Lattice QCD with
Staggered Quarks:
How, Where, & Why (Not)
Andreas S. Kronfeld
f
31 July 2007
Why Are We Here?
Lattice QCD
fπ
fK
•
3mΞ − mN
hep-lat/0303004 showed
results with 2+1 flavors
of staggered sea quarks,
reproducing a wide
variety of quantities.
2mB − mΥ
s
ψ(1P-1S)
Υ(1D-1S)
Υ(2P-1S)
Υ(3S-1S)
Υ(1P-1S)
0.9
1.0
1.1
0.9
1.1
2
2
qmax/mD*
s
2.5
D → Klν
2
1.5
2
f+(q )
•
several papers (e.g., heplat/0408306) used the
same methods to make
predictions, confirmed
later by experiment.
1.0
(nf = 2+1)/experiment
quenched/experiment
1
0.5
0
experiment [Belle, hep-ex/0510003]
lattice QCD [Fermilab/MILC, hep-ph/0408306]
0
0.05
0.1
0.15
0.2
0.25
2
/mD*
s
2
q
0.3
0.35
0.4
0.45
Nobel Reactions
fπ
fK
•
3mΞ − mN
Top plot: “a stunning
achievement.”—
Frank Wilczek
2mB − mΥ
s
ψ(1P-1S)
Υ(1D-1S)
Υ(2P-1S)
Υ(3S-1S)
Bottom plot:
0.9
1.0
1.1
0.9
•
1.0
1.1
(nf = 2+1)/experiment
quenched/experiment
2
“Wow, that’s
impressive!”—
Tini Veltman
2
qmax/mD*
s
2.5
D → Klν
2
1.5
2
f+(q )
•
Υ(1P-1S)
•
David Gross
1
0.5
0
experiment [Belle, hep-ex/0510003]
lattice QCD [Fermilab/MILC, hep-ph/0408306]
0
0.05
0.1
0.15
0.2
0.25
2
/mD*
s
2
q
0.3
0.35
0.4
0.45
The Charge
Dear Andreas,
on behalf of the Local Organizing Committee for Lattice 2007, it is my pleasure to invite you to present
a plenary talk at the conference, which will take place from July 30 to August 4 in Regensburg,
Germany.
The purpose of the talk we would like you to give (which will be 45 minutes including questions) is
twofold:
1. Preceding your talk, there will be a 30-minute talk by Mike Creutz in which he is going to present his
criticism of the rooting procedure for staggered fermions. Your talk should present the response of
the staggered community to this criticism.
2. We ask you to also present a review of recent results obtained with staggered fermions and future
plans of the staggered community (e.g., HISQ). In view of the approaching start-up of LHC it
would also be nice if you could comment on the relevance of lattice QCD for LHC.
There has been a rather lively discussion in the IAC and LOC regarding this talk. The majority opinion
is that it is very important to spent a substantial part of your talk (at least half) on 1. to increase the
confidence of the wider lattice community in the rooting procedure.
The conference web site is ....
Best regards,
Tilo Wettig for the Lattice 2007 LOC
Staggered Community
Those who generate big ensembles
with 2+1 staggered sea quarks
Those who use
these ensembles
Those who submit
papers to hep-lat
Those who think the validity of
staggered quarks is worthy of study
Staggered Community
Those who generate big ensembles
with 2+1 staggered sea quarks
Those who use
these ensembles
Those who submit
papers to hep-lat
Those who think the validity of
staggered quarks is worthy of study
My Talk
• Problems, issues, and resolutions of rooted
staggered quarks:
• theoretical physics for framework;
• numerical simulation for dynamics.
• Speaking for myself.
I’m not an expert in
the rich group-theoretical structure of
staggered fermions...
• ... but I’m making some progress:
• “An expert is a person who has made all
the mistakes that can be made in a very
narrow field.” (Niels Bohr)
Outline
• Why it matters.
• Staggered fermions (without rooting):
• symmetries and the emergence of taste;
• order of continuum and chiral limits.
• Rooting with full SU(4) taste symmetry:
• how it works & where it doesn’t.
• Rooting with Γ taste symmetry:
• violations of locality;
• violations of unitarity.
• Refutation of Mike Creutz’s criticisms.
• New developments.
