Theoretical issues with
staggered fermion simulations
Stephan Dürr
Uni Bern, ITP
based on work together with
Christian Hoelbling (Wuppertal)
Urs Wenger (DESY Zeuthen)
Lattice 05, Dublin, July 25-30, 2005
1.5
st
W
ov
1
0.5
0
−0.5
−1
162, β=3.2, nsmear=0
0
•
•
•
•
S. Dürr, ITP Bern
0.5
1
1.5
2
2.5
3
3.5
−1.5
4
D̂ = iγµpµ + O(p2)
Dst, Dov have no additive mass renormalization, but DW has
†
†
γ5DW,ov γ5 = DW,ov
and η5Dstη5 = Dst
=⇒ det(D) ≥ 0
spec(Dna) = spec(Dst) ⊗ I2 (would be I4 in 4D)
Lattice 05, Dublin, July 25-30, 2005
1/32
Controversy in a nutshell
• In 4D the staggered action yields 4 “tastes” in the continuum, and
3 of these must be excised from ones physical predictions.
• The way how this is done is (in general) different for valence and sea quarks.
• Naively, that difference should leave no trace in the continuum limit.
Question: Is QCD with Nf = 2+1 staggered quarks fundamentally correct ?
[Is it physics from first principles or just a (phenomenologically successful) model of QCD ?]
– In the fundamental theory taste splitting may be suppressed through various
tricks (improved glue, filtering, RG blocking, ...).
– In the effective theory taste splitting effects may be parametrized and thus
“taken away” (approximately) for a variety of observables.
From a conceptual viewpoint either of these improvements is immaterial; if the staggered approach
is correct, it yields the right continuum limit for arbitrary observables without any of these.
⊕: Prove that QCD with Nf = 2+1 staggered quarks is in the right universality class.
ª: Find a single observable where the staggered answer, after continuum extrapolation, is wrong.
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
2/32
• Review: staggered action and taste representation
• Problem: rooting versus locality
• Free case: four constructions
• Interacting theory: spec(Dst) in 4D
• Interacting theory: χsca , χtop in 2D
• Correlation of det1/Nt (Dst,m) and det(Dov,m)
• Low-energy unitarity and SXPT
• Summary
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
3/32
Review: staggered action
Sna
a3 X
†
=
ψ̄(x) γµ [Uµ(x)ψ(x+ µ̂) − Uµ(x− µ̂)ψ(x− µ̂)]
2 x,µ
−→ Naive action describes 4 (2D) or 16 (4D) fermions (in general: 2 d) in the continuum.
ψ(x) = γ(x)χ(x)
Sna/st
ψ̄(x) = χ̄(x)γ(x)
†
γ(x) =
x
x
x
x
γ1 1 γ2 2 γ3 3 γ4 4
ηµ(x) =
P
(−1)µ<ν
xν
a3 X
†
=
χ̄(x) ηµ(x) [Uµ(x)χ(x+ µ̂) − Uµ(x− µ̂)χ(x− µ̂)]
2 x,µ
−→ The 2 (2D) or 4 (4D) components (in general: 2d/2) decouple.
−→ Downgrade to one component, i.e. to 2 (2D) or 4 (4D) “tastes” (in general: N t = 2d/2).
=⇒ KS procedure “thins out d.o.f.”, but distributes/intertwines spinor and taste.
χ(x) → eiθA η5(x)χ(x)
χ̄(x) → e−iθA η5(x)χ̄(x)
(m = 0)
−→ Remainder of SU (2d/2)A in taste space is sufficient to forbid additive mass renormalization.
=⇒ Further (exact) “thinning” impossible, since resulting spectrum is non-degenerate.
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
4/32
Review: taste representation
Nf staggered fields χ = u,d,s,... with 4 tastes each; hypercubic
decomposition χ(x, x+a1̂, ..., x+a1̂+a2̂+a3̂+a4̂) → q(X)
collects 2d/2 tastes with 2d/2 components each in one “blocked node”.
~
With {X} = {N }b and b = 2a the free action takes the form
Sst = b
4
X
X,µ
²¯
±°
b
q̄(X) [∇µ(γµ ⊗I) − 4µ(γ5 ⊗τµτ5)] q(X)
2
with (spinor⊗taste), τµ = γµ∗ ,τ5 = γ5 and the blocked first/second derivative
(∇µq)(X)
=
q(X+bµ̂) − q(X−bµ̂)
2b
(4µq)(X)
=
q(X+bµ̂) − 2q(X) + q(X−bµ̂)
b2
~
¡
ª
? ¡
²¯
z
~¾
±°
p=0
²¯
¡
±°
²¯
~
±°
p=π/a
.
=⇒ In taste basis the taste interactions stem from a dim=5 Wilson-like term.
