La�ice QCD for hadron and nuclear physics: new (and old)

ideas for be�er measurements

Sinéad M. Ryan

Trinity College Dublin

National Nuclear Physics Summer School, MIT July 2016

Lecture Plan

Key ideas to enable current and future physics programmes

Smearing - an (old) and good idea

Distillation - for quark propagation

Spin identification - as discussed earlier

Recent results

The traditional idea - point propagators

�ark propagation from orgin to all sites on the la�ice.

For be�er simulations of hadronic quantities look again at the building blocks: the quark

propagators

Point propagator pros

doesn’t require vast computing resources

C(t,x) = ⟨Tr(γ5M−1

a (x, 0)†γ5ΓM−1

b (x, 0)Γ†)⟩

Point propagator cons

restricts the accessible physics

flavour singlets and condenstates impossible: quark loops need props w sources

everywhere in space

restricts the interpolating basis used

a new inversion needed for every operator that is not restricted to a single la�ice point

entangles propagator calculation and operator construction

throws away information encoded in configurations

Solutions?

Improve the determination point props to access the physics of interest: smearing

Compute all elements of the quark propagator: all-to-all propagators. Problem It’s

expensive - needs an unrealistic number of inversions.

Work around: Use stochastic estimators (with variance reduction). [I won’t talk aboutit here but see refs for details]

or Rethink the problem: combine smearing and propagation ie distillation

I am picking a few methods to focus on.

See references at the end of this lecture for full descriptions of these and other methods.

Smearing

Smearing techniques

Hadrons are extended objects (O(1)fm).

So far the propagator and interpolating fields (operators) are point sources

they can have small overlap with the state of interest: quantified by Zn: (⟨n|OM|0⟩).optimise the projection onto the state we want to study

Gauge-invariant smearing of quark fields:

Ψ(~x, t) =∑

~y

F (~x, ~y,U(t))Ψ(~y, t)

Gaussian smearing: F (~x, ~y,U(t)) = (1 + κsH)nσand H is the la�ice realisation of the

covariant Laplacian in 3d

Variations on a theme: Jacobi, Wuppertal ...

More improvements to gauge noise by smearing the U fields in F :

APE, HYP, Stout

An example from D. Alexandrou at ECT* Trento

Examples of effective mass plots• Quenched at about 550 MeV pions:

amπ

effamN

eff

• Reduce gauge noise by using APE, hypercubic or stout smearing on the links U that enter the smearingfunction F (�x,�y, U(t)).• NF = 2

H. Wittig, SFB/TR16, August, 2009

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Introduction to Lattice QCD Aurora School, ECT* Trento 8 / 26

Smearing

Smeared field: ψ from ψ, the “raw” quark field in the path-integral:

ψ(t) = �[U(t)] ψ(t)

Extract the essential degrees-of-freedom.

Smearing should preserve symmetries of quarks.

Now form creation operator (e.g. a meson):

OM(t) =¯ψ(t)Γψ(t)

Γ: operator in {s, σ, c} ≡ {position,spin,colour}

Smearing: overlap ⟨n|OM|0⟩ is large for low-lying eigenstate |n⟩

Can redefining smearing help?

Computing quark propagation in configuration generation and observable

measurement is expensive.

Objective: extract as much information from correlation functions as possible.

Two problems:

1 Most correlators: signal-to-noise falls exponentially

2 Making measurements can be costly:

Variational bases

Exotic states using more sophisticated creation operators

Isoscalar mesons

Multi-hadron states

Good operators are smeared; helps with problem 1, can it help with problem 2?

Gaussian smearing

To build an operator that projects e�ectively onto a low-lying hadronic state need to

use smearingInstead of the creation operator being a direct function applied to the fields in the

lagrangian first smooth out the UV modes which contribute li�le to the IR dynamics

directly.