• Conclusions and perspective.
4
LGT4LHC
• Strongly interacting fields (or their bound
states) may break electroweak symmetry:
• dynamical susy breaking;
• technicolor (confining GTs);
• topcolor (spontaneously broken GTs);
• extra spatial dimensions.
• If so, LGT will play a role, and the fastest
turnaround enables a trip to Stockholm.
• If not—and even if so—there will be a need
for the best fB & BB, fK, parton densities, etc.
• to explain dark matter, compelling BSMs
have a quantum number implying pair
production of their new particles;
• real
pairs at LHC, but virtual pairs in
meson mixing & penguin decays.
• Therefore, faster computers will not make
LGT with staggered fermions obsolete.
Staggered (and Naïve)
Fermions
Naïve Fermions
of fermions is
• The simplest discretization
!
"
a3
†
S =
ϒ̄(x)γµ Uµ (x)ϒ(x + µ̂) −Ux−µ,µ ϒ(x − µ̂)
∑
2 x,µ
+ m a ϒ̄(x)ϒ(x)
0 ∑
x
i , ϒ̄i
ϒ
Grassmann variables α α on each site.
• Invariant under SU
color(3), translations,
hypercubic rotations, and UV(nf)×UA(nf).
Doubling Symmetry
• The naïve action also has a remarkable
“doubling” symmetry [Karsten&Smit]:
−1
xµ /a
ϒ !→ Bµ ϒ, ϒ̄ !→ ϒ̄Bµ , Bµ = iγµ γ5 (−1)
• Generates a 32-element group (16 physical
transformations × –1).
• Clifford group Γ : {B , B } = 2δ
4
μ
ν
μν.
Ramifications
• Doubling symmetries
A
• map momentum p !→ p + π /a , where π
A
is one of 16 corners of Brillouin zone;
• shuffle Dirac indices.
• For n naïve fermion fields, the gauge
f
coupling runs with β0 = 11 − 16×2nf/3.
• 16 species of fermion!
• The axial anomaly receives contributions
from all 16 species, but they contribute
(π,π,0,0)
(0,0,0,0)
(π,0,0,0)
(π,π,π,0)
(π,π,π,π)
1 − 4 + 6 − 4 + 1 =0
because the UA(1) is exact.
• So it seems that naïve lattice fermion fields
are not what we want for QCD.
Staggering Magic
• But the lattice allows a transformation
n1 n2 n3 n4
ϒ(x) !→ ψ(x) = Ω(x)ϒ(x), Ω(x) = γ1 γ2 γ3 γ4 ,
−1
ϒ̄(x) !→ ψ̄(x) = ϒ̄(x)Ω (x), n = x/a.
[Kawamoto & Smit].
• Ω clobbers the Dirac matrices:
−1
nρ
∑
ρ<µ
Ω (x)γµ Ω(x ± µ̂) = (−1)
=: ηµ (x).
• The naïve action assumes a simpler form:
!
"
3
a
†
S = 2 ∑ ψ̄(x)ηµ (x) Uµ (x)ψ(x + µ̂) −Ux−µ,µ ψ(x − µ̂)
x,µ
+ m0 a ∑ ψ̄(x)ψ(x)
x
• Also attained by diagonalizing a maximal
subgroup of doubling symmetry
[Sharatchandra, Thun, Weisz = STW].
• The Dirac index is now trivial:
!ϒ(x)ϒ̄(y)"U = Ω(x)Ω−1 (y)!χ(x)χ̄(y)"U
where χ—the staggered fermion—has one
component/color/flavor, and action
!
"
3
a
†
S = ∑ χ̄(x)ηµ (x) Uµ (x)χ(x + µ̂) −Ux−µ,µ
χ(x − µ̂)
2 x,µ
+ m0 a ∑ χ̄(x)χ(x)
x
• But now other symmetries are entangled.
• For example, translations become shifts:
S : χ(x) !→ ζ (x)χ(x + µ̂), χ̄(x) !→ ζ (x)χ̄(x + µ̂)
µ
µ
µ
Uν (x) !→ Uν (x + µ̂), ∀ν; ζµ (x) = (−1)∑ρ>µ nρ
(and ψ like χ), so the extra species are tied
up with spacetime.
• Clifford again: {S , S } = 2δ .
