Chiral Perturbation Theory
for
Staggered sea and Ginsparg-Wilson valence quarks
Oliver Bär
University of Tsukuba
in collaboration with
Claude Bernard, Gautam Rupak and Noam Shoresh
2nd ILFTN Workshop
Edinburgh
Mixed lattice theory
We consider Lattice QCD with
• Nf staggered sea quarks
*
N
Ginsparg-Wilson
quarks
V
•
• NV Ginsparg-Wilson ghosts *
* local Dirac operator that satisfies the Ginsparg-Wilson relation
(overlap or domain-wall)
Why mixed lattice theories ?
•
Cost efficient when using existing unquenched configurations
( comparable to quenched overlap / domain-wall simulations )
•
•
Exact chiral symmetry in the valence sector
Naive expectation: Unquenched QCD in the continuum limit
Previous studies
LHP collaboration
UKQCD
see talks by D Richards
G Fleming
But
•
•
•
Unitarity is lost at non-zero lattice spacing
No unquenched sector ( mVal = mSea ) at non-zero lattice spacing
How to match the quark masses in order to reach unquenched QCD
in the continuum limit ?
What can ChPT teach us about mixed lattice theories ?
Symanzik action for the mixed lattice theory
2
SSym = S4 + a S6 + . . .
•
•
•
First term: Continuum PQ QCD action with 4Nf sea quarks
No terms linear in the lattice spacing (incompatible with the symmetries)
Three types of terms quadratic in the lattice spacing
•
•
•
Type 1: Involve sea fields only
known
Type 2: Involve valence fields only
Type 3: Involve both
new !
known
Lee, Sharpe, Bernard, Aubin
OB, Rupak, Shoresh
Mixed 4 - fermion operators
General structure of mixed 4 - fermion operators
(6)
OMix
•
•
•
= ψ S (γSpin ⊗
a
tColor )ψS
ψ V (γSpin ⊗
a
tColor )ψV
Allowed operators are products of a sea-sea and a val-val bilinear
The sea-sea bilinear is trivial in taste space !
In total there are four types of these operators
γSpin : vector or axial vector
taColor : colour group generator or identity
Spontaneous Chiral Symmetry Breaking
Symmetry group of PQ QCD
GPQ QCD = SU (4Nf + NV |NV )L ⊗ SU (4Nf + NV |NV )R ,
•
Assumption: spontaneously broken to vector subgroup
light pseudo scalar Goldstone bosons
described by a chiral Lagrangian
•
Follow standard procedure for constructing the chiral Lagrangian
( spurion analysis for the mass and the symmetry breaking terms in S6 )
The pseudo scalar fields
In the following we consider
3 sea quark flavors: u, d, s
2 valence quark flavors: x, y
Σ = exp(2iΦ/f )






Φ=




U
π−
K−
Q†ux
†
Quy
···
···
π+
D
K¯0
†
Qdx
†
Qdy
···
···
K+
K0
S
Q†sx
†
Qsy
···
···
Qux
Qdx
Qsx
X
−
P
Rx̃x
Rỹx
Quy
Qdy
Qsy
P+
Y
Rx̃y
Rỹy
···
···
···
†
Rx̃x
†
Rx̃y
X̃
P̃ −
···
···
···
†
Rỹx
†
Rỹy
P̃ +
Ỹ











