Recent progress in hadron spectroscopyon the lattice

Mike PeardonSchool of Mathematics, Trinity College Dublin, Ireland

Electron-Nucleus Scattering XII,Marciana Marina, Elba, Italy 25th June 2012

In collaboration with ...

JLab:Jo Dudek, Robert Edwards, David Richards

TIFR Mumbai:Nilmani Mathur

Trinity College Dublin:Liuming Liu, Graham Moir, Sinéad Ryan,Christopher Thomas, Pol Vilaseca

U Maryland:Steven Wallace

The myths about lattice spectroscopy

Myths: lattice QCD can . . .

. . . only study hadronic ground-states

. . . not study states with high spin

. . . not study isoscalar meson with precision

. . . not deal with resonances or compute scatteringproperties

• Where do these myths come from?• How close to solving these problems are we?• New results: most of these myths need to be re-examined

Where do these myths come from?

Mostly restrictions with standard techniquesused to perform numerical simulations, par-ticularly those needed to study quarks

Are we close to solving these problems?

• New methods have enabled many of thesechallenges to be overcome.

• Can study excited and high spin states reliably• New data on isoscalar mesons are almost asprecise as isovector states

• Many collaborations publishing results onscattering and resonances.

Methods forlattice spectroscopy

Lattice regularisation

• Lattice provides a non-perturbative, gauge-invariant regulator for QCD

• Quarks live on sites• Gluons live on links• a - lattice spacing• a ∼ 0.1 fm

Quark fields

on sites

on links

Gauge fields

• The Nielson-Ninomiya theorem means chirally symmetricquarks are missing, but can discretise quarks by trading-oUsome symmetry

• In a Vnite volume V = L4, Vnite number of degrees offreedom

Finite V: path-integral is an ordinary (but large) integral. Makepredictions from the QCD lagrangian byMonte Carlo

Spectroscopy in lattice QCD

• Energies of colourless QCD states can be extracted fromtwo-point functions in Euclidean time

C(t) = 〈0| Φ(t)Φ†(0) |0〉

• Euclidean time: Φ(t) = eHtΦe−Ht so C(t) = 〈Φ|e−Ht|Φ〉.Insert a complete set of energy eigenstate and:

C(t) =∞∑k=0

|〈Φ|k〉|2e−Ekt

• limt→∞ C(t) = Ze−E0t, so if observe large-t fall-oU, thenenergy of ground-state is measured.

Euclidean metric very useful for spectroscopy; it provides a wayof isolating and examining low-lying states

Excited states

• Excited-state energies can be measured by correlatingbetween operators in a bigger set, {Φ1,Φ2, . . . ,ΦN}

Cij(t) = 〈0| Φi(t)Φ†j (0) |0〉

• Solve generalised eigenvalue problem:

C(t1) v = λ C(t0) v

for diUerent t0 and t1 [Lüscher & WolU, C. Michael]

• Then lim(t1−t0)→∞ λn = e−En(t1−t0)

• Method constructs optimal ground-state creation operator,then builds orthogonal states.

Excited states accessed if basis of creation operators is used andthe matrix of correlators can be computed

Spin on the lattice

• Lattice breaks O(3)→ Oh

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

• Continuum: subduce O(3) irreps→ Oh

• Look for degeneracies. Problem: spin-4has same pattern as 0⊕ 1⊕ 2.

• Better spin assignment by constructing operators fromlattice representation of convariant derivative.

• Start in continuum with operator of deVnite J, thensubduce this into Oh and then replace derivatives withtheir lattice equivalent. Measure 〈0|Φ| JPC〉 and look forremnants of continuum symmetries.