• Rotations: S transform like a vector.
• Spatial inversion: I S = –S I so correlators
μ
ν
μν
s μ
–1
μ s
μ
produce desired states and parity partners.
[Golterman & Smit; Golterman; Kilcup and Sharpe]
• Representations of shift symmetry:
iπ
(b)
A
ip
a
A
• bosonic: D (Sµ) = sµ e , sµ = e
(
f
)
ip
a
• fermionic: D (Sµ) = ξµe , {ξµ, ξν} = 2δµν.
• momentum pµ ∈ (−π/2a, π/2a]
A, ξ
s
• The factors µ µ next to the exponential
A
µ
µ
µ
denote a new quantum number, “taste.”
• Taste (defined group-theoretically) tied to
momentum.
• Physically suggestive quark field:
Ψαt (y) = ∑ Ωαt (r)U(y, y + r)χ(y + r)
r
y labels 24 hypercubes, r runs over corners
of hypercubes, U(y,x) is parallel transport,
and α & t are (emergent) Dirac & taste
indices [Gliozzi; Kluberg-Stern et al.].
Hadronic Examples
• Stag-stag mesons have bosonic taste A.
• Stag-Wilson mesons (including heavy-light)
have fermionic taste t = 1, ..., 4.
• Stag-stag-stag baryons have a plethora of
possibilities [Bailey].
• Stag-stag-Wilson baryons have bosonictaste diquark [Gottlieb, Na, Nagata].
• Clifford Γ : four fermionic tastes, with index
4
t, are degenerate.
•
Rotations: taste degeneracies for bosonic sA
• I: (0,0,0,0);
• V: (1,0,0,0), (0,1,0,0), (0,0,1,0); (0,0,0,1);
• T: (1,1,0,0), (0,1,1,0), (1,0,1,0); blah
(1,0,0,1), (0,1,0,1), (0,0,1,1);
• A: (0,1,1,1), (1,0,1,1), (1,1,0,1); (1,1,1,0);
• 5: (1,1,1,1).
Emergence
• One must conjecture that taste emerges, in
the continuum limit, as a flavor-like index of
a Dirac fermion, with SUV(4nf)×SUA(4nf)
combined flavor-taste symmetry.
• Not especially transparent: e.g., the rotation
symmetry:
SW ⊂ [SO(4) × SO(4)] ⊂ SO(4) × SU (4n )
4
V
f
diag
Γ4 ⊂ SUV (4n f )
• Axial SU (4n ) should also emerge, but the
A
f
non-anomalous symmetry
χ(x) !→ eiθε(x) χ(x), ε(x) = (−1)n1 +n2 +n3 +n4 ,
iθε(x)
χ̄(x) !→ e
χ̄(x),
should be a taste-nonsinglet part of SUA(4nf).
• STW inexact axial current with the correct
anomaly should become taste-singlet UA(1).
Its chirality should feel topology of the gauge
field via the index theorem [Smit & Vink].
• Some papers blur the distinction between
staggered and naïve fermions.
• Should not do this.
• The emergent axial symmetries require the
staggered interpretation, otherwise one is
led to use naïve chirality instead of tastesinglet chirality.
•
†
Propagator Ωαt (x)Ωtβ !χ(x)χ̄(y)"
over tastes, and that it fine.
sums
Symanzik LEL
• With the desired continuum limit, one can
build up a Symanzik effective field theory.
• Dimension 4 terms have flavor-taste
symmetry SUV(4nf)×SUA(4nf) (by design).
• Dimension-6 4-quark operators with taste
exchange break these and SO(4) rotations
down to staggered symmetry group [Lee &
Sharpe; Aubin & Bernard].
• The emergence of a (4n )-species version of
f
QCD is a dynamical assumption.
• It is borne out by extensive numerical
simulations.
• In particular, improved actions and lattice
spacing dependence establish that (most of?)
the taste-exchange (or taste-breaking)
interactions are dimension 6.
MILC
} Λ4a2
coarse lattice
a = 0.121 fm
m2A − m25 ≈ (280 MeV)2
m2T − m2A ≈ (240 MeV)2
hep-lat/0407028
needs confirmation
m
Λ2
2
I
V
T
A
} Λ4a2
5
Q=±
I = 1, Q = 0
I=0
Topology
• With improved actions, eigenvalues iλ of
the Dirac operator appear in quartets.