LO chiral Lagrangian
2
2
f2
f
B
m
0
†
†
†
2
2
Lχ =
!∂µ Σ∂µ Σ " −
!ΣM + M Σ " +
!Φ" + a V.
8
4
6
• !. . ." = supertrace
• M = quark mass matrix
• f, B = usual LO low energy constants
• singlet explicitly left in the Lagrangian (for convenience)
later we take the limit
•
•
V
m0 → ∞
2
= potential terms proportional to a
assumption:
mq ≈ a2 Λ3QCD
The potential
The potential can be written as a sum of three terms
V = US +
•
!
US
+ UV
!
U
+
U
The first two terms S
S
•
are the same terms as for the staggered ChPT case
example:
•
•
†
ˆ
ˆ
C1 !ξ5 PS Σ ξ5 PS Σ "
PS : projector on sea fields
ξˆ5 : taste matrix
in total 8 terms with 8 unknown low-energy constants
all terms break the SU(4) taste symmetry
( explicit appearance of taste matrices )
Lee, Sharpe,
Bernard, Aubin
The potential UV
2
!
UV = −a CMix τ3 Στ3 Σ
"
†
τ3 = (1S , −1V )
•
•
•
•
One term with one low-energy constant
UV originates in the mixed 4 - fermion operators
UV mixes sea and valence fields
UV preserves the SU(4) taste symmetry
( expected since the mixed four-fermion operators are trivial in taste )
•
Consequence: Taste symmetry violations start at 2 loops
LO masses
Val-Val:
2
mP
= B(mx + my )
Vanishes for zero quark mass because of exact chiral symmetry
Sea-Sea:
2
mF F ! ,b
Taste splittings ∆(ξb )
are the same as in SChPT
= B(mF +
!
mF )
2
+ a ∆(ξb )
F != F ! : Sea quark flavor
ξb : taste label
Taste splittings in the MILC simulations
plot by C. Bernard
LO masses
Sea-Val:
2
mF x
2
= B(mF + mx ) + a ∆Mix
16CMix
≡
f2
•
This mass depends on the new low energy constant: ∆Mix
•
This mass can be directly measured from the propagator of a mixed
meson
•
This mass will enter the 1-loop expression for the decay constants
(see later)
NLO results
•
We computed to one loop:
•
•
Val - Val pseudo scalar masses
Val - Val pseudo scalar decay constants
• Done for the most general case with mu != md != ms != mx != my
• The following results hold for
mu = md ≡ m̂
• 2+1 sea quark flavors
mx = my
• degenerate valence quarks
• simulations employing the “4th root trick”
1
factor
4
“quark flow” diagram
The pseudo scalar mass
NLO 2
(mP + )
2Bmx
1
2
=1+
16π 2 f 2 3
!
[2,2]
[2]
[2]
2
˜
R1 ({MX,I }; {µI }) "(mX )
+
2
!
"
[2,2]
[2]
[2]
2
Dj,1 ({MX,I }; {µI }) !(mj )
j=1
+ analytic
l(m2 ), ˜l(m2 ) : chiral logs
R, D : residue functions
ratios of products
involving LO masses
(m2UI − m2X )(m2SI − m2X )
R1 =
m2ηI − m2X
mX = mLO
P+
Question
How to choose the quark masses such that the result resembles
the full (unquenched theory) ?
Criterion: Bring the coefficients of the chiral logs to continuum form
i.e. simplify the residue functions
This is achieved by choosing
( at this order )
mX = mπI0
2
2Bmx = 2B m̂ + a ∆(ξI )
mx > m̂
Explicit result
NLO 2
(mP + )
2Bmx
1
=1+
16π 2 f 2
$
!
" 2 # 1 " 2 #
" mπ0 − " mηI +
I
3
32B
16B
2
(2L
−
L
)
(2m
)
+
(2L
−
L
)
(2
m̂
+
m
)
+
a
C
8
5
x
6
4
s
2
2
f
f
•
Simplified residues, but still not unquenched
•
•
Masses in the chiral logs are taste singlet masses
•
Matching the Goldstone pion looks more partially quenched
The mixed result has always some remnant of partial quenching
Note: Our criterion for trivial residues leads to a matching that involves
the taste singlet pion, not the Goldstone pion !
Decay constant
fPNLO
!
1
2
2
−
2"
m
−
"
m
=f 1+
ux
sx
16π 2 f 2
"
#
$
#
$%
16B
8B
2
+ 2 L5 (2mx ) + 2 L4 (2m̂ + ms ) + a F
f
f
•
Involves the mixed meson masses in the chiral logs
( different from the pion mass )
•
•
No residue functions ( cancellation for the case mx = my )
No obviously preferred way to define an unquenched pion decay constant
Number of unknown constants
1. Provided one uses the MILC configurations and the MILC results for
the sea-sea meson masses
2. Provided one has measured the mixed meson mass
•
•
The pion mass has one unknown low energy constant
The decay constant has one unknown low energy constant
in addition to the usual continuum ChPT constants
The mixed results are highly constrained
Comments on the quark mass matching
•
Pion mass result suggests
mP + = mπ +
I
for the quark mass matching
•
Nevertheless
Using any mπb on the right hand side leads to unquenched QCD
in the continuum limit
( all
•
mπ b
become degenerate in the continuum limit )
But
PQ effects at non-zero lattice spacing are probably much more
pronounced
The scalar propagator
( LHP collaboration )
Quark mass matching using
mP + = mπ +
5
negative propagator
clear sign of
partial quenching !
What happens for
mP + = mπ +
I
plot by K. Orginos
?
Some MILC numbers
a
mπ5 /mρ
mπI /mρ
≈ 0.12fm
0.3
0.59
≈ 0.09fm
0.49
0.57
≈ 0.09fm
0.38
0.48
extracted from
C. Aubin et. al.
hep-lat/0402030
mπI /mρ is significantly larger than mπ5 /mρ
Summary and outlook
•
1-loop results for pseudo scalar masses and decay constants in
mixed staggered - GW simulations are now available
hep-lat/0503???
•
What one can do next:
Vector mesons
Baryons
...
•
What one should do next:
Analyze existing data with the fit forms presented here
Does it work ???