−→ Do they go away with a → 0 without any trace ? (order of limits?)
p=0
p=-π/a
Interacting theory:
x-space: different tastes (and individual components of each) see slightly different local gauge field.
p-space: gluons with p ∼ π/a may kick field from one taste to another (flavor exact with Nf fields).
−→ Identification staggered=physical flavor may work only, if taste interactions minimized/eliminated.
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
5/32
|q|=0
|q|=1
|q|=4
see text
0.5
staggered
overlap
0
-0.5
0
1
3
0
1
3
0
1
# smearing steps
3
0
3
(Schwinger model, β = 4.0, 20 2 )
SD, C.Hoelbling, PRD 69, 034503 (2004) [hep-lat/0311002]
S. Dürr, ITP Bern
1
Lattice 05, Dublin, July 25-30, 2005
6/32
Problem: rooting versus locality
Marinari,Parisi,Rebbi (1981):
“On the lattice the action SG−14 tr log(Dst) will produce a violation of fundamental axioms, but we expect
the violation to disappear in the continuum limit and then recover the theory with a single fermion.”
• With any undoubled Dirac operator, e.g. D = DW , one has
Z
Z
PR
ψ̄Dψ
−S [U ]−
N
−S [U ]
Z|Nf = D[U,ψ̄,ψ] e G
= D[U ] det f (D) e G .
• With a 4-flavor Dirac operator like Dst, one may formally define
Z
N /4
−S [U ]
Z|Nf = D[U ] det f (Dst) e G
• but it is not clear whether there is a 1-taste operator Dca such that
Z
R
1/4
− ψ̄Dca ψ
det (Dst) = D[ψ̄,ψ] e
.
(A) Quenched QCD: quark loops neglected
(B) Full QCD
To guarantee locality/causality of arbitrary Green’s functions (thus to discuss renormalizability and
universality) a “candidate” operator Dca should exist with K.Jansen, NPPS 129, 3 (2004) [hep-lat/0311039]
(1)
(2)
S. Dürr, ITP Bern
a↓0
det(Dca ) −→ const·det1/4(Dst)
||Dca(x, y)|| < C e−ν|x−y|/a
with C, ν independent of U.
Lattice 05, Dublin, July 25-30, 2005
7/32
• naive rooting of Dst
P
†
†
†
Dst is a normal operator ( [Dst, Dst
] = 0 ), hence Dst =
λ λ ψ(x)ψ (y) = U Λ U
with U unitary (R/L-eigenvectors in columns) and Λ diagonal, thus f (D st) = U f (Λ) U †.
2
Dst
D1/2
st
D1/4
1
st
Im(λ)
(D+ D )1/4
st st ee
0
−1
−2
−2
1/2
−1
0
Re(λ)
1
2
3
1/4
=⇒ Dst ,Dst (in 2D, 4D) unacceptable, since D̂ = iγµpµ +O(p2) violated (and non-analytic in p).
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
8/32
†
1/2
• explicit non-locality of (Dst
Dst)ee
¯
Measure of localization: f (r) = sup{ ||ψ(x)||2 ¯ ||x−y||1 = r } where
P
ψ(x) =
Dca(x, y)η(y) with η normalized random vector at y .
−→ For local Dca one has f (r) ∝ e−r/rloc , ideally with rloc ∝ a (in any case ξphys /rloc → ∞).
B.Bunk, M.DellaMorte, K.Jansen, F.Knechtli, NPB 697, 343 (2004) [hep-lat/0403022]
1
β=6.0
β=6.2
β=6.5
6
10
0.9
0
0.8
4
0.7
10
loc 0
0.6
r /r
<f(r)>/<f(rref)>
r/r =1.86
0
r/r =3.72
2
10
0.5
0.4
0
0.3
10
0.2
−2
0.1
10
0
2
4
r/r
6
8
10
0
0
0.05
0
0.1
a/r0
0.15
0.2
†
1/2
2
Rooted operator Dca = (Dst,m
Dst,m )ee
= (−Dst
+m2)1/2
ee > 0 as a first test.
−→ f (r) and rloc (r) are finite quantities; they scale even though (for a local Dca) they should not.
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
9/32
−→ Bunk et al. show: rloc = fcn(r/r0)/Mπ in the interacting theory (numerically).
−→ Bunk et al. show: rloc = const/m in the free theory (analytically).
†
−→ In physical units rloc constant under β → ∞, thus (Dst
Dst)1/2
is non-local.
ee
A.Hart, E.Müller, PRD 70, 057502 (2004) [hep-lat/0406030]
2
2
ONE-LINK
FAT7TAD
ASQTAD
β=5.8, 12
3
β=6.0, 12 32
β=6.2, 24
1.5
rloc/r0
1
eff
eff
rloc/r0
1.5
0.5
0
4
4
1
0.5
0
2
4
r/r0
6
8
0
0
2
4
r/r0
6
8
−→ Same conclusion (of course) with improved/filtered staggered quarks.