• When Q ≠ 0, some quartets are near-zero
modes, λ ~ Λ3a2.
• Numerical simulations [FHD=Follana, Hart,
Davies] show that all 4 in quartet have the
same taste-singlet chirality.
Staggered χPT
• Staggered chiral perturbation theory [LS,
AB] includes leading effects of order Λ2a2.
• Can lead to complicated (ugly?) fits: e.g., the
π-π threshold splits into several.
• The complications come not from rooting,
but from the rich symmetry structure and
the desire for best results.
Other Continuum
Limits
• Another possible continuum limit arises if
the quark mass is taken to zero too fast.
•
Then only π5 is a pseudo-Goldstone boson,
other πA remain massive particles.
• Much numerical evidence, e.g., eigenvalue
spectrum compared to random matrix.
• Finite volume: ε regime vs. ε´ regime [FHD].
Rooting
A Clever Trick
• Suppose someone with a good imagination
found a way to speed up “your favorite”
fermions by substituting
1/4
det1 (D
/ + m) = {det4 [(D
/ + m) ⊗ 14 ]}
with four “tastes.”
• This would be fine if det is real and positive.
• So it doesn’t work for m < 0, or μ ≠ 0.
Go to the Source
• One can introduce sources:
{det [(D
1/4
4 / + m + J + J5 ) ⊗ 14 ]}
a
a
a
a
(J
,
J
)
, T γ5 )ψ.
where
5 is source for ψ̄(T
• Now generalize the sources:
1/4
{det4 [(D
/ + m) ⊗ 14 + J + J5 ]}
which means “ask more.”
• Start withZ(DU = gauge-field measure)
Z(J, J5 ) =
DU {det4 [(D
/ + m) ⊗ 14 + J + J5 ]}
1/4
• All correlators taken in original, tastesymmetric ensemble.
• Legendre transform
A
J
A
σ ,
A
J5
A
π , and
→
→
derive mass matrices (for constant fields)
2
2
∂ Γ
∂ Γ
,
A
B
∂σ ∂σ
∂πA ∂πB
• Find usual pattern of spontaneous breaking.
•
This formulation has (4nf)2 – 1 pseudoGoldstone bosons, instead of (nf)2 – 1.
• The extra ones are phantoms—a figment
of the algorithm’s imagination.
• Their total contribution to any tasteless
correlation function must cancel.
• Not unitary; not worrisome either.
• A safe house for phantom Goldstones.
Synthesis
The Actual Trick
• Simulate sea! quarks with "the Ansatz
1/4
det4 (D
/stag + m)
since (valence) taste-breaking is small.
• Use correlators whose meaning is based on
rooting in the absence taste exchange.
• It is possible to construct unphysical ones,
leading to confusion or mistakes.
• With fermionic taste, use any single index
or average over all four.
• With bosonic taste,
• use taste singlets.
• use any taste for flavor non-singlets:
continuum-limit symmetry relates them.
Taste Breaking
• Taste-exchange interactions break the
symmetry of the idealized rooted setup:
SU
(4n
)
×
SU
(4n
)
→
U
(n
)
×
U
(n
)
×
Γ
L
f
R
f
f
r
f
4
!
• As in the idealized situation, the phantom
particles are not unitary.
• But pseudoscalar masses are now split, so
the cancellation is approximate (at best).
• The result of turning on taste-breaking is
non-locality [BGS]. The taste-singlets are
more massive than their phantom partners.
• It is not a non-locality that breaks cluster
decomposition.
the non-locality of a propagator
• It is not−1/4
/ + m)
(D
.
• Nevertheless, non-degenerate phantoms
are ugly, perhaps even scary.
Scalar Propagators
• The phantoms show up in low-lying states
(of a given flavor combo) in RSχPT loops.
• Extensive evidence that this works.
• In higher mass particles, like scalar mesons,
phantoms are more singular [Prelovsek].
• Taste-singlet correlators contain bubbles of
other-taste mesons.
a0 propagator, fit to formula
derived in RSχPT [DeTar et al.]
