Studying hadron excitations with latticeQCD

Mike PeardonSchool of Mathematics, Trinity College Dublin, Ireland

ECT? Trento 14th January 2014

Overview

• Motivation and challenges• Excited hadrons• Scattering and resonances

• Progress and new directions• Framework for measurements - distillation• Light mesons• Charmonium• Scattering

• Summary and outlook

Motivation and challenges

Hadrons beyond the quark model

• QCD allows for states beyond the simple mesons andbaryons, including those with intrinsic gluonic excitationssuch as hybrids and glueballs.

• The “charmonium renaissance” found many newresonance above the open-charm threshold, some arerather narrow.• X(3872) — 1++ resonance close to DD∗ threshold and verynarrow (Γ < 3MeV)

• Z+(4430) – charged state so cannot be cc

• Where are the gluonic excitations? A few hybrid mesonand glueball candidates. Very little consensus.

• SigniVcant experimental activity — new data will becoming soon...

The GlueX experiment at JLab

• 12 GeV upgrade toCEBAF ring

• New experimental hall:Hall D

• New experiment: GlueX

• Aim: photoproduce mesons, in particular hybrids• Expected to start operation this year?

Panda@FAIR, GSI

• Extensive new construction atGSI Darmstadt

• Expected to start operation201x?

PANDA: Anti-Proton ANnihilationat DArmstadt• Anti-proton beam from FAIRon Vxed-target.

• Physics goals includecharmonium and searches forhybrids and glueballs.

Can Lattice QCD study these states?

• Lattice calculations of glueballs and hybrids have beencarried out since 1980s. Focus was ground-states.

• Most experimental candidates for non-QM states arehigh-lying resonances, “hidden inside” a simpler QMspectrum. In QCD, states mix.

• For lattice calculations to be relevant, need:• Precision data on highly-excited states, including reliablespin resolution.

• scattering information, including data above inelasticthresholds

• to study QCD with quarks

• Most of these demands require new ideas and methods toemerge

Excitations and the variational method

• Suppose for a particular set of quantum numbers, we candeVne a basis of interpolating Velds φa for a = 1 . . .N

Variational method

If we can measure Cab(t) = 〈0|φa(t)φ†b(0)|0〉 for all a, b and

solve generalised eigenvalue problem:

C(t) v = λC(t0) v

then

limt−t0→∞

λk = e−Ekt

• v contains overlaps of operators onto states 〈0|φa|k〉• For this to be practical, we need: a ‘good’ basis set thatresembles the states of interest

Scattering

Scattering matrix elements not directly accessible from Eu-clidean QFT [Maiani-Testa theorem]

• Scattering matrix elements:asymptotic |in〉, |out〉 states.〈out |eiHt| in〉 → 〈out |e−Ht| in〉

• Euclidean metric: project ontoground-state

In

States

Out

States

• Lüscher’s formalism: information on elastic scatteringinferred from volume dependence of spectrum

• Requires precise data, resolution of two-hadron andexcited states.

Hadrons in a Vnite box: scattering• On a Vnite lattice with periodic b.c., hadrons have quantisedmomenta; p = 2π

L

{nx, ny, nz

}• Two hadrons with total P = 0 have a discrete spectrum• These states can have same quantum numbers as those created byqΓq operators and QCD can mix these

• This leads to shifts in thespectrum in Vnite volume

• This is the same physics thatmakes resonances in anexperiment

• Lüscher’s method - relateelastic scattering to energyshifts

Toy model

H =

(m gg 4π

L

)

6 8 10 12 14 16 18 20

mL

0

0.5

1

1.5

2

E/m

g/m=0.1

g/m=0.2

Progress and new directions

New measurement framework — distillation

• Better creation operators are smeared• In lattice calculation, don’t have direct access to quarkVelds, so using complicated operators tricky

• Observation: good smearing operators reduce eUectivedegrees of freedom on a time slice by many orders ofmagnitude.

• Re-deVne smearing as projection operator into low-rankvector space of smooth quark Velds. This enables eUectivecalculations of many otherwise inaccessible correlationfunctions

�(x, y) =

NV∑k=1

vk(x) v∗k(y)

• Problem — expensive, with steep dependence on volume

Spin on the lattice

• Lattice states classiVed by quantumletter, R ∈ {A1,A2, E, T1, T2}.

