Wilson chiral perturbation theory for Nf=2+1
Shinji Takeda
Sinya Aoki, Oliver Bär and Tomomi Ishikawa
University of Tsukuba
"From Actions to Experiment"
The 2nd International Lattice Field Theory Network Workshop
@ e-Science Institute
March 10, 2005
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.1/31
Outline
Introduction
Pseudo scalar meson masses
Vector meson masses
Conclusion
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.2/31
Introduction
Motivation
Chiral perturbation theory for lattice fermions
Spurious analysis for WChPT
Counting scheme
Purposes of this talk
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.3/31
Motivation
Nf = 2 + 1 lattice QCD simulation
CP-PACS/JLQCD Collaboration
T. Ishikawa talk
with non-perturbative O(a) improved Wilson fermion
MP S /MV ' 0.60 − 0.78
heavier than physical value
Provide fit form for chiral extrapolation
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.4/31
Chiral perturbation theory for lattice fermions
Continuum ChPT
Lattice QCD data is not described well in many stages.
Of course, the quark masses are heavy.
Exciplicit chiral symmetry breaking term by non-zero a
non-zero lattice spacing effect can be included
Sharpe et al. ’98
two step matching procedure
Sharpe et al. ’98, Lee et al. ’99
Lattice QCD
⇓
Symanzik effective theory
⇓
Chiral perturbation theory
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.5/31
Spurious analysis for WChPT
Symanzik effective theory for Wilson fermion
SSymanzik =
Z
1
ψ̄(D
/+m)ψ+
4
Z
2
trFµν +ac
Z
ψ̄iσµν Fµν ψ+O(a2 )
Generalized chiral transformation
Rewrite m and a terms
ψ̄L M ψR + ψ̄R M † ψL ,
ψ̄L Aiσµν Fµν ψR + ψ̄R A† iσµν Fµν ψL
Invariant under generalized(spurious) chiral transformation
ψR −→ RψR , ψL −→ LψL , M −→ LM R† , A −→ LAR† ,
Original one is recovered by M
= M † = m, A = A† = aI
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.6/31
Spurious analysis for WChPT
Contruction of effective chiral Lagrangian
Leff [Σ, ∂µ Σ, M , A],
Σ = exp (2iπ/f )
Leff should have lattice symmetries and be invariant under
generalized(spurious) chiral transformation
Σ −→ LΣR† , M −→ LM R† , A −→ LAR† ,
After that, replacements M
= M † = m, A = A† = aI
Leff [Σ, ∂µ Σ, m, a]
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.7/31
Counting scheme
Previous work for WChPT
1.
O(a) effects are included
2.
O(a2 ) in NLO Lagrangian Bär et al. ’03
1. 2. =⇒ produce almost identical results to continuum ChPT
apart from mass shift mq −→ mq + O(a)
3.
Rupak et al. ’02
O(a2 ) in LO Lagrangian, Nf = 2
Aoki ’03
Aoki’s phase senario
Modification of chiral logarithm
O(am, ap2 ) : MP4 S ln MP2 S , −→ (1 + ac)MP4 S ln MP2 S ,
O(a2 ) : appearance of a2 MP2 S ln MP2 S
We adopt third counting scheme for Nf
=2+1
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.8/31
Purposes of this talk
Derive the meson mass formular as a function of the quark masses
and lattice cutoff in WChPT
with Nf
= 2 + 1,
mu =md = m6=ms
pseudo scalar meson and vector meson
up to NLO (loop correction)
including O(a2 ) to LO Lagrangian (with counting scheme
p 2 ∼ a2 )
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.9/31
Pseudo scalar meson masses
Order counting
LO Lagrangian
Critical mass at tree level
NLO Lagrangian
Mass formula
Short summary for pseudo scalar meson mass
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.10/31
Order counting
Double expansion parameters
p
,
Λ
aΛ
Order counting for Lagrangian
LO
:



 O(a),
O(p2 , mq , a2 ),



O(ap2 , amq ),

 O(p4 , p2 m , m2 , a2 p2 , a2 m ).
q
q
q
NLO :
 O(ap4 , ap2 mq , am2 ).
q
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.11/31
LO Lagrangian
LLO
=
+
+
+
+
+
2
f2
f 2 B0
f 2 c1 a
†
†
h∂µ Σ∂µ Σ i −
hM Σ + Σ M i −
hΣ + Σ† i −→O(a)
4
2
4
f 2a
[c0 hΣ + Σ† − 2ih∂µ Σ∂µ Σ† i + c̃0 h(Σ + Σ† − 2)∂µ Σ∂µ Σ† i] −→O(ap2 )
4
f 2 a2
[c2 hΣ + Σ† i2 + c̃2 h(Σ + Σ† )2 i] −→O(a2 )
16
f 2 B0 a
[2c3 hΣ + Σ† ihM Σ + Σ† M i + c̃3 h(Σ + Σ† )(M Σ + Σ† M )i] −→O(am)
8
f 2 a2
[c4 hΣ − Σ† i2 + c̃4 h(Σ − Σ† )2 i] −→O(a2 )
16
f 2 B0 a
[2c5 hΣ − Σ† ihM Σ − Σ† M i + c̃5 h(Σ − Σ† )(M Σ − Σ† M )i] −→O(am)
8
0
6i B
B
Σ = exp 6
4f @
π0
√
+ η/ 3
√ −
2π
√ −
2K
√
−π 0
2π +
√
+ η/ 3
√ 0
2K̄
√
2K +
√
2K 0
√
−2η/ 3
13
C7
C7 ,
A5
2
6
M =6
4
3
m
m
ms
7
7
5
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.12/31
Critical mass at tree level
Pseudo scalar meson mass at tree level
LLO
for φ
= π, K, η
1X
=
[(∂µ φ)2 + M̃φ φ2 ] + · · · ,
2 φ
2
c3 + ac1 − 3a2 c2 − a2 c̃24 ,
M̃φ2 = Mφ2 (1 − 3ac3 − ac̃35 ) − 3aMav
1X 2
2
Mav =
Mφ , c̃ij = c̃i + c̃j
8 φ
Critical mass
condition
for m
= ms
M̃P2 S = MP2 S (1 − 6ac3 − ac̃35 ) + ac1 − 3a2 c2 − a2 c̃24 = 0
⇓
1 ac1 − 3a2 c2 − a2 c̃24
mc = −
2B0 1 − 6ac3 − ac̃35
Subtracted masses
m̃ = m − mc ,
m̃s = ms − mc
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.13/31
NLO Lagrangian
Terms contributing to the mass only
LN LO = Lp2 m + Lm2 + La2 p2 + La2 m + Lap2 m + Lam2
Lp2 m : L4 , L5 terms
Lm2 : L6 , L7 , L8 terms
La2 p2 : 7 terms
La2 m : 6 terms
Lap2 m : 9 terms
Lam2 : 9 terms
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.14/31
Mass formula
Mπ 2
n
1h X
= x + 2y + 2
Lφ Aπφ x + Bφπ y +
f φ=π,K,η
| {z }
give aMP4 S ln MP2 S
C πφ
|{z}
give a2 MP2 S ln MP2 S
n
oi
π 2
π
− Dx x + Dπy y + Dxx x2 + Dyy
y + Dxy
xy
o
π
π
Aπφ , Bφπ , Dxx , Dyy
, Dxy
∼ w0 + w1 a + O(a2 )
C πφ , Dx , Dπy ∼ v0 a2 + O(a3 )
X
Y
x = 2B0 (2m̃ + m̃s ), y = 2B0 (m̃ − m̃s ), X, Y ∼ 1 + O(a),
3
6
M̃φ2
2
Lφ =
ln
M̃
φ , for φ = π, K, η
2
16π
For fixed a, 15 independent fit parameters
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.15/31
Short summary for pseudo scalar meson mass
Large number of fit parameters (15)
We found chiral logarithms aMP4 S ln MP2 S and
a2 MP2 S ln MP2 S which are typical one in our counting scheme
and originate from O(ap2 , am) and O(a2 ) terms in LLO
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.16/31
Vector meson masses
Effective theory for vector meson
Heavy vector meson formalism
Order counting of WChPT for vector meson
Vector meson fields
Lagrangian
Loop contribution
Mass formula
Discussion
Short summary for vector meson mass
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.17/31
Effective theory for vector meson
There are a variety of attempts to describe the vector meson.
Relativistic vector meson (naive)
=⇒ no chiral counting scheme, MV MP S
Hidden local symmetry
Bando et al. ’85, Harada et al. ’01
=⇒ chiral counting scheme OK
but the vector meson is taken as massless gauge boson
Heavy vector meson
Jenkins et al. ’95
=⇒ chiral counting scheme OK
···
We adopt heavy vector meson formalism
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.18/31
Heavy vector meson formalism
Basics
Jenkins et al. ’95
vector number preserve process V
−→ V 0 P
ρ −→ ππ can not be treated
1/MV expansion ←− MV ∼ Λ ∼ 1GeV
=⇒ Systematice counting scheme
=⇒ Double expansion parameters
p/Λ,
p/MV
p : momentum of the pseudo scalar meson or residual
momentum rµ of the vector meson.
kµ = M V v µ + r µ ,
kµ
:
four-momentum,
vµ
:
four-velocity,
v2 = 1,
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.19/31
Heavy vector meson formalism
Derivation of kinetic term
Bijnens et al. ’97
Relativistic
1
1 2
2
LV = − (∂µ Vν − ∂ν Vµ ) + MV Vµ V µ
4
2
⇓

 Relation V = √ 1
−iMV v·x
iMV v·x
†
e
Wµ + e
Wµ ,
µ
2MV
 1/MV expansion
Heavy vector meson
LV
−→ −iWµ† (v · ∂)W µ + O(1/MV ),
v · Wµ = 0
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.20/31
Order counting of WChPT for vector meson
Triple expansion parameters
p
,
Λ
aΛ,
p
MV
Counting scheme
LO
NLO
NNLO
: O(p), O(a),
: O(mq ), O(p2 ), O(ap), O(a2 ),
: O(mq p), O(p3 ), O(amq ), O(ap2 ), O(a2 p), O(a3 ),
Increase in unit of p
This counting scheme means p
∼ a, i.e. p2 ∼ mq ∼ a2
=⇒ consistent with pure pseudo scalar meson sector
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.21/31
Vector meson fields
Vector fields