Formula displays explicitly how
phantoms cancel.
ml/ms = 0.007/0.05
0.004
0.002
C(t)
Same happens with
mixed action (here
DWF valence on
staggered sea)
[Aubin, Laiho,
Van de Water]
mv = 0.01
mv = 0.02
mv = 0.03
mv = 0.04
mv = 0.05
0
-0.002
-0.004
-0.006
0
2
4
t
6
8
10
My Favorite D
• We would like to argue that the taste
exchange interactions are small, and treat
them as perturbations.
• This is clearly OK at the tree level, and very
likely at all orders in perturbation theory.
• Not proven at levels of Reisz’s theorems,
but see Giedt’s paper.
• What this really entails is to establish the
Symanzik LEL not by Ansatz, but by
derivation.
• Upcoming paper by Bernard, Golterman,
and Shamir, shows a new way to do this.
• Based on Shamir’s RG approach, covered
thoroughly in Sharpe’s review last year.
talk
• After n blocking steps Shamir arrives at a
blocked staggered operator
(D
/ + m)n ⊗ I4 + a f ∆n
with taste-breaking in the defect Δn.
• BGS develop an expansion in the spacing of
the underlying (fine) lattice, af.
• Match to a continuum LEL that reproduces
the Lee-Sharpe LEL for staggered fermions.
• A full discussion is necessarily technical
(and given yesterday in parallel session).
• Key take home messages:
• valence quarks control the symmetries
and the field content of the LEL, so
they’re oblivious to setting nr = ¼;
• sea quarks depend on n
non-trivially, but
they do not alter taste-flavor symmetry.
r
• This is a very strong result.
• It underscores the importance of checking
numerically that Δn scales as expected.
• Indeed, Δ -Δ
contracted with a hard gluon
generates all the taste-breaking interactions
found by Lee&Sharpe in the Symanzik LEL.
n
n
A Clean Kill
• If you want to “kill” staggered fermions,
show that taste-breaking does not go away
in the continuum limit.
• Then we are stuck with a non-local, nonunitary theory.
• Contrary to the physical picture and the
numerical evidence accumulated so far.
vs.
Creutz
anti-Creutz
Bibliography
•
M. Creutz, “Flavor extrapolations and staggered
fermions,” hep-lat/0603020; “Chiral anomalies and
rooted staggered fermions,” hep-lat/0701018
(original title: “The evil that is rooting”), PLB.
•
C. Bernard, M. Golterman,Y. Shamir, S.R. Sharpe,
“Comment on [...],” hep-lat/0603027, PLB.
•
M. Creutz, “The author replies,” arXiv:0704.2016,
PLB.
•
M. Creutz, preceding talk.
Order of limits (a → 0, m fixed; m → 0)
is “peculiar.”
1. Computers have a finite memory, so
we take a → 0 at fixed volume, L3.
No SSB in finite volume, so must keep
m ≠ 0 when L < ∞. Take L → ∞, then
m → 0.
2. Continuum with pseudo-Goldstones
always subtle.
Quark mass dependence is mutilated:
rooted staggered theory the same for
mass m and –m, therefore ChPT f(m2).
Rooting (nothing to do with staggered)
turns m into (m4)1/4 = |m|.
Odd powers come from (near-)zero
modes iλi, λi = ci Λ3a2
!
2 + c2 a2 Λ3
"
c
1/4
2 3 2
2 2 3 2
2
1
2
[c1 Λ a + m ][c2 Λ a + m ]
= |m| +
4
|m|
and arise with correct order of limits.
const

m/g
parameter of SSB)
β=7.2, 24^2, nsmear=1 ("thick link")
0.3
red, Nf=0
red, Nf=1
red, Nf=2
p, Nf=0
p, Nf=1
p, Nf=2
5
(Nf = 1)
(Nf = 2)
staggered, Nf=0
staggered, Nf=1
staggered, Nf=2
overlap, Nf=0
overlap, Nf=1
overlap, Nf=2
0.25
χ’/g
0.2
0.15
0.1
0.05
0.3
f , and lim
m→0
0
0
0.05
Nf =1
χsca /g
0.1
0.15
m/g
0.2
0.25
Dürr, Lattice
2005 for β ≥ 4.
consistent
with 0.1599...
0.3
Conventional anomaly is cancelled.
