Propagator of the lattice domain wall fermion and the staggered fermion

0.21 MB·English

by  Sadataka Furui

Few-Body Systems 0, 1–15 (2009) Few-
Body
Systems
(cid:13)c by Springer-Verlag 2009
PrintedinAustria
Propagator of the lattice domain wall
9
0 fermion and the staggered fermion
0
2

n
Sadataka Furui
a
J

3 School of Science and Engineering, Teikyo University.
1 1-1 Toyosatodai, Utsunomiya, 320-8551 Japan∗

]
t
a
l
- Abstract.Wecalculate thepropagator ofthedomainwallfermion(DWF) of
p
the RBC/UKQCD collaboration with 2+1 dynamical flavors of 163×32×16
e
h lattice in Coulomb gauge, by applying the conjugate gradient method. We
[
find that the fluctuation of the propagator is small when the momenta are

5 taken along the diagonal of the 4-dimensional lattice. Restricting momenta in
v
this momentum region, which is called the cylinder cut, we compare the mass
5
2 function and the running coupling of the quark-gluon coupling αs,g1(q) with
3 those of the staggerd fermion of the MILC collaboration in Landau gauge.
0
In the case of DWF, the ambiguity of the phase of the wave function
.
1
is adjusted such that the overlap of the solution of the conjugate gradient
0
8 method and the plane wave at the source becomes real. The quark-gluon
0 coupling α (q) of the DWF in the region q > 1.3GeV agrees with ghost-
: s,g1
v gluoncouplingα (q)thatwemeasuredbyusingtheconfigurationoftheMILC
s
Xi collaboration, i.e. enhancement by a factor (1+c/q2) with c ≃ 2.8GeV2 on
the pQCD result.
r
a
In the case of staggered fermion, in contrast to the ghost-gluon coupling
α (q) in Landau gauge which showed infrared suppression, the quark-gluon
s
coupling α (q) in the infrared region increases monotonically as q → 0.
s,g1
Above 2GeV, the quark-gluon coupling α (q) of staggered fermion calcu-
s,g1
lated by naive crossing becomes smaller than that of DWF, probably due to
the complex phase of the propagator which is not connected with the low
energy physics of the fermion taste.
1 Introduction
Inthecalculation ofquark-gluonvertices intheinfraredregion,non-perturbative
renormalization is possible by calculating the quark propagator in a fixed gauge.
The calculation of the quark propagator on lattice is reviewed in [1]. In our
previous paper [2], we studied the quark propagator of staggered fermion in
∗E-mail address: [email protected]
2 Propagator of thelattice domain wall fermion and thestaggered fermion
Landau gauge using the full QCD configurations of relatively large lattice (243×
64) of MILC collaboration [4] available from the ILDG data base [3]. In the last
year, full QCDconfigurations of the domain wall fermion (DWF) of medium size
(163×32×16)werereleasedintheILDGandinthisyearlargesize(243×64×16)
were released [5] from the RBC/UKQCD collaboration [6].
In these configurations the length of the 5th dimension was fixed to be 16.
In this paper we show the results of the medium size DWF configurations and
compare with the results of the large size staggered fermion.
Charcteristic features of infrared QCD are confinement and chiral symmetry
breaking. The confinement is related to the Gribov copy i.e. non gauge unique-
ness,whichmakesthesharpevaluation ofphysicalquantities difficult,andwetry
to fix the gauge in the fundamental modular region [7]. Chiral symmetry break-
ing is speculated to be related to instantons [8]. The Orsay group discussed that
the infrared suppression of the triple gluon coupling is due to instantons[9, 10].
Therunningcoupling fromthe quark-gluon coupling in quenched approximation
also showed similar infrared behavior [11]. Our simulation of the ghost-gluon
