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?