Oµ = 

ρ0µ
√
2
+
(8)
φµ
√
ρ−
µ
Kµ∗−
6
ρ+
µ
ρ0µ
− √2
+
(8)
φµ
√
6
K̄µ∗0
Under chiral transformation SU (3)L
Oµ −→ U Oµ U † ,

Kµ∗+ 
∗0 
,
K

(8)
2φ
− √µ6
× SU (3)R , ξ =
Sµ
√
Σ
Sµ −→ Sµ
ξ −→ LξU † = U ξR†
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.22/31
Lagrangian
LO Lagrangian O(p), O(a)
Lp = −iSµ† (v · ∂)S µ − ihOµ† {(v · ∂)Oµ + vν [V ν , Oµ ]}i
+ig1 (Sµ† hOν Aλ i − Sµ hOν† Aλ i)vσ µνλσ
+ig2 h{Oµ† , Oν }Aλ ivσ µνλσ
La = α1 hW+ iSµ† S µ + α2 (hOµ† W+ iS µ + hOµ W+ iS µ† )
+α3 h{Oµ† , Oµ }W+ i + α4 hW+ ihOµ† Oµ i
Vµ =
W+ =
1
(ξ∂µ ξ † + ξ † ∂µ ξ),
2
a
(Σ + Σ† )
2
i
Aµ = (ξ∂µ ξ † − ξ † ∂µ ξ)
2
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.23/31
Lagrangian
NLO Lagrangian O(m), O(a2 )
O(m) : 4 terms
O(a2 ) : 10 terms
NNLO Lagrangian O(am)
O(am) : 12 terms
terms contributing to the mass only
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.24/31
Loop contribution
PS
V
V
PSfrag replacements
O(p)
∼ MP3 S
O(p)
∼ aMP2 S ln MP2 S
PSfrag replacements
O(a)
Chiral log appears as lattice artifacts
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.25/31
Mass formula
For ρ,
mρ = mO (a) + λx (a)x + 2λy (a)y
|
{z
}
polynomial correction
i
1 h 2 2 2 3
2
2
3
2
3
M̃
M̃
M̃
−
+
)
+
2g
+
g
g
(g
2
K
12πf 2 1 3 2 π
3 2 η
{z
}
|
continuum loop correction ∼mq3/2
2 1 i
a h
−
3η1 + η2 Lπ + 4η1 − η2 LK + η1 − η2 Lη ,
2
f
3
3
{z
}
|
Lattice artifact chiral logarithm ∼amq ln mq
x
Lφ
X
Y
= 2B0 (2m̃ + m̃s ), y = 2B0 (m̃ − m̃s ),
3
6
M̃φ2
2
=
ln M̃φ , for φ = π, K, η
2
16π
For fixed a, 7 fit parameters:
m O , λx , λy , g 1 , g 2 , η 1 , η 2
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.26/31
Mass formula
Chiral symmetry breaking by mass term
→
Mixing for φ(8) and singlet,


m(88) m(80)
m(08) m(00)


Additive 5 fit parameters
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.27/31
Discussion
Large N
mS (a = 0) − mO (0) −→ 0,
λx (0) − λy (0) −→ 0,
2
g1 −→ √ g2 ,
3
σx (0) −→ 0,
√
σy (0) −→ 2 2λy (0),
Reduction of the number of fit parameters
In reverse, if the fit works well, one may check how good
Large N is and determine the low energy constants for heavy
vector meson sector
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.28/31
Short summary for vector meson mass
Small number of fit parameters(7 for ρ)
Chiral logarithm aMP2 S ln MP2 S appears as lattice artifacts
Large N may help to reduce the number of fit parameter
If the fit works well, the low energy constants for heavy vector
meson sector can be determined from lattice data
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.29/31
Conclusion
We have derived the chiral extrapolation form of the pseudo scalar and
vector meson for the Wilson type fermion up to loop order with counting
scheme O(a2 )
∼ O(p2 )
Pseudo scalar meson sector
Large number of fit parameters(15)
Typical chiral logarithms aMP4 S
ln MP2 S and a2 MP2 S ln MP2 S
appear from O(ap2 , am) and O(a2 ) in LLO
Vector meson sector
Small number of fit parameters(7 for ρ)
Chiral logarithm aMP2 S
ln MP2 S appears as lattice artifacts
We will perform the fit with the obtained fitting form
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.30/31
Next targets
AWI quark masses
Decay constant for pseudo scalar and vector meson
Baryon and heavy meson masses and · · ·
···
with our counting scheme p2
∼ m q ∼ a2
"From Actions to Experiment" The 2nd International Lattice Field Theory Network Workshop (@ e-Science Institute, March 10, 2005) – p.31/31