E1(0)| > |M2(0)|

E1, M2

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Axial 1++ – Vector 1--

c.f. quark models (23P1) 14 – 71 keV

E1, M2

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More charmonium results

Exotic 1-+:

Very large for M1 transition (typical 2 keV)

Vector (1--) hybrid candidate:

Vector – Psuedoscalar

Scalar – Vector

Axial – Vector

Tensor – Vector

Dudek, Edwards & CT, PR D79 094504 (2009)

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