coupling in Landau gauge obtained by configurations of the MILC collaboration
showed infrared suppression, but in Coulomb gauge the running coupling α (q)
I
of MILC and of RBC/UKQCD did not show suppression [12]. Thus, it is in-
teresting to check the difference of the Coulomb gauge and the Landau gauge,
staggered fermion and DWF, and the ghost-gluon coupling and the quark-gluon
coupling.
The domain wall fermion (DWF) was first formulated by Kaplan in 1992
[13, 14] by assuming that the chiral fermion couples with the gauge field in the
fifth dimension. The model was improved by Narayanan and Neuberger [15] and
Shamir [16, 17], such that the gauge field are strictly four dimensional and are
copied to all slices in the fifth dimension. The model was applied in the finite
temperature simulation of 83×4 lattice with L from 8 to 32 lattices [18] and to
s
quenched simulation of 83×32,123 ×32, and 163 ×32 lattices with L from 16
s
to 64 [20].
The fermionic part of the Lagrangian formulated for the lattice simulation is
[16, 17, 21],
S (ψ¯,ψ,U) = − ψ¯ (D ) ψ , (1)
F x,s F x,s;y,s′ y,s′
x,s;y,s′
X
where
(D ) = δ Dk +δ D⊥ . (2)
F x,s;y,s′ s,s′ x,y x,y s,s′
The interaction Dk contains the gauge field and the interaction in the fifth di-
mension defined by D⊥ does not contain the gauge field[20].
The bare quark operators are defined on the wall at s = 0 and s = L −1 as
s
q = P ψ +P ψ , (3)
x L x,0 R x,Ls
1+γ 1−γ
5 5
whereP = andP = aretheprojection operator.For theDirac’s
R L
2 2
γ matrices, we adopt the convention of ref. [18], in which γ is diagonal.
5
In the DWF theory, a Lagrangian density in the fermion sector
L = iψ¯(∂/−iA/)ψ+ψ¯(MP +M†P )ψ (4)
1 R L
SadatakaFurui 3
with an operator M acting on the left-handed and right-handed field was pro-
posed [15]. In this method, the free fermion propagator becomes
1
[p/−M†P −MP ]−1 = (p/+M)P
L R Lp2−M†M
1
+(p/+M†)P . (5)
Rp2−MM†
On the lattice, one introduces the bare quark mass m that mixes the two
f
chiralities, and 5 dimensional mass M . The lattice simulation of DWF propaga-
5
tor is performed by introducing a Hamiltonian, whose essential idea is given in
Appendix. This formalism was adopted in the Schwinger model [22] and in the
4-dimensional lattice simulation [18, 23, 6].
Instead of using the transfer matrix method, we calculate the quark prop-
agator by using the conjugate gradient method in five dimensional spaces. We
interpret the configuration at the middle of the two domain walls in the fifth di-
mension as the physical quark wave function. We measure the propagator of the
domainwallfermionusingtheconfigurationsoftheRBC/UKQCD,andcompare
the propagator with that of the configurations of MILC [2]. We measure also the
quark-gluon coupling from the quark propagator, by applyingthe Ward identity.
The organization of this paper is as follows. In sect. 2, we present a formula-
tion of the lattice DWF and its numerical results are shown in sect.3. In sect.4
the lattice calculation of the staggered fermion propagator which we adopted in
[2] is summarized and in sect. 5, a comparison of the DWF fermion-gluon and
the staggered fermion-gluon is given. Conclusion and discussion are given in the
sect.6. Some comments on the Hamiltonian is given in the Appendix.
2 The lattice calculation of the DWF propagator
In this section we present the method of calculating the DWF propagator.
Using the D defined in eq.2 we make a hermitian operator D = γ R D ,
F H 5 5 F
where (R ) = δ is a reflection operator as
5 ss′ s,Ls−1−s′
−m γ P γ P γ (Dk−1)
f 5 L 5 R 5
γ P γ (Dk−1) γ P
5 R 5 5 L
 