Remnants of continuum spin can be found on the lattice if webuild operators more carefully and can measure their correlators

Isoscalar meson correlation functions

• Isovector mesons: Wickcontraction gives

• Isoscalar meson correlator has extra diagram. Wickcontraction:

〈ψiψjψkψl〉 = M−1ij M−1kl −M−1il M

−1jk

〈0|Φ(I=0)(t)Φ†(I=0)(0)|0〉 =

〈0|Φ(I=1)(t)Φ†(I=1)(0)|0〉 − 〈0|Tr M−1Γ(t)Tr M−1Γ(0)|0〉

Measuring isoscalar meson correlation functions means alsocomputing the disconnected Wick graphs by Monte Carlo.

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

8 9 10 11 12 13 14 15 16 17 18 19L

1.0

1.5

2.0

2.5

3.0

Ene

rgy

[D. McManus, P. Giudice & MP]

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

• Requires precise data, resolutionof two-hadron and excited states.

Monte Carlo sampling the QCD lattice vacuum

Variance of Monte Carlo estimators is huge unless use impor-tance sampling in Euclidean space-time

• In a Euclidean metric:

C(t1, t0) =∫DUDψDψ ψu(t1)Γψd(t1) ψd(t0)Γψu(t0) e−SG−ψuMψu−ψdMψd∫

DUDψDψ e−SG−ψuMψu−ψdMψd

• Hard to deal with Grassmann algebra

. . . so integrate out quark Velds• Quenched approximation was to ignore det M2

• Nf = 2 importance sampling measure. Non-negative,thanks to Euclidean metric

Monte Carlo sampling the QCD lattice vacuum

Variance of Monte Carlo estimators is huge unless use impor-tance sampling in Euclidean space-time

• In a Euclidean metric:

C(t1, t0) =∫DU Tr {ΓM−1(t1, t0)ΓM−1(t0, t1)} detM[U]2 e−SG∫

DU detM[U]2 e−SG

• Hard to deal with Grassmann algebra

. . . so integrate out quark Velds• Quenched approximation was to ignore det M2

• Nf = 2 importance sampling measure. Non-negative,thanks to Euclidean metric

Monte Carlo sampling the QCD lattice vacuum

Variance of Monte Carlo estimators is huge unless use impor-tance sampling in Euclidean space-time

• In a Euclidean metric:

C(t1, t0) =∫DU Tr {ΓM−1(t1, t0)ΓM−1(t0, t1)} detM[U]2 e−SG∫

DU detM[U]2 e−SG

• Hard to deal with Grassmann algebra

. . . so integrate out quark Velds• Quenched approximation was to ignore det M2

• Nf = 2 importance sampling measure. Non-negative,thanks to Euclidean metric

The numerical tool-kit for quarks

• Physics focus of LQCD has been matrix elements, notspectroscopy.

• Traditionally, quark propagationcomputed starting with point source:η(x, t) = δt,0δx,0

• Solve Mψ = η, then ψ is one columnof M−1

• QCD is translationally invariant

• With this trick, make simple mesons and baryons cheaply.• Not so well suited to studying isoscalar mesons,higher-spin states, hybrids, large operator bases . . .

The “point-to-all” propagator has limited the scope of physicslattice QCD has addressed. Better calculations need “all-to-all”

New methods: distillation

• We can ameliorate the problems coming from measuringquark propagation by looking more carefully at howhadrons are constructed most eXciently

• Smeared Velds: determine ψ from the “raw” Veld in thepath-integral, ψ:

ψ(t) = �[U(t)]ψ(t)

• Extract conVnement-scale degrees offreedom while preserving symmetries

• Build creation operators on smeared Velds• Re-deVne smearing to be a projectionoperator into a small vector space smoothVelds: distillation

Results

Spin identiVcation — J = 3 example

0

1

2

3

4

[Dudek et.al Phys.Rev.D82:034508,2010]

Isovector meson spectroscopy

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

[Dudek et.al. Phys.Rev.D82:034508,2010]

• mπ = 400 MeV• No 2-meson operators

Should be a dense spectrum oftwo-meson states:

— Not seen at all

Light quark mass dependence

0.1 0.2 0.3 0.4

500

1000

1500

2000

2500

3000

0.1 0.2 0.3 0.4

500

1000

1500

2000

2500

3000

0.1 0.2 0.3 0.4

500

1000

1500

2000

2500

3000

0.1 0.2 0.3 0.4

500

1000

1500

2000

2500

3000

[Dudek et.al. Phys.Rev.D82:034508,2010]

Isoscalar mesons

0.5

1.0

1.5

2.0

2.5

exotics

isoscalar

isovector

YM glueball

negative parity positive parity

[Dudek et.al. Phys.Rev.D83:111502,2011]• mπ = 400 MeV, Vnite a• No 0++ data presented• No glueball or two-mesonoperators

Statistical precision:η 0.5 %η′ 1.9 %

N and ∆ spectroscopy

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

[Edwards et.al. Phys.Rev.D84:074508,2011]

Light quark mass dependence — the Roper?

1.0

1.5

2.0

2.5

3.0

0 0.05 0.1 0.15 0.2 0.25

CSSMBGR

this paper

[Edwards et.al. Phys.Rev.D84:074508,2011]

Charmonium spectrum

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

Hadron Spectrum: arXiv:1204.5425• Resolve states up to J = 4; most Vt into quark model• 1S, 1P, 2S, 1D, 2P, 1F, 2D all seen• Beyond quark model: exotic (and non-exotic) hybrids seen

Charmonium hyperVne structure

T1--T1--

20

40

60

80

100

120

140

160M

-M

Ηc

HMeV

L

A1++A1++ T1

++T1++ T2

++T2++ E++E++ T1

+-T1+-

400

450

500

550

600

T1-+T1-+

1200

1250

1300

1350

1400

Hadron Spectrum: arXiv:1204.5425

• HyperVne structure is sensitive to Vnite-lattice-spacingartefacts

• Change lattice action to investigate their inWuence• Spin-exotic 1−+ moves only about 50MeV

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

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 scattering using distillation

[C.Lang et.al. arXiv:1105.5636]

• Number of groups have measured Γρ on the lattice.• Need non-zero relative momentum of pions in Vnal state(P-wave decay)

• New calculation using distillation

2 4 6 8 10 12 14 16 18t

10-2

10-1

100

101

102

103

104

105

C(t)

<ππ|ππ> (Fig. 1a+1c)

<ππ|ππ>dis

(Fig. 1c)

Im[<O1

qq |ππ>] (Fig. 1e)

- Im[<ππ|O1

qq >] (Fig. 1d)

<O1

qq |O

1

qq > (Fig. 1b)

P=(0,0,1)

0.4

0.6

0.8

1

En a

1 2 3 4 5 6 7 8

1

0.4

0.6

0.8

1E

n a

1 2 3 4 5 6 7 8interpolator set

0.4

0.6

0.8

1

En a

interpolator set:

qq ππ

1: O1,2,3,4,5

, O6

2: O1,2,3,4

, O6

3: O1,2,3

, O6

4: O2,3,4,5

, O6

5: O1 , O

6

6: O1,2,3,4,5

7: O1,2,3,4

8: O1,2,3

P=(1,1,0)

P=(0,0,1)

P=(0,0,0)

with ππ without ππ

I = 1 ππ phase shift

[C.Lang et.al. arXiv:1105.5636]

• mπ ≈ 266 MeV• Better resolution bystudying moving ρ aswell

• ρ resonance resolvedclearly, withmρ = 792(7)(8) MeV

• gρππ = 5.13(20)0.1 0.15 0.2 0.25 0.3 0.35 0.4

s

0

50

100

150

δ1

Conclusions

• Precision spectroscopy from the lattice is improvingrapidly:• Variational methods have enabled excited states to bestudied

• More sophisticated operator construction allows us todisentangle higher spins in lattice data

• Isoscalar mesons are determined at similar precision• Lüscher’s method links the spectrum in Vnite volume toscattering properties.

• These developments have been enabled by extending thetoolkit for measuring quark propagation on the lattice

• Look out for better data on scattering soon . . .• . . . but inelastic thresholds remain a challenge