=⇒ Question: Is there one local Dca with det(Dca ) = const·det1/4(Dst) or modulo cut-off effects ?
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
10/32
Free case: four candidates
In the free case spec(Dst) highly degenerate, thus “thinning” of d.o.f. much easier.
•
D.Adams, hep-lat/0411030
In the taste basis Dst,m = ∇µ(γµ ⊗I4) − 2b 4µ(γ5 ⊗τµτ5) + m(I4 ⊗I4) on the blocked lattice
may be used to build an operator which is simultaneously diagonal in spinor⊗taste
†
Dst,m Dst,m
b2 2
2
= [−∇ + 4 + m ](I4 ⊗I4) .
4
2
On the blocked lattice a free generalized Wilson operator Dca,m = ∇µγµ + 2b W + m yields
b
†
2
2
Dca,m Dca,m = [−∇ + ( W + m) ](I4 ⊗1) .
2
†
†
With det(Dst,m
) = det(Dst,m ) and det(DW,m
) = det(DW,m ) it follows that
b
b2 2
2
2
4 + m = ( W + m)
4
2
=⇒
−→
q
2
Dca,m = ∇µγµ + b4 42 + m2
p
a↓0
rloc = 8a/m −→ 0
S. Dürr, ITP Bern
=⇒
det
1/4
(Dst,m ) = det(Dca,m ) .
(in the free theory, on the blocked lattice)
(i.e. local, but requires m > 0)
Lattice 05, Dublin, July 25-30, 2005
11/32
•
F.Maresca, M.Peardon, hep-lat/0411029
On the blocked lattice a free 1-taste operator Dca = iγµ Pµ +Q with local (i.e. not ultra-local) Pµ, Q
†
†
yields det(Dca) = det1/4(Dst), if Dca
Dca ⊗I4taste = Dst
Dst. With Pµ = Pµ†, Q = Q† trivial in spinor
(!)
†
space and [Pµ, Q] = 0 one thus requires that Dca
Dca = Pσ Pσ + Q2 = [−4 +
have range r .
Solution 1:
Without further constraints, optimizing
locality of Dca yields spectrum and fall
r
off pattern of ωp,µ
, ωqr shown on the left.
0
-0.5
[Sol. 1 for m > 0 even better localized.]
S. Dürr, ITP Bern
0
1
0.5
1.5
0
Re λ
1
0.5
1.5
Re λ
0
10
10
-2
pµ, on-axis
10
-4
pµ, diagonal
-6
q, on-axis
q, diagonal
10
10
-2
10
-4
-8
10
|ωR(x,y)|
|ωp,q(x,y)|
0
-0.5
0
Solution 2:
Ditto, but restriction to Q = −4R with
local R yields spectrum and (slower)
fall off pattern shown on the right and
{Dca , γ5} = Dca 2Rγ5 Dca.
] ⊗I4spinor .
0.5
Im λ
where
r≥0 |d|≤r
r
ωp,µ
, ωqr
0.5
Im λ
Ansatz:
P P r
Pµ =
ωp,µ (x, y)
r≥0 |d|≤r
P P r
ωq (x, y)
Q =
b2 2
44
-10
10
-12
10
10
-6
10
-14
10
-8
-16
10
10
-18
10
-20
10
0
1
2
3
4
5
6
7
Field separation, |x-y|
Lattice 05, Dublin, July 25-30, 2005
8
9
10
-10
10
0
1
2
3
4
5
6
7
8
9
10
Field separation, |x-y|
12/32
•
Y.Shamir, PRD 71, 034509 (2005), hep-lat/0412014
Main idea: Improve taste symmetry through RG blocking. Infinitely many blocking steps would achieve
Dn → D∞ ⊗I4, while Dca after n steps satisfies det1/4(Dst) = det(Dca) det1/4(T ) where T contains
only cut-off excitations and should maintain Symanzik class, i.e. det(T ) = const·(1+O(a 2)).
Note: If one is satisfied with a = 0.4 fm for Dca (optimistic view), then original lattice with a ∼ 0.1 fm
allows for 2 steps; for 5 steps original lattice must have a ∼ 0.01 fm (cf. talk by F. Maresca).
The massless staggered action on the original lattice satisfies {D0, (γ5 ⊗ τ5)} = 0 or equivalently
(γ5 ⊗τ5)D0(γ5 ⊗τ5) = D0†. After n ≥ 1 RG steps (parameter αn) one has the generalized GW relation
−1
{Dn , (γ5 ⊗τ5)} = const·δx,y
or
{Dn, (γ5 ⊗τ5)} = Dn
2
(γ5 ⊗τ5) Dn .