··· ··· ···
D =
H  ··· ··· ··· 
 
 γ P γ (Dk−1) γ P 
 5 R 5 5 L 
 γ (Dk−1) γ P −m γ P 
 5 5 L f 5 R 
 
where P = (1±γ )/2, and -1 in (Dk−1) originates from D⊥ .
R/L 5 s,s′
The quark sources are sitting on the domain walls as
q(x) = P Ψ(x,0)+P Ψ(x,L −1). (6)
L R s
We take PLΨ(x,s) ∝ e−(rs)2, PRΨ(x,s) ∝ e−(Ls−rs−1)2, (s = 0,1,···,Ls−1) with
r = 0.842105, which corresponds to (r/L )/(1+r/L )= 0.05 [24].
s s
4 Propagator of thelattice domain wall fermion and thestaggered fermion
In the case of quenched approximation, a condition on the Pauli-Villars reg-
ularization mass M for producing a single fermion with the left hand chirality
bound to s= 0 and the right bound to s = L −1 is 0 < M < 2. In the free the-
s
ory, the condition that the transfer matrix along the 5th dimension be positive
yields a restriction 0 < M < 1 [15]. However, in a quenched interacting system,
M = 1.8 was adopted [19] and since in the conjugate gradient method there is
no M < 1 constraint, we adopt the same value.
We define the base of the fermion as
Ψ(x)= t(φ (x,0),φ (x,0),···,φ (x,L −1),φ (x,L −1)) (7)
L R L s R s
wheret meansthetransposeofthevector,andtheφ (x,l )containscolor3×3
L/R s
matrix, spin 2×2 matrix and n ×n ×n ×n site coordinates. We measure
x y z t
9·4 matrix elements on each site at once.
The wave functions φ (x,s) are solutions of the equation
L/R
φ (x,s)
γ (Dk−1) L
5 φ (x,s)
R
(cid:18) (cid:19)
1 0 −B C φ (x,s)
= L
0 −1 −C† −B φ (x,s)
R
(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
−B C φ (x,s)
= L (8)
C† B φ (x,s)
R
(cid:18) (cid:19)(cid:18) (cid:19)
where
4
1
B = (5−M )δ − (U (x)δ +U† (y)δ ), (9)
5 xy µ x+µˆ,y µ x−µˆ,y
2
µ=1
X
and
4
1
C = (U (x)δ −U† (y)δ )σ . (10)
µ x+µˆ,y µ x−µˆ,y µ
2
µ=1
X
Here
−M s < Ls−1
M = Mθ(s−L /2) = 2 (11)
5 s M s ≥ Ls−1
(cid:26) 2
is the mass introduced for the regularization. Using the γ matrices as defined in
[18], we obtain
4
1
C(x,y)P = (U (x)δ −U† (y)δ )ΣP ,
R µ x+µˆ,y µ x−µˆ,y R
2
µ=1
X
4
1
C†(x,y)P = (U† (x)δ −U (y)δ )Σ†P
L µ x+µˆ,y µ x−µˆ,y L
2
µ=1
X
(12)
where
iσ −iσ
1 1
−iσ iσ
Σ =  2  and Σ† =  2 .
iσ −iσ
3 3
 −I   −I 
   