The naïve anomaly vanishes, but
conventional staggered anomaly
[STW], a taste singlet, is correct.
Near-zero modes appear in quartets,
all with the same taste-singlet chirality
[FDHM].
Don’t confuse the two axial charges!
Topology bad: quartets must “transit”
at boundary of topological sectors.
Yes, transiting from quartets occurs.
But such gauge fields (should) have a
large gluon action and are, hence,
suppressed.
Cancellation of phantoms contrived.
Automatic if taste exchange vanishes.
Recall phantom sector discussed
earlier.
Rank (number of diagonal generators)
of chiral symmetry “wrong.”
Rank is indeed larger. For mesons
(especially for flavor singlets) the
physical sector is the taste-singlet
sector; others are phantoms.
Cancellation of phantom modes can be
tested by monitoring how the 2PI part
of scalar correlators scales with a2.
’t Hooft vertices wrong, stemming
from ± chirality near-zero modes.
Naïve chirality is ±. Staggered tastesinglet chirality of near-zero quartets
tracks Q, including sign [FHD+Mason].
Keeping m ≠ 0, near-zero modes’
powers of |m| from det1/4 and
propagators cancel as they should.
Confusion stems from a ≠ 0 w/ m = 0.
Near-zero eigenvalues appear in the numerator from the det
and in the denominator from propagators.
In most cases, this is only slightly tricky, but one can worry
that too many masses end up in the denominator.
It is tricky for taste-flavor singlets. Starting from taste-symmetric ro
theory, the correct taste-singlet correlator is
1
1
!ψ̄ΓUψ(x)ψ̄ΓUψ(x)" = − Tr[G(x, y)ΓUG(y, x)Γ] + Tr[G(x, y)Γ
4
16
Factors of four cancel. Key is approximate orthogonality of eigenve
exact as a → 0.
Cancellations occur if you take a → 0 before m → 0.
Summary
• The phantoms have a safe house as a → 0.
• Mike’s other criticisms hinge on one of two
incorrect steps:
• incorrect order of limits (obdurately
forcing m → 0 before a → 0);
• mixing up the (exact) taste non-singlet
chirality and SV taste-singlet chirality.
New Developments
HISQ
• In the past year, HPQCD has started to use
“highly-improved staggered quarks” (HISQ).
• They are designed to reduce taste-exchange
4-quark interactions.
• Verified that (one-loop) coefficients are 100
times smaller than with asqtad.
• Decay constants: Jüttner, Della Morte.
• MILC and HPQCD are discussing HISQ sea.
E. FOLLANA et al.
•
•
Fμ is FAT7 smearing
•
HYP is hypercube
smearing
•
FμUFμ is FAT7,
unitarize, FAT7
•
HISQ removes O(a2)
from FμUFμ
Our resu
errors fro
!s " 1=
HISQ ac
competit
change a
ASQTAD removes
O(a2) from Fμ
avg 4-quark coeff
FIG. 4. The splitting between the 3-link and Goldstone (0-link)
pions plotted versus the average value of the coefficients of the
taste-exchange operators (Eq. (33)) from one-loop perturbation
The m
actions i
3– 4 tim
Consequ
measure
mately b
results f
spacing)
HISQ m
The ASQ
a very e
taste-exc
This p
because
correctio
observed
could be
tive corr
Conclusinos
Conclusions
• The validity of rooted staggered fermions
• requires several common assumptions;
• hinges on (novel, but not unprecedented)
non-unitary effective field theories;
• needs a few more gaps to be filled, e.g.,
• mass of the η´;
• scaling of Shamir defect Δ .
n
• Those who use rooted staggered quarks to
help us learn about QCD would like
• the gaps filled;
• failing that a “clean kill” of the method
(e.g., Shamir’s defect not scaling; a 4-quark
operator of dimension < 6).
• If you have arguments against staggered
quarks: please be like Mike and write it up!
• That being said, it is hard to see how Δ
could fail to scale:
• it seems to scale for det = 1;
• it ought to scale for n = 1;
• why should it not scale for n = 1/4?
r
r
• A homework assignment for the wider
lattice community.
n
Thanks
• LOC and IAC
• Claude, Maarten, & Yigal
• Eduardo Follana
• Bill Bardeen
• Mike Creutz