A popular gauge-covariant smearing algorithm — Gaussian smearing: Apply the

linear operator

�J = exp(σ∇2)

∇2is a la�ice representation of the 3-dimensional gauge-covariant laplace operator

on the source time-slice

∇2

x,y = 6δx,y −3∑

i=1

Ui(x)δx+ι,y + U†i (x − ι)δx− ι,y

Correlation functions look like Tr �JM−1�JM−1

�J . . .

Distillation [0905.2160]

“distill: to extract the quintessence of” [OED]

Distillation: define smearing to be explicitly a very low-rank operator. Rank is

ND(� Ns × Nc).

Distillation operator

�(t) = V (t)V †(t)

with V ax,c(t) a ND × (Ns × Nc) matrix

Example (used to date): �4 the projection operator into D4, the space spannedby the lowest eigenmodes of the 3-D laplacianProjection operator, so idempotent: �

2

4 = �4limND→(Ns×Nc) �4 = I

Eigenvectors of ∇2not the only choice. . .

Using eigenmodes of the gauge-covariant laplacian preserves la�ice symmetries.

Distillation

Distillation: a redefinition of smearing as explicitly a low-rank operator.

E�ect: project out eigenmodes that do not contribute to hadronic physics.

In the low-rank space M−1can be calculated exactly.

Consider an isovector meson two-point function with {s, σ, c} for position, spin,

colour.

CM(t1 − t0) = ⟨⟨u(t1)�t1Γt1�t1

d(t1) d(t0)�t0Γt0�t0

u(t0)⟩⟩

Integrating over quark fields yields

CM(t1 − t0) = ⟨Tr{s,σ,c}�

�t1Γt1�t1

M−1(t1, t0)�t0Γt0�t0

M−1(t0, t1)�

⟩

Substituting the low-rank distillation operator � reduces this to a much smallertrace:

CM(t1 − t0) = ⟨Tr{σ,D} [Φ(t1)τ(t1, t0)Φ(t0)τ(t0, t1)]⟩

Φα,aβ,b and τα,aβ,b are (Nσ × ND)× (Nσ × ND) matrices.

Φ(t) = V†(t)Γt V(t) τ(t, t′) = V†(t)M−1(t, t′)V(t′)

The “perambulator”

Good news: precision spectroscopy (2)

Iso

scalar

meso

ns

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Correlation functions for ψγ5ψ operator, with di�erent flavour content (s, l).

163

la�ice (about 2 fm). [arXiv:1102.4299]

Limitation

Distillation does not give direct access to all modes of the Dirac operator, only those

low-modes relevant for spectroscopy

Cannot use the method to calculate eg the strangeness content of the nucleon.

⟨N(tf , ~q)|∑

xe−i~q·~x s(t′, ~x)Γs(t′, ~x)|N(0,~0)⟩

Use standard all to all instead.

Bad news: the bill!

For constant resolution distillation space scales with Ns

The cost of a calculation scales with V 2

The problem:

To maintain constant resolution, need ND ∝ Ns

Budget:

Fermion solutions construct τ O(Ns2)

Operator constructions construct Φ O(Ns2)

Meson contractions Tr[ΦτΦτ] O(Ns3)

Baryon contractions BτττB O(Ns4)

Ok for reasonable la�ices (eg with Ns = 163,ND = 64) but scaling this to a 32

3volume

requires ND = 512. Numerically costly.

Distillation does not preclude stochastic estimation - use both for large V . [See refs

for more on stochastic distillation methods and other methods.]

Interpolating Operators

The interpolating operators

We have spent some time looking at methods for quark propagation

What about the operators O = Ψiα(~x, t)ΓαβΨiβ (~x, t)?

The simplest objects are colour-singlet local fermion bilinears:

Oπ = dγ5u, Oρ = dγiu, ON = εabc�

uaCγ5db�

uc ,

O∆ = εabc�

uaCγnudb�

uc

or more correctly!

OA1= dγ5u, OT1

= dγiu, OG1= εabc

�

uaCγ5db�

uc ,

OH = εabc�

uaCγnudb�

uc

Access to JPC = 0−+, 0++, 1−− , 1++, 1+− , 1/2, 3/2

Extended operators

We would like to access states with J > 1

Would like many more operators that all transform irreducibly under some irrep

enabling variational analysis.