   
SadatakaFurui 5
The conjugate gradient method for solving the 5 dimensional DWF propa-
gator is a simple extension of the method we used in the staggered fermion [2],
since the degrees of freedom in the 5th dimension can be treated as if they are
internal degrees of freedom on each 4 dimensional sites.
As in the transfer matrix method, we define
1
M¯ = I + D , (13)
H
5−M
(cid:18) 5 (cid:19)
0 0
L = (14)
− 1 D 0
(cid:18) 5−M5 Heo (cid:19)
and
0 − 1 D
U = 5−M5 Hoe (15)
0 0
(cid:18) (cid:19)
such that
I 0
(1−L)−1M¯(1−U)−1 = , (16)
0 I − 1 D D
(cid:18) 5−M5 Heo Hoe (cid:19)
where even-odd decomposition is done in the 5 dimensional space. We solve the
equation for
φ′
φ= o
φ′
(cid:18) e (cid:19)
using the source
1 ρ′
ρ= ρ′ = o ,
5−M ρ′
5 (cid:18) e (cid:19)
1 1
I − D D φ = ρ′ − D ρ′. (17)
(5−M )2 Heo Hoe e e 5−M Heo o
(cid:18) 5 (cid:19) 5
The solution on the odd sites is calculated from that of even sites as
1
φ = ρ′ − D φ′. (18)
o o 5−M Hoe e
5
In the process of conjugate gradient iteration, we search shift parameters
for αL [27] for φ and αR for φ and in the first 50 steps we choose α =
k L k R k
Min(αL,αR) and shift φL =φL−α φL and φR = φR−α φR and in the last
k k k+1 k k k k+1 k k k
25 steps we choose α = Max(αL,αR), so that the stable solution is selected for
k k k
both φ and φ .
L R
The convergence condition attained in this method is about 0.5×10−4. One
can improve the condition by increasing the number of iteration, but the overlap
of the solution and the plane wave do not change significantly.
To evaluate the propagator, we measure the trace in color and spin space of
the inner product in the momentum space between the plane waves
χ(p) = t(χ (p,0),χ (p,0),···,χ (p,L −1),χ (p,L −1))
L R L s R s
and the solution of the conjugate gradient method
Ψ(p) = t(φ (p,0),φ (p,0),···,φ (p,L −1),φ (p,L −1))
L R L s R s
6 Propagator of thelattice domain wall fermion and thestaggered fermion
as
Trhχ¯(p,s)P Ψ(p,s)i = Z (p)(2N )B (p,s),
L B c L
Trhχ¯(p,s)P Ψ(p,s)i = Z (p)(2N )B (p,s) (19)
R B c R
and
Trhχ¯(p,s)i/pP Ψ(p,s)i = Z (p)/(2N )ipA (p,s),
L A c L
Trhχ¯(p,s)i/pP Ψ(p,s)i= Z (p)/(2N )ipA (p,s) (20)
R A c R
1 2πp¯
i
where p = sin (p¯ = 0,1,2,···,N /2).
i i i
a N
i
On the lattice at each s the 4-dimensional torus is residing. We perform the
Fourier transformin the4-dimensional space,buttake themomentum inthe 5th
direction to be zero since it corresponds to the lowest energy state. Z (p) and
A
Z (p) are the wave function renormalization factor.
B
When p = 0, the term B(p,s) are given by the matrix elements of hχ ,Ψ i
4 R L
and hχ ,Ψ i. The operator p/ yields matrix elements of hχ,ΣΨ i and hχ,ΣΨ i.
L R L R
The propagator is parametrized as
−i/p+M†(pˆ) −i/p+M(pˆ)
S(p)= [ P ]+[ P ] (21)
p2+M(pˆ)M†(pˆ) L p2+M†(pˆ)M(pˆ) R
where
Re[B (p,L /2)]
R s
M(pˆ) =
Re[A (p,L /2)]
R s
and
Re[B (p,L /2)]
M†(pˆ) = L s .
Re[A (p,L /2)]
L s
2 πp¯
i
The momentum assignment pˆ = sin is introduced for removing doublers
i
a N
i
using the Wilson prescription.
TheM(pˆ) has zero eigenfunction and dim(KerM) = n = 1 and the M†(pˆ)
R
does not have zero eigenfunction and dim(KerM†)= n = 0.
L
3 Numerical results of the DWF propagator
The configurations of the RBC/UKQCD collaboration are first Landau gauge
fixed and then Coulomb gauge fixed (∂ A = 0) as follows[12]. We adopt
i i
the minimizing function F [g] = ||Ag||2 = tr Ag †Ag , and solve
U x,i x,i x,i
∂gA (x,t)= 0 using the Newton method. We obPtain ǫ =(cid:16) 1 ∂ A(cid:17)from the eq.
i i −∂D i i
∂ A +∂ D (A)ǫ = 0.Puttingg(x,t) = eǫ inUg (x,t) = g(x,t)U (x,t)g†(x+i,t)
i i i i i i
we set the ending condition of the gauge fixing as the maximum of the diver-
gence of the gauge field over N2 − 1 color and the volume is less than 10−4,
c
Max (∂ A )a < 10−4. This condition yields in most samples
x,a i x,i
1
(∂ Aa )2 ∼ 10−13.
8V i x,i
a,x
X
SadatakaFurui 7
We leave the remnant gauge on A (x) unfixed, but since the Landau gauge
0
preconditioning is done, it is not completely random. We leave the problem of