• Start with continuum: ψΓDiDj . . . ψ andsubduce O(3) irreps→ Oh

• Example:Φij = ψ

(γiDj + γjDi − 2

3δijγ · D)ψ

• Lattice: substitute D→ Dlatt

• Now have a reducible representation:

ΦT2 = {Φ12,Φ23,Φ31} & ΦE ={

1√2(Φ11−Φ22),

1√6(Φ11+Φ22−2Φ33)

}• Look for signature of continuum symmetry:

〈0|ΦT2|2++(T2)〉 = 〈0|ΦE|2++(E)〉Remnants of continuum spin can be found on the lattice. Buildoperators in continuum and measure overlaps to Vnd patterns

Isoscalar/isovector light meson spectrum

500

1000

1500

2000

2500

3000

Caveat: mπ ≈ 400MeV [Dudek et.al Phys.Rev.D88 094505]

Excitation spectrum of charmonium

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

1500M

-M

Ηc

HMeV

L

• Quark model: 1S, 1P, 2S, 1D, 2P, 1F, 2D, . . . all seen.• Not all Vt quark model: spin-exotic (and non-exotic)hybrids seen

[Liu et.al. arXiv:1204.5425]

Gluonic excitations in charmonium?

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

-M

Ηc

HMeV

L

• See states created by operators that excite intrinsic gluons• two- and three-derivatives create states in the open-charmregion.

[Liu et.al. arXiv:1204.5425]

I = 2 π − π phase shift

0.10

0.15

0.20

0.25

0.30

0.35

0.40

• Lüscher’s method: Vrstdetermine energy shiftsas volume changes

• Data forL = 16as, 20as, 24as

• Small energy shifts areresolved

• Measured δ0 and δ2 (δ4 is very small)• I = 2 a useful Vrst test - simplest Wick contractions

Dudek et.al. [Phys.Rev.D83:071504,2011, arXiv:1203.6041]

I = 2 π − π phase shift

-50

-40

-30

-20

-10

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Dudek et.al. [Phys.Rev.D83:071504,2011, arXiv:1203.6041]

I = 1 π − π phase shift (near the ρ)

0

20

40

60

80

100

120

140

160

180

800 850 900 950 1000 1050

Dudek et.al. [arXiv:1212.0830]

I = 3/2, Dπ in a Vnite volume

0.40

0.45

0.50

0.55

aE t

A1+ P = (0,0,0)

[2,0,0][-2,0,0]

[1,1,1][-1,-1,-1]

[1,1,0][-1,-1,0]

[1,0,0][-1,0,0]

[0,0,0][0,0,0]

D πP P

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Inelastic

0.40

0.45

0.50

0.55

aE

t

A1 P = (1,0,0)

[1,0,0][0,0,0]

[0,0,0][-1,0,0]

[1,1,0][0,-1,0]

[0,1,0][-1,-1,0][2,0,0][-1,0,0]

[1,1,1][0,-1,-1]

[0,1,1][-1,-1,-1]

[1,0,0][-2,0,0]

P PD π

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _Inelastic

0.40

0.45

0.50

0.55

aE

t

A 1 P = (1,1,0)

[2,0,0][-1,-1,0]

[1,1,0][-2,0,0][1,0,1][0,-1,-1]

[0,0,1][-1,-1,-1]

[0,0,0][-1,-1,0][1,1,0][0,0,0]

[1,1,1][0,0,-1]

[1,0,0][0,-1,0]

P PD π

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _Inelastic

0.40

0.45

0.50

0.55

aE

t

A1 P = (1,1,1)[2,0,0][-1,-1,-1]

[1,1,1][-2,0,0]

[0,0,0][-1,-1,-1]

[1,0,0][0,-1,-1]

[1,1,0][0,0,-1]

[1,1,1][0,0,0]

P PD π

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _Inelastic

I = 3/2, Dπ scattering phase shift

PRELIMINARY

60

50

40

30

20

10

00.40 0.41 0.42 0.43 0.44 0.45 0.46

_

_

_

_

_

_

δ0(

deg)

a tEcm

P = (0,0,0)P = (1,0,0)P = (1,1,0)P = (1,1,1)

Summary

• To study states beyond the quark model (hybrids,tetraquarks, glueballs) on the lattice requires precisiondeterminations of excitation spectra

• New framework show this can be achieved, but is stillexpensive.

• Precision data on excitations in charmonium and lightmeson spectra.

• Studying elastic scattering using Lüscher formalism isworking well.

• Approaching the physical point — more thresholds open• Next challenge — what to do above inelastic thresholds?