Chiral Perturbation Theory
at non-zero lattice spacing
Oliver Baer
University of Tsukuba
Matching Light Quarks to Hadrons
Workshop at the BENASQUE CENTER FOR SCIENCE
27 July 2004
Introduction: Chiral extrapolation
Some numbers I heard at Lattice 2004:
Collaboration
UKQCD
SPQcdR
qq+q
CP-PACS
CP-PACS
MILC
RBC
Fermion type
Wilson
Wilson
Wilson
Wilson
Wilson
Staggered
Domain Wall
Nf
2
2
2
2
2+1
2+1
2
smallest mπ /mρ
Lattice 2004
Talk/poster by:
~0.44
~0.66
~0.47
~0.35
~0.62
~0.3
~0.53
physical value: 0.18
Still needed: rather long extrapolation in the light quark mass
Tarantino
Scholz
Namekawa
Ishikawa
Bernard
Izubuchi
Chiral Perturbation Theory (ChPT)
Analytic guidance is required for the extrapolation
⇒ provided by ChPT
ChPT: Effective theory for low-energy QCD
Weinberg, Gasser, Leutwyler
⇒ quark mass dependence of observables
Example:
!
m2π = M02 1 +
M02
16π 2 Nf f 2
M02 = (mu + md )B
Question:
ln M02 −
8LM02
f2
"
f, B, L : low-energy constants
Does ChPT describe the lattice data ?
Smoking gun: curvature due to chiral logs
Panel discussion
Lattice 2002, Boston
A potential problem
ChPT formulae derived for a = 0 only !
Lattice data
Step 1:
continuum limit
−→
a→0
Continuum
−→
Step 2:
chiral extrapolation
mq → mq,phys
Real world QCD
A potential problem
ChPT formulae derived for a = 0 only !
Lattice data
.......
−→
a→0
Step 2:
continuum limit
−→
Continuum
−→
−→
Step 1:
chiral extrapolation
Step 1:
continuum limit
Step 2:
chiral extrapolation
mq → mq,phys
Real world QCD
What about the opposite order ?
Desirable: ChPT for lattice theories at non-zero lattice spacing
ChPT at non-zero lattice spacing:
Main strategy
Two-step matching to effective theories:
1. Lattice theory
Symanzik’s effective theory
→
2. Symanzik’s effective theory
Lee, Sharpe ‘98
Sharpe, Singleton ‘98
continuum theory making
the a-dependence explicit
→
ChPT
including the a-dependence
⇒ Chiral expressions for mπ , fπ ... with explicit a-dependence
Symanzik's action for Lattice QCD with Wilson fermions
Locality and symmetries of the lattice theory
!
2
Sef f = SQCD + a c ψ iσµν Gµν ψ + O(a )
⇒
•
At O(a) only one additional operator ( making use of EOM )
• c
•
: unknown coefficient (”low-energy constant”)
O(a2 ) : dim-6 operators:
- fermion bilinears
- 4-fermion operators
O(4) rotation invariance is broken at this order
Sheikholeslami, Wohlert
Reminder: Chiral Lagrangian
Fields:
Lagrangian:
Σ(x) = exp
!
2i a
π (x)T a
F
"
!
"
! !
"
!
Lef f Σ, M = Lef f Σ , M
!
Σ = LΣR
†
M ! = LM R†
Ta:
Group generators
M:
Quark mass matrix
L, R
: Left, Right
transformations
Expand in powers of derivatives and masses: Lef f = L2 + L4 + . . .
2
"
"
f2 !
f
B ! †
†
†
L2 =
tr ∂µ Σ∂µ Σ −
tr Σ M + M Σ
4
2
f, B : undetermined low-energy constants
Chiral Lagrangian including a
Sef f = SQCD + a c
!
2
ψ iσµν Gµν ψ + O(a )
Pauli term breaks the chiral symmetry exactly like the mass term in SQCD
⇒ a enters chiral Lagrangian exactly like the mass term
⇒
2
"
"
f2 !
f
B ! †
†
†
L2 =
tr ∂µ Σ∂µ Σ −
tr Σ M + M Σ
4
2
!
"
f 2 W0
†
−
a tr Σ + Σ
2
Sharpe, Singleton ‘98
Rupak, Shoresh ‘02
W0 : new undetermined low-energy constant
2
includes c = c(g0 )
not really a constant (weak a dependence)
L4 -Lagrangian:
L4 = L4 (p4 , p2 m, m2 ) + L4 (p2 a, ma) + L4 (a2 )
Gasser, Leutwyler ‘85
Rupak, Shoresh ‘02
•
• Total number of low-energy constants:
Rupak, Shoresh, OB ‘03
Aoki ‘03
No O(4) symmetry breaking terms in L4 (a2 ) ( start at O(a2 p4 ) )
10 Li + (5 + 3) Wi = 18 !
Li are physical ChPT parameters → values are interesting
( independent of a )
Role of Wi : parameterize a dependence → values are less interesting
Note: Wi are not universal ( depend on the lattice action )
Power counting
The power counting is non-trivial because of
1. the additive mass renormalization
2. two symmetry breaking parameters
⇒ their relative size matters
1
∝
a
a , mquark
Leading order pion mass ( degenerate case )
M02 = 2mB + 2aW0
m = mu = md
Leading a -effect: Shift in the pion mass
But: This shift might already be absorbed in mc
e.g.