αn
If one could establish (γ5⊗τ5)-hermiticity of Dn, one would easily obtain
Dn +Dn† = Dn α2n Dn† , i.e. spectrum on a circle.
P
With Dn = j λj uj vj† and vi†uk = δi,k , upon sandwiching vi†(GW)uk ,
one obtains λivi†(.⊗.)uk +vi†(.⊗.)uk λk = λivi† α2n (.⊗.)uk λk and
thus for arbitrary pair (i,k) of L,R-eigenmodes vi†,uk that vi†(γ5 ⊗τ5)uk = 0 or λi +λk − α2n λiλk = 0 .
In particular for i = k it follows that vj†(γ5 ⊗τ5)uj 6= 0 implies 2λj − α2n λ2j = 0 or λj ∈ {0, αn}.
κf
Re λ
S. Dürr, ITP Bern
H.Neuberger, PRD 70, 097504 (2004), hep-lat/0409144
•
J.Giedt, hep-lat/0507002
•
Re λ
κf
Im λ
Im λ
Similar concepts – exploratory discussion of interacting case.
Issue cast into local field-theoretical framework in 6D.
Lattice 05, Dublin, July 25-30, 2005
13/32
Interacting theory: spec(Dst) in 4D
• concept of filtering
− Replace covariant derivative, e.g. Uµ(x)ψ(x+ µ̂) − ψ(x) → UµHYP (x)ψ(x+ µ̂) − ψ(x).
− Redefines (diminishes) cut-off effects without changing Symanzik class, i.e. O(a 2) → O(a2).
− Designed to render Dst “immune” against p ∼ π/a gluons, impact on taste “symmetry” ?
• chirality of low-lying eigenmodes
Continuum: Dψ = λψ ⇐⇒ Dγ5ψ = −λγ5ψ and ψ †γ5ψ = 0 (for λ 6= 0).
Dζ = 0 and ζ †γ5ζ = ±1 characteristic signature of zero-modes.
A.Hasenfratz, http://www.rccp.tsukuba.ac.jp/lat03/Dat/OHP/a.hasenfratz.ps
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
14/32
• near-degeneracy of staggered quartets
Wilson glue, a ' 0.1 fm
0.8
CHIRALITY
0.7
0.6
0.5
HISQ
HYP
ASQTAD
ONE-LINK
0.6
0.4
0.3
0.2
0.2
0.1
0.1
0
0
-0.1
-0.1
4
6
8
10
12
14
16
2
4
6
8
10
12
14
16
10
12
14
16
EIGENVALUES
0.08
0.07
0.01
0.06
0.008
0.05
0.006
0.04
0.004
0.03
0.02
0.002
0
0
0.09
EIGENVALUES
0.012
HISQ
ASQTAD
ONE-LINK
0.5
0.3
2
CHIRALITY
0.7
0.4
0
Symanzik glue, a ' 0.1 fm
0.8
0.01
0
2
4
6
8
10
12
14
16
0
0
2
4
6
8
E.Follana, A.Hart, C.T.H.Davies, PRL 93, 241601 (2004) [hep-lat/0406010]
E.Follana, NPPS 140, 141 (2005) [hep-lat/0409062]
−→ Fine lattice plus filtering makes separation into near-zero modes and non-zero modes visible.
−→ Near-zero modes have chirality |ζ †γ5ζ| ' 1, non-zero modes have ψ †γ5ψ ' 0.
=⇒ Approximate index theorem for staggered fermions (once taste “symmetry” visible).
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
15/32
• comparing with overlap spectrum on individual configurations
CU 0001: staggered Nt = 2 simulation (β = 5.7, m = 0.01 → a ' 0.1 fm)
0.08
CU_0001, conf_000, ρ=1.0
0.08
overlap
staggered
0.06
0.06
0.04
0.04
0.02
0.02
0
0
−0.02
−0.02
−0.04
−0.04
−0.06
−0.06
−0.08
orig
1 APE
3 APE
1 HYP
3 HYP
−0.08
CU_0001, conf_006, ρ=1.0
overlap
staggered
orig
1 APE
3 APE
1 HYP
3 HYP
SD, C.Hoelbling, U.Wenger, PRD 70, 094502 (2004) [hep-lat/0406027]
−→ Filtering pushes λst out and pulls λ̂ov in (in general: drives ZS → 1).
−→ Manifest staggered quartet to single overlap mode correspondence (modulo different Z S factor).
=⇒ In particular: 4|q| staggered near-zero modes on (typical) configuration with overlap charge q .
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
16/32
• cut-off dependence of taste-splitting
Matched lattices (β = 5.66,5.79,6.00,6.18, L4 = 64,84,124,164) with V = (1.12 fm)4 = 1.57 fm4.