La�ice operators are bilinears with path-ordered products between the quark and

anti-quark field; di�erent o�sets, connecting paths and spin contractions give

di�erent projections into la�ice irreps.

Meson operators examples

Oαβ = Oiαβ = Oij

αβ =∑

x ψα(x)ψβ(x)∑

x ψα(x)Ui(x)ψβ(x + ι)∑

x ψα(x)Ui(x)Uj(x + ι)ψβ(x + ι+ ȷ)

Extended baryon operators

The same idea for baryons gives prototype extended operators

� ��uuusingle-

site

muu usingly-

displaced

eu uutriply-

displaced

With thanks: 0810.1469

We can make arbitrarily complicated operators in this way

An early success was glueball calculations

Glueballs

QCD nonAbelian ⇒ allows bound states of glue

Candidates observed experimentally: f0(1370), f0(1500), f0(2220)

Glueballs can be calculated in la�ice QCD

The interpolating fields are purely gluonic, built from Wilson loops

++ −+ +− −−PC

0

2

4

6

8

10

12

r 0m

G

2++

0++

3++

0−+

2−+

0*−+

1+−

3+−

2+−

0+−

1−−

2−−

3−−

2*−+

0*++

0

1

2

3

4

mG (

Ge

V)

Morningstar and Peardon, 1999. �enched

Good operators

What makes a good operator?

An operator of definite momentum that transforms under a la�ice irrep

An operator that has strong overlap with the (continuum) state you are interested in.

An operator is not noisy ie that produces an acceptable correlator

Note that smearing and distillation are rotationally symmetric operations and do not

change the quantum numbers.

But recall from earlier that subduction leads to

La�ice irrep, Λ Dimension Continuum irreps, J

A1 1 0, 4, ...

A2 1 3, 5, ...

E 2 2, 4, ...

T1 3 1, 3, ...

T2 4 2, 3, ...

G1 3 1/2, 7/2, ...

G2 3 5/2, 7/2, ...

H 4 3/2, 5/2, ...

So a correlator C(t) = ⟨0|ϕ(t)ϕ†(0)|0⟩ contains in principle information about all

(continuum) spin states subduced in ΛPC.

Operator basis — derivative construction

A closer link to (or “memory” of) the continuum would be good

There are di�erent approaches to optimise la�ice operators. This is one.

Start with continuum operators, built from n derivatives:

Φ = ψ Γ�

Di1 Di2 Di3 . . .Din�

ψ

Construct irreps of SO(3), then subduce these representations to Oh

Now replace the derivatives with la�ice finite di�erences:

Djψ(x)→1

a

�

Uj(x)ψ(x + ȷ)− U†j (x − ȷ)ψ(x − ȷ)�

On a discrete la�ice covariant derivative become finite displacements of quark fields

connected by links

arXiV:0707.4162

Operator basis — derivative construction

A closer link to (or “memory” of) the continuum would be good

There are di�erent approaches to optimise la�ice operators. This is one.

Start with continuum operators, built from n derivatives:

Φ = ψ Γ�

Di1 Di2 Di3 . . .Din�

ψ

Construct irreps of SO(3), then subduce these representations to Oh

Now replace the derivatives with la�ice finite di�erences:

Djψ(x)→1

a

�

Uj(x)ψ(x + ȷ)− U†j (x − ȷ)ψ(x − ȷ)�

On a discrete la�ice covariant derivative become finite displacements of quark fields

connected by links

arXiV:0707.4162

Example: JPC = 2++

meson creation operator

Trying to gain more information to discriminate spins. Consider continuum operator

that creates a 2++

meson:

Φij = ψ�

γiDj + γjDi −2

3

δijγ · D�

ψ

La�ice: Substitute gauge-covariant la�ice finite-di�erence Dla� for D

A reducible representation:

ΦT2 = {Φ12,Φ23,Φ31}

ΦE =

�

1

p2

(Φ11 − Φ22),1

p6

(Φ11 + Φ22 − 2Φ33)

�

Look for signature of continuum symmetry:

Z = ⟨0|Φ(T2)|2++(T2)⟩ = ⟨0|Φ(E)|2++(E)⟩

up to rotation-breaking e�ects

This idea appears to work well

E.g. in charmonium - arXiv:1204.5425

Spin-3 identification

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à

à

àà à

à

à

àà

à

à

à

àà

à

çç

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çç

àà

àà

àà

atmA2= 0.6770H7L

atmT1= 0.676H1L

atmT2= 0.6768H7L

atmA2= 0.774H2L

atmT1= 0.767H3L

atmT2= 0.769H3L

A2 T1 T2 A2 T1 T2 A2 T1 T2 A2 T1 T2 A2 T1 T2

´10

´10

´50

´10

0

1

2

3

4

5

Z

J = 3 in A2,T1,T2

J = 4 in A1,T1,T2, E

Spin-4 identification

çççç

çç

çç

çç

çç

çç

çç

atmA1= 0.763H4L

atmT1= 0.776H3L

atmT2= 0.777H2L

atmE= 0.771H4L

A1 T1 T2 E0.0

0.5

1.0

1.5

2.0

Z

operators of definite JPCconstructed in step 1 are

subduced into the relevant irrep

a subduced irrep carries a “memory” of continuum spin J

from which it was subdduced - it overlapspredominantly with states of this J.

J 0 1 2 3 4A1 1 0 0 0 1

A2 0 0 0 1 0

E 0 0 1 0 1

T1 0 1 0 1 1

T2 0 0 1 1 1

Using Z = ⟨0|Φ|k⟩, helps to identify continuum spins

For high spins, can look for agreement between irreps

Data below for T−−1

irrep, colour-coding is Spin 1, Spin 3 and Spin 4.

0.53726H4L0.53726H4L0.53726H4L0.53726H4L0.53726H4L0.53726H4L0.53726H4L0.53726H4L0.667H3L 0.646H1L0.646H1L0.646H1L0.646H1L0.646H1L0.646H1L0.646H1L0.646H1L0.667H3L 0.6713H5L0.6713H5L0.6713H5L0.6713H5L0.6713H5L0.6713H5L0.6713H5L0.6713H5L0.667H3L 0.676H1L0.676H1L0.676H1L0.676H1L0.676H1L0.676H1L0.676H1L0.676H1L0.667H3L 0.727H5L0.727H5L0.727H5L0.727H5L0.727H5L0.727H5L0.727H5L0.727H5L0.667H3L 0.753H2L0.753H2L0.753H2L0.753H2L0.753H2L0.753H2L0.753H2L0.753H2L0.667H3L 0.759H7L0.759H7L0.759H7L0.759H7L0.759H7L0.759H7L0.759H7L0.759H7L0.667H3L 0.767H3L0.767H3L0.767H3L0.767H3L0.767H3L0.767H3L0.767H3L0.767H3L0.667H3L 0.776H3L0.776H3L0.776H3L0.776H3L0.776H3L0.776H3L0.776H3L0.776H3L0.667H3L

Spectroscopy - selected results

Single hadron states: Charmonium exotics

Precision calculation of high spin (J ≥ 2) and exotic states is relatively new

Caveat Emptor

Only single-hadron operators

Physics of multi-hadron states

appears to need relevant

operators

No continuum extrapolation

mπ ∼ 400MeV ← already

changing

Charmonium

from HSC 2012

→ Expect improvements now methods established

Single-hadron states: light exotics

0.5

1.0

1.5

2.0

2.5

exotics

isoscalar

isovector

YM glueball

negative parity positive parity

from HSC 2010

Single-hadron states: baryons @ 396MeV

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

from HSC 2011

Hybrids

Expect a large overlap with operators O ∼ Fμν

1.0

1.5

2.0

2.5

3.0

from HSC

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

1500

M-

MΗ

cHM

eVL

cc from HSC

Lightest hybrid supermultiplet and excited hybrid supermultiplet same pa�ern and scale

in meson and baryon, heavy and light[HadSpec:1106.5515]

sectors.