2
mπ
for
=0
!
m = Zm (m0 − mcr ) = 0
⇒ Express the observables mπ , fπ in terms of m!
Note: additional shifts enter at
2
a , ...
Two expansion parameters:
2Bm!
(4πf )2
Both need to be small
2W0 a
(4πf )2
Relative size determines which terms are LO, NLO etc.
Example:
2
L2 (p2 , m! )
→
→
L4 (a )
!
"
†
f B m tr Σ + Σ
! "
#$
† 2
W a tr Σ + Σ
2
m !a
LO
NLO
m! ≈ a2
LO
LO
2
!
!
2
The proper power counting depends on the relative size of m and a
!
Different power countings have been discussed:
If m! ! a2 → continuum like ChPT + small O(an ) corrections
Rupak, Shoresh, OB ‘03
If m! ≈ a2 →
qualitatively different !
Non-trivial phase diagram
Sharpe, Singleton ‘98
Modification of chiral logs
Aoki ‘03
Non-trivial phase diagram
Potential energy:
(Nf = 2)
Sharpe, Singleton ‘98
!
"
# !
"$
†
2
† 2
V = −c1 m tr Σ + Σ + c2 a tr Σ + Σ
!
c1 (f, B)
c2 (f, B, Wi )
A: sign c2 = +1
⇒
Σvacuum != ±1
Aoki ’85
Aoki phase
flavor and parity are broken
massless pions at a != 0
B: sign c2 = −1
⇒
Σvacuum = ±1
no flavor/parity breaking
no massless pions
Question: Scenario A or B ?
Depends on the underlying lattice action
Plaquette action + unimproved Wilson: Scenario B
Data:
Theory:
β = 5.2
0.7
3
0.6
2
(amPS )2
m2 f2" / 2|c2|
0.5
0.4
0.3
0.2
1
123 × 24 low
123 × 24 high
163 × 32 low
163 × 32 high
0.1
0
-2
Sharpe, Singleton ‘98
-1
0
1
2
0.0
2.895
2.9
2.905
2.91
!
F. Farchioni et.al. ‘04
Questions:
•
•
•
Which scenario is realized for improved fermions ?
Can we find a lattice action with very small c2 ?
What are the consequences for overlap fermions ?
2.915
2.92 2.925
1/(2κ)
2.93
2.935
2.94
2.945
similar findings by
Ilgenfritz et. al. ‘03
Pseudo-scalar mass to 1-loop (2 deg. flavors)
Aoki ‘03
!
"
!
!
2
!
m (2B + w1 a)
2Bm
w0 a
2Bm
2
!
mπ = 2Bm 1 +
log
+
log
2
2
2
2
2
32π f
Λ
32π f
Λ2
+ analytic
•
•
•
•
a → 0 gives the correct continuum expression
w0 , w1 : combinations of low-energy constants Wi
!
!
m
ln
m
The coefficient of
is no longer universal at non-zero a
The lattice spacing generates an a2 ln m! contribution
⇒ This term dominates [....] for small m!
But: A resummation of the (ln m ) terms can be performed ....
! n
Resummend Pseudo-scalar mass
Aoki ’03
2
2 2
!
"
#
$
w̃
a
/32π
f
0
!
!
!
m (2B + w1 a)
2Bm
2Bm
2
!
mπ = 2B̃m 1 +
log
log
2
2
2
32π f
Λ
Λ̃2
+ analytic
a2
Expansion of {. . .}
gives result on previous slide
Assumption: Aoki phase scenario
Similar modifications for fπ and mAWI
Feature in WChPT:
The coefficients of the chiral logs can be altered by O(a) corrections
Question: What does m ≈ a2 precisely mean for a given lattice action ?
Crude dimensional argument:
a = 0.15fm
m ≈ a2 Λ3QCD
⇒
ΛQCD = 300MeV
Current lattice simulations do not satisfy
m ≈ 15MeV
m ! a2
Applying to numerical data
Two groups analyzed their data using WChPT:
1. CP-PACS :
Tadpole improved clover quark action
Namekawa et.al ‘04
a = 0.2f m ,
−→
2. qq+q :
Mπ
0.35 ≤
≤ 0.8
Mρ
( 8 Sea quark masses )
Good fits with WChPT assuming Aoki power counting, i.e. m ≈ a2
Unimproved Wilson quarks
Farchioni et.al. ‘03/‘04
−→
a ≈ 0.2f m ,
Mπ
0.47 ≤
≤ 0.76
Mρ
Good fits with continuum ChPT
Current numerical data is not conclusive
( 4 Sea quark masses )
Staggered fermions
+
+
Fast to simulate
-
Fermion doubling: Each flavor comes with 4 tastes
Exact U(1) symmetry for massless quarks
Taste reduction on the lattice:
1. The
√
4
2. The
√
4
det D →
√
4
det D
“fourth root trick”
det D -theory has no local lattice action: Universality ?
det D -theory has no local Symanzik action: How to include
a in ChPT ?