4
4
8 , β=5.79, conf_000, ρ=1.0
16 , β=6.18, conf_000, ρ=1.0
0.25
0.2
0.1
0.15
0.1
1 HYP over
1 HYP bare stag
1 HYP ren. stag
0.05
0
0
2
4
6
8
10
12
14
16
0.05
0
1 HYP over
1 HYP bare stag
1 HYP ren. stag
0
2
4
6
8
10
12
14
16
0.25
0.2
0.1
0.15
0.1
3 HYP over
3 HYP bare stag
3 HYP ren. stag
0.05
0
0
2
4
6
8
10
12
14
16
0.05
0
3 HYP over
3 HYP bare stag
3 HYP ren. stag
0
2
4
6
8
10
12
14
16
SD, C.Hoelbling, U.Wenger, PRD 70, 094502 (2004) [hep-lat/0406027]
−→ On matched lattices taste “symmetry” (quartet near-degeneracy) improves with a → 0.
−→ Rescaling with ZSov /ZSst: quantitative agreement of staggered quartet with single overlap mode.
=⇒ Explicit tests passed by Dov (index theorem, RMT agreement, Banks-Casher, ...) transfer to Dst.
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
17/32
• explicit check that hλii/hλj i agrees with RMT prediction
7
Q=0
|Q|=1
|Q|=2
6
HISQ
HYP
ASQTAD
ONE-LINK
5
4
3
2
1
2/1
4/1
4/2
3/1
3/2
4/3
2/1
4/1
4/2
3/1
3/2
4/3
2/1
4/1
4/2
3/1
3/2
4/3
E.Follana, A.Hart, C.T.H.Davies, PRL 93, 241601 (2004) [hep-lat/0406010]
−→ Sectoral hλii/hλj i agrees with RMT prediction (up to small finite volume effects ?).
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
18/32
• explicit check that CED(λmin ) agrees with RMT prediction
1
Q=1
0.6
0.4
Q=2
0.2
UFat7 Asq
0
0
2
4
6
8
0.4
Q=2
0.2
0
10
2
4
ζ
6
8
Q=2
0
2
4
6
8
10
a = 0.077 fm
1
Q=3
0.8
0.6
0.4
0.2
0
Overlap
q = 0, quartets 1,2,3,4
Pc(ζ)
Pc(ζ)
Q=1
Q=2
ζ
a = 0.077 fm
Q=0
0
10
mode 1, q = 0, 1, 2, 3
0.8
0.4
ζ
K.Y.Wong, R.M.Woloshyn, PRD 71, 094508 (2005) [hep-lat/0412001]
1
Q=1
0.6
0.2
HYP
0
Q=0
0.8
Q=1
0.6
none
Dov
1
Q=0
0.8
Pcumm (ζ)
Pcumm (ζ)
1
Q=0
0.8
HYP
Dst
Pcumm (ζ)
UFat7×Asq
Dst
k=1
k=2
k=3
k=4
0.6
0.4
0.2
0
2
4
ζ
6
8
10
0
0
2
4
6
8
ζ
10
12
14
16
E.Follana, A.Hart, C.T.H.Davies, Q.Mason, hep-lat/0507011
=⇒ On fine enough lattices (and with filtering) agreement with RMT for individual topological sectors.
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
19/32
• evidence that taste-splitting is O(a2) effect
∆ [fuzziness of first would−be zero mode]
0
0.35
orig
1 HYP
3 HYP
0.3
∆ [fuzziness of first positive mode]
0.5
0.2
0.4
0.15
0.3
0.1
0.2
0.05
0.1
0.005
0.01
0.015
orig
1 HYP
3 HYP
0.6
0.25
0
0
1
0.7
0
0
0.02
0.005
0.01
0.015
0.02
SD, C.Hoelbling, U.Wenger, PRD 70, 094502 (2004) [hep-lat/0406027]
|Q|=1
|Q|=2
20
|Q|=1
Q=0
30
ONE-LINK
ASQTAD
FAT7xASQ
|Q|=2
ONE-LINK
ASQTAD
FAT7xASQ
15
<δΛ1>Q / MeV
<Λ0>Q / MeV
20
10
10
5
0
0
0.01
0
0.01
2
(Lattice spacing, a / fm)
0
0
0.01
0
0.01
0
2
(Lattice spacing, a / fm)
0.01
E.Follana, A.Hart, C.T.H.Davies, Q.Mason, hep-lat/0507011
=⇒ Minimal a for taste breaking to possibly be O(a2) effect seems enlarged through filtering.
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
20/32
Interacting theory: χsca , χtop in 2D
• In 2D rooting issue exists for Nf = 1, 3, ..., since Dst yields 2 fermions in the continuum.