Energy scale for hybrids

0

500

1000

1500

2000

m0 = mρ for mesons and m0 = mN for baryons.

The nucleon mass - unexpected behaviour!

A.W

alke

r-Lo

ud@

Lat2

014

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

mπ /(2√

2πf0 )

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

MN

[G

eV

]

physical

LHPC 2008

χQCD 2012

RBC: Preliminary DSDR

RBC: a−1 =1.75(3) GeV

RBC: a−1 =2.31(4) GeV

A ruler plot ie linear in quark mass

Taking this seriously then MN (MeV) = αN0

+ αN1

mπ = 800 + mπ parameterises the

numerial results in the available range and agrees with the physical point!

But it predicts the wrong quark mass dependence at/near the chiral limit.

Nucleon size with physical quark masses

0.3

0.4

0.5

0.6

0.7

0.10 0.15 0.20 0.25 0.30 0.35 0.40

(r2 1)s

(fm

2)

mπ (GeV)

PDGµp

coarse 483×48coarse 323×96coarse 323×48

coarse 323×24coarse 243×48coarse 243×24

fine 323×64

LHPC 2014 at multiple la�ice spacings & volumes and physical pion masses.

Nucleon strangeness - snapshot of activity

needs disconnected diagrams - distillation not suitable so other all-to-all methods

needed.

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Shanahan et al 1403.6537: deducing disconnected

e�ects from experiment and la�ice data.

Direct calculations underway.

A complilation - Nucleon strangeness

From Junnakar & Walker-Loud (PRD.87 (2013))

0.00 0.05 0.10

fs

Feyn

man

-Hel

lman

n

0.053(19) present work0.134(63) [35] nf = 2 + 1, SU(3)0.022(+47

−06) [34] nf = 2 + 1, SU(3)0.024(22) [33] nf = 2 + 1, SU(3)0.076(73) [32] nf = 2 + 10.036(+33

−29) [31] nf = 2 + 10.033(17) [21] nf = 2 + 1, SU(3)0.023(40) [27] nf = 2 + 10.058(09) [30] nf = 2 + 10.046(11) [28] nf = 2 + 10.009(22) [27] nf = 2 + 10.035(33) [36] nf = 2 + 10.048(15) [26] nf = 2 + 10.014(06) [25] nf = 2 + 1 + 10.012(+17

−14) [24] nf = 20.032(25) [22] nf = 20.063(11) [29] nf = 2 + 1

fs

0.043(11) lattice average (see text)

Dire

ctE

xclu

ded

Their average: ms⟨N |ss|N⟩ = 48± 10± 15MeV and fs = 0.051± 0.011± 0.016.

Summary

New ideas are enabling rapid progress.

Lots of precision and pioneering calculations of hadronic and nuclear quantities.

Next - multi-hadron, many-body systems. A rapidly moving field!

References

APE smearing APE collaboration, M. Albanese et al., Glueball masses and stringtension in la�ice QCD, Phys. Le�. B192 (1987) 163.

Hyper-cubic (HYP) smearing A. Hasenfratz and F. Knechtli, Flavor symmetry and thestatic potential with hypercubic blocking, Phys. Rev. D64 (2001) 034504.

Stout-link smearing C. Morningstar and M.J. Peardon, Analytic smearing of SU(3) linkvariables in la�ice QCD, Phys. Rev. D69 (2004) 054501.

All-to-all propagators Foley et al, hep-lat/0505023 and references 1-13 therein.

Distillation Peardon et al, arXiV:0905.2160

Stochastic distillation: LaPH, Morningstar et al, arXiv:1104.3870

All mode averaging (AMA), Shintani et al, arXiv:1402.0244