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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
ç ç
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ç
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ç
ç ç
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ç
à
à
àà à
à
à
àà
à
à
à
àà
à
çç
çç
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