• In 2D scale may be set through fundamental coupling: [g] = [e] = 1 (no UV running), β = 1/(ag) 2.
µ
¶
anomaly
induced
• Analytic Nf = 1 result: limm=0hψ̄ψi/g = eγ/(2π 3/2) = 0.1599...
J.Schwinger (1962)
overlap fermions (ρ = 1):
χov
sca
g
=
N
ov ) P 1 i
√ hdet f (Dm
β
λ̂+m
N
V
ov )i
hdet f (Dm
ov
det(Dm
)=
λ̂ =
staggered fermions:
(bare)
Q
((1− m
2 )λ + m)
1
(1/λ−1/2)
χst
sca
g
=
√ hdetNf /2 (D st ) P 1 i
m
β
λ+m
2V
Nf /2 st
hdet
(Dm )i
st
det(Dm
)=
(=stereogr. proj.)
Q
(bare)
(λ + m)
− Sample quenched, compute all λ [LAPACK] and build observables (variable m) in analysis program.
− In plots below theory is unitary (at least with Dov ); we use msea = mval throughout.
− Details in
S. Dürr, ITP Bern
SD, C.Hoelbling, PRD 69, 034503 (2004) [hep-lat/0311002]& PRD 71, 054501 (2005) [hep-lat/0411022].
Lattice 05, Dublin, July 25-30, 2005
21/32
• χsca with Dst and Dov at β = 7.2
Continuum:

 g/m
χsca /g ∝
const

m/g
(Nf = 0)
(Nf = 1)
(Nf = 2)
β=7.2, 24^2, nsmear=0 ("thin link")
0.3
staggered, Nf=0
staggered, Nf=1
staggered, Nf=2
overlap, Nf=0
overlap, Nf=1
overlap, Nf=2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
Overlap:
0.05
0.1
0.15
m/g
0.2
0.25
staggered, Nf=0
staggered, Nf=1
staggered, Nf=2
overlap, Nf=0
overlap, Nf=1
overlap, Nf=2
0.25
χ’/g
0.2
χ’/g
β=7.2, 24^2, nsmear=1 ("thick link")
0.3
0.25
0
(in 2D not an order
parameter of SSB)
0.3
0
0
0.05
0.1
0.15
m/g
0.2
0.3
0.25
N =1
f
qualitatively correct behavior ∀Nf , and lim χsca
/g consistent with 0.1599... for β ≥ 4.
m→0
Staggered: qualitatively wrong behavior in chiral limit for Nf = 0,1, since lim χsca /g = 0 for any β ,
m→0
but filtering shifts point where staggered answer fails more chiral (rel. to taste splitting ?).
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
22/32
Question: can one obtain χsca /e = 0.1599... with staggered fermions, if one
first extrapolates to the continuum, taking m → 0 afterwards ?
• χsca with Dst and Dov in the continuum
overlap (nsmear=1)
0.2
L=12 β=1.8
L=16 β=3.2
L=20 β=5
L=24 β=7.2
0.19
0.18
0.17
0.18
0.17
0.16
χ’/e
χ’/e
L=12 β=1.8
L=16 β=3.2
L=20 β=5
L=24 β=7.2
0.19
0.16
0.15
0.15
0.14
0.14
0.13
0.13
0.12
0.12
0.11
0.11
0.1
staggered (nsmear=1)
0.2
0
0.1
m/e
0.2
0.3
0.1
0
0.1
m/e
0.2
- small cut-off effects
- large cut-off effects
- for m/e ≤ 0.3 already β ≥ 1.8
sufficient to enter scaling window
- for small m/e ever larger β
needed to enter scaling window
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
0.3
23/32
0.165
exact result
staggered
overlap
0.16
χ’subt/e
0.155
0.15
0.145
0.14
0.135
0.13
0
0.05
m/e
• overlap (“universal behavior”):
0 ov (m/e,a2 )
χsca
lima→0 limm→0
e
0
ov
χsca (m/e,a2 )
limm→0 lima→0
e
=
=
eγ
2π 3/2
eγ
2π 3/2
0.1
0.15
• staggered (“non-commutativity”):
st (m/e,a2 )
χ0sca
e
st (m/e,a2 )
χ0sca
limm→0 lima→0
e
lima→0 limm→0
=0
=
eγ
2π 3/2
=⇒ Staggered fermions see chiral anomaly, but only if lima→0 is taken first and limm→0 thereafter.