whether a random gauge transformation, or the remnant gauge fixing on A (x)
0
modify the propagators, but we do not expect drastic corrections will happen.
UsingthegaugeconfigurationsofRBC/UKQCDcollaborationafterCoulomb
gauge fixing, we calculate Trhχ(p,s)φ (p,s)i and Trhχ(p,s)i/pφ (p,s)i and
L L
Trhχ(p,s)φ (p,s)i and Trhχ(p,s)i/pφ (p,s)i at each 5-dimensional slice s. Num-
R R
ber of samples is 49 for each mass m = 0.01, 0.02 and 0.03. We measured in
f
certain momentum directions of m = 0.01, 149 samples.
f
In our Lagrangian there is a freedom of choosing global chiral angle in the
5th direction,
ψ → eiηγ5ψ, ψ¯→ ψ¯e−iηγ5ψ. (22)
WeadjustthisphaseofthematrixelementsuchthatbothTrhχ(p,0)φ (p,0)i
L
and Trhχ(p,L −1)φ (p,L −1)i are close to a real number. Namely, we define
s R s
Trhχ(p,0)φ (p,0)i
eiθL = L ,
|Trhχ(p,0)φ (p,0)i|
L
Trhχ(p,L −1)φ (p,L −1)i
e−iθR = s R s
|Trhχ(p,L −1)φ (p,L −1)i|
s R s
and sample-wise calculate eiη such that
|eiθLeiη +1|2+|eiθRe−iη −1|2 (23)
is minimum. When p¯is even and the momentum is not along the diagonal of the
four dimensional system, we also calculate eiη′ such that
|eiθLeiη′ −1|2+|eiθRe−iη′ −1|2 (24)
is minimum, but the final results by multiplying eiη and eiη′ are similar.
Inthecalculation of B ,we definematrix elements multiplied by thephase
L/R
as
^
hχ(p,s)φ (p,s)i = hχ(p,s)φ (p,s)ie−iη,
L L
^
hχ(p,s)φ (p,s)i = hχ(p,s)φ (p,s)ieiη.
R R
and correspondingly denote B (p,s) multiplied by the phase e−iη and eiη as
L/R
B˜ (p,s), respectively.
L/R
In the calculation of A (p,s), we diagonalize
L/R
^ ^
[hχ(p ,s)φ (p ,s)iσ +hχ(p ,s)φ (p ,s)iσ
x L x 1 y L y 2
^ ^
+hχ(p ,s)φ (p ,s)iσ +hχ(p ,s)φ (p ,s)iiI] (25)
z L z 3 z L z
and
^ ^
[hχ(p ,s)φ (p ,s)iσ +hχ(p ,s)φ (p ,s)iσ
x R x 1 y R y 2
^ ^
+hχ(p ,s)φ (p ,s)iσ +hχ(p ,s)φ (p ,s)iiI], (26)
z R z 3 z R z
8 Propagator of thelattice domain wall fermion and thestaggered fermion
where I is the 2×2 diagonal matrix, and define the A (p,s) multiplied by the
L/R
phase as A˜ (p,s).
L/R
The term B is a sum of color-spin diagonal scalar, while the term A
L/R L/R
is a color-diagonal but momentum dependent spinor and we take the positive
eigenvalue.
In order to minimize the artefact due to violation of rotational symmetry
of the lattice we restrict the momentum configuration to be diagonal in the 4-d
lattice. This prescription which is called cylinder cut [28] is already adopted in
ghost propagator [30] and in quark propagator [2, 1] calculations.
In general, there is a mixing between φ and φ and there is a sign problem
L R
i.e. the sign of Re[B˜ (p,L /2)] and Re[A˜ (p,L /2)] becomes random. The
L/R s L/R s
sign is related to the sign of the source at s = 0 and s = L −1. But, when the
s
cylinder cut is chosen, the sign problem does not seem to occur.
In the calculation of the propagator of DWF, the mass originates not only
from the mid-point matrix Q(mp) defined as
Q(mp) = P δ δ +P δ δ (27)
s,s′ L s,Ls/2 s′,Ls/2 R s,Ls/2−1 s′,Ls/2−1
but also from Q(w) defined as
Q(w) =P δ δ +P δ δ (28)
s,s′ L s,0 s′,0 R s,Ls−1 s′Ls−1
At zero momentum the numerator B (p = 0,s = 0) becomes 1 and it gives
L
a contribution of m Q(w) = m . Since there is no pole mass in φ (s,l ), the
f f R s
value of B (p = 0,s = L − 1) is not physical. In the midpoint contribution
R s
Re[B˜ (p,L /2)]
L/R s
,wetakeintoaccountthatthenumeratorofthemassfunction
Re[A˜ (p,L /2)]
L/R s
contains (2N )×(2N ) coherent contributions and divide by the multiplicity.
c c
The Fig.1 is the mass function of m = 0.01/a = 0.017GeV. The momenta
f
correspond to p¯= (0,0,0,0),(1,1,1,2),(2,2,2,4),(3,3,3,6) and (4,4,4,8). The
dotted lines are the phenomenological fit
cΛ2α+1 m
f
M(pˆ) = + (29)
p2α+Λ2α a
Since the pole mass Q(w) is not included in the plots, m is set to be 0 here. The
f
corresponding values of 0.02/a = 0.034GeV and 0.03/a = 0.050GeV are similar.
In the χ2 fit, we choose α equals 1,1.25 and 1.5 and searched best values for
c and Λ. We found the global fit is best for α = 1.25. The fitted parameters are
given in Table 1.