Staggered Chiral Perturbation Theory (SChPT)
Bernard ‘02
Aubin, Bernard ‘03
Main strategy:
1. Lattice theory with
Nf staggered fermions
2. Symanzik’s effective theory
→
→
Symanzik’s effective theory
with 4Nf fermions
ChPT
3. Compute physical observables like Mπ2 , fπ , . . .
4. Adjust by hand to one taste per flavor
= include factors of 1/4
for sea quark loops
1
No factor
4
1
One factor
4
“quark flow” diagrams
Pseudo-scalar mass to 1-loop
(3 deg. flavor)
Aubin, Bernard ‘02/03
!
2
mπ + )
In the Continuum limit :
2
m
1
πI
2
5
=1+
mπI ln 2 + analytic
Continuum expression
−→
2
2
2Bm
24π f
Λ
!
"
2
2
m
!
m πV
4
ηV
2
2
−→
0
+
m
−
m
ln
)
! ln
π
η
V
V
24π 2 f 2
Λ2
Λ2
!
"
2
2
m
!
m πA
4
ηA
2
2
2
−→
0
+
m
ln
−
m
ln
)
+
a
C
!
π
η
A
2
2
2
2
A
24π f
Λ
Λ
•
•
Reduces to the continuum expression for a → 0
•
MILC data strongly suggests the presence of these contributions
−→ talk by Claude Bernard
( not easy to see here )
Non-zero a : Additional log contributions involving other particles
Continuum log behaviour may be changed significantly !
Question:
Is the 4th-root trick really legitimate ?
•
Can we analytically understand why it works ?
•
Can we find additional cross checks ?
( Besides comparing with experimental results )
tmQCD
Twisted mass term:
m! eiωγ5 τ3 = m + iµγ5 τ3
ω:
twist angle
Why tmQCD:
•
•
•
•
No exceptional configurations
π
Automatic O(a) improvement at maximal twist ω =
2
No exceptional configurations
2
m
!
a
But: O(a)
Seemsimprovement
to work only if at maximal twist
Why ?
Automatic
Frezzotti, Rossi ‘03
Twisted mass term on the lattice
Mass term + Wilson term on the lattice :
ψ̄(x)
!"
$
r# !
−a
∇µ ∇µ + Mcr (r)
2 µ
%
+ mq exp(iwγ5 τ3 ) ψ(x)
Field redefinition:
bare quark mass
→
→
mq = m0 − Mcr (r)
critical quark mass
ω
ψph = exp(i γ5 τ3 )ψ,
2
ω
ψ̄ph = ψ̄ exp(i γ5 τ3 )
2
ψ̄ph (x)
!"
$
%
r# !
−a
∇µ ∇µ + Mcr (r) exp(−iwγ5 τ3 ) + mq ψph (x)
2 µ
Wilson average and O(a) improvement
The Wilson average
!O"W A (r, mq , ω) ≡
can be shown to be O(a) improved:
Crucial assumption:
!
1
2
"
!O"(r, mq , ω) + !O"(−r, mq , ω)
= !O"cont (mq ) + O(a2 )
Mcr (−r) = −Mcr (r)
Automatic O(a) improvement at maximal twist
Consider the twist average:
!
"
π
1
π
π
!O"T A (r, mq , ω = ) ≡ !O"(r, mq , ω = ) + !O"(r, mq , ω = − )
2
2
2
2
π
π
exp(−i γ5 τ3 ) = − exp(i γ5 τ3 )
2
2
1!
π
π "
= !O"(r, mq , ω = ) + !O"(−r, mq , ω = )
2
2
2
For observables even in
ω (e.g. masses):
π
π
TA
!O"(r, mq , ω = ) = !O" (r, mq , ω = ) = !O"cont (mq ) + O(a2 )
2
2
O(a) improvement without taking an average !
Using WChPT you can explicitly show ( in the Aoki phase scenario with
r
Mcr (r) = M1 (ra) + a2 r2 M2 (ra)
a
c2 > 0 )
M1,2 (ra) :
even polynomials in ra !
⇒
Mcr (r) != −Mcr (−r)
π TA
!O"(r, mq , ω = )
2
m!q
⇒
!
1
"
π
π
!
= !O"(r, mq , ω = ) + !O"(−r, mq , ω = + ω ! )
2
2
2
=
!
m2q + (2a2 r2 M2 (ar))2
2 2
2a
r M2 (ar)
!
tan ω =
mq
2
m
!
a
Automatic O(a) improvement only if q
New definition for the twist angle
Define:
Mcr (r) − Mcr (−r)
M cr (r) =
= −M cr (−r)
2
Mcr (r) + Mcr (−r)
∆Mcr (r) =
= ∆Mcr (−r)
2
⇒
New definition for the twist angle:
! "
$
%
r# !
ψ̄ph (x) − −a
∇µ ∇µ + M cr (r) exp(−iwγ5 τ3 ) + mq + ∆Mcr (r) ψph (x)
2 µ
You can show:
Automatic O(a) improvement at maximal twist without restrictions on mq