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
24/32
• χsca with Dst and Dov in hybrid mode
β=7.2, 24^2, nsmear=0
β=7.2, 24^2, nsmear=1
0.3
0.3
stag-val, Nf=0
stag-val, Nf=1 ov-sea
stag-val, Nf=2 ov-sea
ov-val, Nf=0
ov-val, Nf=1 stag-sea
ov-val, Nf=2 stag-sea
0.25
0.25
0.2
χ’/g
χ’/g
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
0.05
0.1
0.15
m/g
0.2
0.25
stag-val, Nf=0
stag-val, Nf=1 ov-sea
stag-val, Nf=2 ov-sea
ov-val, Nf=0
ov-val, Nf=1 stag-sea
ov-val, Nf=2 stag-sea
0.3
0
0
0.05
0.1
0.15
m/g
0.2
0.25
0.3
−→ Warning for all “hybrid action” studies:
−→ Deficiencies of Dst in sea/valence sector may overwhelm good properties of Dov in other sector.
=⇒ Failure of χscal in staggered Nt = 1 case cannot be attributed to sea or valence sector alone.
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
25/32
• χsca with Dst and Dov in the quenched case
χsca /e ∝
Reminder:
½
e/m
m/e
Chiral condensate for Nf=0
(Nf = 0)
(Nf = 2)
Chiral condensate for Nf=2
β=7.2, N=24, nsmear=1, coeff. quenched divergence
β=7.2, N=24, nsmear=1, coeff. linear piece
0.04
2
overlap
staggered
0.03
0.02
χ/m
mχ
overlap
staggered
1
0.01
0
0
0.1
m/e
0.2
0
0
0.1
m/e
0.2
−→ Even in the quenched theory a staggered non-commutativity (in an artificial observable) found.
Earlier paper: J.Smit, J.C.Vink, NPB 286, 485 (1987).
=⇒ Non-commutativity not genuinely tied to rooted determinant, more likely due to mismatch in
sea and valence sector; compare discussion in C.Bernard, PRD 71, 094020 (2005) [hep-lat/0412030].
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
26/32
• χtop with Dst and Dov at β = 4
ov
χtop
β
=
V
ov
)q 2i
hdetNf (Dm
ov )i
hdetNf (Dm
st
χtop
β
=
V
st
)q 2i
hdetNf /2(Dm
st )i
hdetNf /2(Dm
nsmear=0 (“thin link”)
0.04
χtop/g
2
0.03
2
χtop/g
ind = − 12 tr(γ5Dov )
R
1
F12 d2x
2π
nsmear=1 (“thick link”)
0.04
0.03
q=
(
0.02
staggered, Nf=1
staggered, Nf=2
overlap, Nf=1
overlap, Nf=2
0.01
0
0
Overlap:
0.1
0.2
0.3
m/g
0.4
0.5
0.6
0.02
staggered, Nf=1
staggered, Nf=2
overlap, Nf=1
overlap, Nf=2
0.01
0.7
0
0
0.1
0.2
0.3
m/g
0.4
0.5
0.6
0.7
small O(a2) effects, fairly insensitive to filtering (overlap yields good IR↔UV separation).
Staggered: large O(a2) effects, increase towards chiral limit, substantially reduced through filtering.
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
27/32
• χtop with Dst and Dov in the continuum
¯
Matched lattices: β = 1.8, 3.2, 5.0, 7.2, 9.8 with L = 12, 16, 20, 24, 28 yields LM η0 ¯ N =1,m=0 ' 5.
f
0.026
0.024
staggered
0.014
nsmear=0
nsmear=1
nsmear=3
overlap Nf=1 χtop/e2 at m/e=0.1
staggered Nf=1 χtop/e2 at m/e=0.1
0.028
0.022
0.02
0.018
0.016
0.014
0.012
0.01
0.008
0
0.1
0.2
0.3
1/β
0.4
0.5
0.6
0.0135
overlap
nsmear=0
nsmear=1
nsmear=3
0.013
0.0125
0.012
0.0115
0.011
0.0105
0.01
0.0095
0
0.1
0.2
0.3
1/β
0.4
0.5
0.6
−→ Staggered extrapolation much steeper than with overlap (different scales).
−→ Combined fit with several filtering levels yields cost-effective continuum extrapolation.
N =1
f
=⇒ Results for χtop
suggest universal continuum limit, in spite of det1/2(Dst).
−→ Similar agreement in other continuum extrapolated quantities, e.g. for F HQ.
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
28/32
Correlation of 12 log det(Dst,m) and log det(Dov,m) in 2D
Determinant ratio:
λ1 λ2 ... ¯¯
λ01 λ02 ... ov
'
β=9.8
β=5.0
β=1.8
st0/ov0
γ1 γ2 ... ¯¯
0 γ 0 ... st
γ1
2
?