In the case of staggered fermion in Landau gauge, we adopt in this work
the staple plus Naik action on m = 0.0136GeV and 0.027GeV configurations[2,
f
29]. We fixed the parameter α = 1 and obtained Λ = 0.82GeV and 0.89GeV,
respectively. In general Λ becomes larger for larger α, but Λ of RBC/UKQCD
seems larger than that of MILC, which is also observed in the quenched overlap
fermionpropagator[31].InthecaseofMILC,Λbecomessmallerforsmallermass
m , butin the case of RBC/UKQCD, it is opposite. Analytical expression of the
f
quark propagator in Dyson-Schwinger equation is formulated in [32, 33] and a
SadatakaFurui 9
m /a m /a c Λ(GeV) α
ud s
DWF 0.01 0.04 0.24 1.53(3) 1.25
01
DWF 0.02 0.04 0.24 1.61(5) 1.25
02
DWF 0.03 0.04 0.30 1.32(4) 1.25
03
MILC 0.006 0.031 0.45 0.82(2) 1.00
f1
MILC 0.012 0.031 0.43 0.89(2) 1.00
f2
Table 1. The fitted parameters of mass function of DWF(RBC/UKQCD) and staggered
fermion (MILC) with thestaple plusNaik action.
comparison with theselattice data are given in[34,35].Themass functionof the
staggered fermion is close to that of the Dyson-Schwinger equation of N = 3,
f
but larger than that of the N = 0.
f
0.6
0.5
0.4
æà
L
p
H 0.3
M
0.2 æà
0.1
æà
æà æà
0.0
0 1 2 3 4 5
pHGeVL
Figure1.ThemassfunctioninGeVofthedomainwallfermionasafunctionofthemodulusof
Euclidean four momentum p(GeV).m =0.01. (149 samples). Blue disks are m (left handed
f L
quark)and red boxes are m (right handed quark).
R
The error bars are taken from the Bootstrap method after 5000 re-samplings
[36, 37]. The re-sampling method reduces the error bar by about a factor of 10
as compared to the standard deviation of the bare samples.
We measured also momentum points (p¯,0,0,0), (0,p¯,0,0), (0,0,p¯,0) and
(p¯,p¯,p¯,0) with p¯ = 1,2,3 and 4, but the error bars of the mass function are
foundtobelargeespecially at (2,2,2,0). We observed systematic difference of the
magnitude of mass functions of (p¯,0,0,0) with even p¯and odd p¯. Such problems
would be solved by improving statistics, and systematically correcting deviation
from the spherical symmetry, but at the moment these problems can be evaded
by adopting the cylinder cut.
4 The lattice calculation of the staggered fermion propagator
The lattice results of fermion propagator using staggered (Asqtad) action and
overlap action are reviewed in [1]. We calculated the propagator of staggered
fermion of the MILC collaboration using the conjugate gradient method [2]. The
10 Propagator of thelattice domain wall fermion and thestaggered fermion
0.6
0.5
0.4
L
p
0.3
H
M0.2
0.1
0
1 2 3 4 5
pHGeVL
Figure 2. The mass function in GeV of the staggered fermion MILC as a function of the
f2
modulusof Euclidean four momentum p(GeV). Thestaple plusNaik action is adopted
inverse quark propagator is expressed as
9 1
S−1(p,m) = i (γ¯ ) sin(p )− sin(3p ) +mδ¯ (30)
αβ µ αβ 8 µ 24 µ αβ
µ (cid:20) (cid:21)
X
where α = 0,1, β = 0,1 and δ¯ = δ . The momentum of the
µ µ αβ µ αµβµ|mod2
staggered fermion k takes values k = p +πα where
µ µ µ µ
Q
2πm L
µ µ
p = , m = 0,···, −1. (31)
µ µ
L 2
µ
The γ matrix of staggered fermions is
(γ¯ ) = (−1)αµδ¯ (32)
µ αβ α+ζ(µ),β
where
1 if ν < µ
ζ(µ) = (33)
ν 0 otherwise
(cid:26)
The A(p) is defined as
i (−1)αµp Tr[ S (p)] = 16N p2A(p) (34)
µ αβ c
α µ β
XX X
where 16 is the number of taste.
The staggered fermion incorporating the lattice symmetries including parity
and charge conjugation by introducing a general mass matrix is formulated in
[42]. In our model, we do not incorporate the general mass matrix, but take the
same mass as MILC collaboration.
The chiral symmetry of the staggered fermion of the MILC collaboration is
currently under discussion [44, 45, 46]. In our simple model, the charge conju-
gation operator can be taken as C = −iγ and CγTC† = −γ . We interpret p
4 µ µ
in the original as q in crossed channel and since staggered actions are invariant
under translation of 2a, we modify the scale by a factor of 1/2:
1 9 1
p = sin(p )− sin(3p ) (35)
µ µ
2a 8 24
(cid:20) (cid:21)