(γk =
st1/ov1
p
λ2k−1 λ2k geometric staggered mean)
ov1/ov0
st1/st0
4
2
0
-2
-4
4
2
0
-2
-4
4
2
0
-2
-4
-6 -4 -2 0 2 4
-6 -4 -2 0 2 4
-6 -4 -2 0 2 4
-6 -4 -2 0 2 4
1APE
−→ At fixed quark mass det1/2(Dst,m
) in 2D generates ensemble that is closer to the one from
1APE
none
det(Dov,m
) than the latter would be to det(Dov,m
), and the agreement improves with β .
=⇒ Maybe, Dov,m is a “candidate” operator with det(Dca,m ) = const·det1/2(Dst,m )(1+O(a2)) .
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
29/32
Low-energy unitarity and SXPT
• continuum chiral perturbation theory (XPT)
Observation: In QCD with p ¿ 1 GeV chiral symmetry constrains interactions of low-energy
degrees of freedom with each other and with heavier particles (e.g. nuclei).
Consider ππ forward
scattering (s = 0,I = 0)
at low momentum:
Fππ (ν)
s=0
I=0
I=0
=
T
[0, 2Mπ (Mπ +ν), 2Mπ (Mπ −ν)]
=
Mπ2
4
− 2 + O(p )
Fπ
Mπ ,p↓0
−−−→
0
J.Gasser, H.Leutwyler, Ann.Phys. 158, 142 (1984), NPB, 250, 465 (1985)
Taste splitting makes
¯ 5 ⊗T )u
most d(γ
combinations become
non-Goldstone bosons:
(Mπ r0)2
• staggered chiral perturbation theory (SXPT)
6
p
p
p
p
p
p
p
p
p p
p
p
p
p p
p
p p p p
p p
p p
p p
p
p
p
p p p
p p
p p p p p p
p p p p
p
p
pp p pp p p p p p p p p p
pp p p p p p p
p pp pp pp pp p p p p p p p p p p p p p p p
p
pu
p
P
pup
p
A
pup
p
T
pup p
V
pup p
S
(a/r0)
-
2
W.J.Lee, S.R.Sharpe, PRD 60, 114503 (1999) [hep-lat/9905023]
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
30/32
Assumption: With Nf flavors of (4-taste) quark fields the pattern of SSB is
SU (4Nf )L ×SU (4Nf )R → SU (4Nf )V leading to 16Nf 2 −1
pseudo-Goldstone bosons, collected in the 12×12 matrix (Nf = 3)
U =e
iΦ/f

9,16
Φu π + K +
X
λa b
ab
−
0
Φ=π
Φ
T
Φd K  =
2
a,b=1
K − K̄ 0 Φs


m u I4
0
0
M = 0
0 
m d I4
0
0
m s I4

that transforms as U → VLU VR† under chiral rotations with unitary VL,VR .
With f ' 122 MeV and Σ ' (270 MeV)3 the LO-Lagrangian (counting scheme p2 ∼ m ∼ a2) reads
f2
Σ
2m20
†
†
2
L=
tr(∂µU ∂µU ) − tr(M U+M U ) +
(Φu,TS +Φd,TS +Φs,TS ) + a VTB
8
2
3
C.Aubin, C.Bernard, PRD 68, 034014 (2003) [hep-lat/0304014] & PRD 68, 074011 (2003) [hep-lat/0306026]
and allows for systematic treatment of quantities covered by XPT, e.g. M π , fπ , MK , fK .
⊕ SXPT analysis includes taste breaking effects.
⊕
⊕
ª
ª
Overall fits with horrific covariance matrices (“fitting herds of elephants”) yield acceptable χ 2/d.o.f.
Some tests [Nt = 1.28(12) per flavor, SXPT logs] successful, more [e.g. Sharpe, van de Water] to come.
What about physical observables not covered by (S)XPT ?
Unphysical tastes excised from predictions, but differently in valence and sea sector (unitarity?).
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
31/32
Summary
• Full QCD with Nf = 2+1 staggered fermions is controversial, since the Boltzmann weight det 1/4(Dst)
assumes a taste symmetry which is only approximate.
• Formally, the taste symmetry breaking is due to a dimension 5 Wilson-type term in the taste basis and
should thus go away in the continuum limit.
• Weak coupling, filtering, RG blocking reduce the taste splitting and give staggered quarks more
appealing features, but there is no guarantee that no trace is left1 in the continuum.
• One legal 1-flavor Dca with det(Dca ) = const·det1/4(Dst)(1+O(a2)) is sufficient to re-interpret
existing MILC ensembles as being generated with a local action.
• The problem of (exact) unitarity in the fundamental theory remains, unless same D ca is used in
valence sector, too. Otherwise “partially quenched” situation with unitarity restored with a → 0 ?
1 Caveat: at least in some theories the cases m = 0 and m > 0 might be different in this respect.
S. Dürr, ITP Bern
Lattice 05, Dublin, July 25-30, 2005
32/32