JLQCD's dynamical overlap fermion project Norikazu Yamada - - PowerPoint PPT Presentation
JLQCD's dynamical overlap fermion project Norikazu Yamada (KEK/GUAS) for JLQCD Collaboration Seminar@Toyama Univ. 2008.04.18 JLQCD Collaboration KEK: S. Hashimoto, H. Ikeda, T. Kaneko, H. Matsufuru, J. Noaki, E. Shintani, N.Y. Tsukuba: S.
Seminar@Toyama Univ. 2008.04.18
KEK: S. Hashimoto, H. Ikeda, T. Kaneko, H. Matsufuru, J. Noaki, E. Shintani, N.Y. Tsukuba: S. Aoki, K. Kanaya, N. Ishizuka, K. Takeda,
Yoshie NBI: H. Fukaya (KEK) YITP: H. Ohki, T. Onogi Hiroshima: K.-I. Ishikawa, M. Okawa (Taiwan: T.W. Chiu, T.H. Hsieh, K. Ogawa)
IBM BlueGene/L (57.3 TFlops) Hitachi SR11000/K1 (2.15 TFlops)
Lattice formulation with the exact chiral symmetry is clearly suitable.
which do not exist in the continuum.
The uncertainty due to the chiral extrapolation is reduced.
(normal regime)
(Supuer-computer + improvements of algorithms) makes it possible! Keyword : Chiral Symmetry on the lattice
Discretize the space-time and define “fields” on a lattice
xµ = anµ = a × (nx, ny, nz, nt) a−3/2ψ(x), a−3/2 ¯ ψ(x), Uµ(x) = eigaAb
µ(x)tb ∈ SU(Nc)
Pµ,ν(x) = Tr
µ)U †
µ(x + ˆ
ν)U †
ν(x)
6
[3 − RePµ,ν(x)] − →
4F b
µνF b µν + O(a2)
(where β = 6/g2)
μ ν x
a
Gauge transformation: ψ(x) → V (x)ψ(x), ¯ ψ(x) → ¯ ψ(x)V †(x), Uµ(x) → V (x)Uµ(x)V †(x + ˆ µ)
propagator: 1 aS(p) = −iγµ sin(apµ)
This gives you a gauge-invariant regularization of Quantum Field Theory.
1 aS(p) = −iγµ sin(apµ)
µ (cos(apµ) − 1)
2
⇒
Swt = −r
¯ ψ(x)
µ) − 2ψ(x) + U †
µ(x − ˆ
µ)ψ(x − ˆ µ)
→
2 ¯ ψ(x)D2ψ(x) + O(a3)
2a
¯ ψ(x)γµUµ(x)ψ(x + ˆ µ) − ¯ ψ(x)γµU †
µ(x − ˆ
µ)ψ(x − ˆ µ)
→
ψ(x)D / ψ(x) ?
SWilson = Snaive + Swt− →
ψ(x)
/ − ar 2 D2 + O(a2)
In the continuum, QCD is invariant for massless quarks. ⇒Chiral Symmetry SU(Nf)L×SU(Nf)R Since we want to study SχSB using Lattice QCD, it is clearly better that it has the symmetry. If the symmetry is violated explicitly from the beginning, the study will encounter many difficulties.
¯ ψ(x) → ¯ ψ(x)eiθbtaPL/R, ψ(x) → e−iθbtaPR/Lψ(x), PL/R = (1 ∓ γ5)/2
The axial part of chiral transformation, is not the symmetry because of the Wilson term even in the massless llimit. Chiral symmetry is explicitly violated for Wilson fermion!
This difficulty is rather general, and known as “No-go theorem”. [Nielsen,Ninomiya(1981,1981)]
Long standing problem in Lattice QCD for ~25 years.
¯ ψ(x) → ¯ ψ(x)eiθbtaγ5, ψ(x) → eiθbtaγ5ψ(x),
SWilson =
ψ(x)
/ + mq − ar 2 D2 + O(a2)
where 0 < m0 < 2 (m0=1.6).
which is equivalent to satisfying Ginsberg-Wilson relation,
[Neuberger (1998)]
Dov =
2
2
δ ¯ ψ(x) = ¯ ψ(x)iγ5θbT b, δψ(x) = iθbT bγ5 (1 − aDov) ψ(x),
Dovγ5 + γ5Dov = aDovγ5Dov
HW(−m0) = γ5 (DW − m0) sgn[X] = X √ X†X
With G-W relation Suppose that Dov uk = λk uk, one can prove that
eigenvectors for those are given by (uk, γ5uk) (except for zero- modes)
[Neuberger (1998)]
Dovγ5 + γ5Dov = aDovγ5Dov
the propagator 1/(Dov+mq).
be calculated for very light quark mass.
⇒ algorithm is breakdown for light quarks.
Lattice simulation is essentially doing Path Integral numerically. e.g.) pion two-point correlation function ⇒ fπ & mπ We can thus calculate various hadron masses, decay constants, transition matrix elements through lattice calculation.
A4(x) = ¯ u(x)γ4γ5d(x),
A4(x)A†
4(0)
= Z−1
A4(x)A†
4(0)
=
→ f 2
πmπ
2 e−mπt + (excited states)
Simulating lighter quark masses becomes possible.
Quark mass dependence of the decay const
[PRL98(2007)172001,PRD76(2007)054503,arXiv:0711.4965 (appear in PRD)]
[PRL98(2007)172001,PRD76(2007)054503,arXiv:0711.4965 (appear in PRD)]
ǫK = A(KL → (ππ)I=0) A(KS → (ππ)I=0), |ǫK|exp = 2.28(2) × 10−3 ǫK = ¯ ηA2 ˆ BK ×
ρ) + 0.31(5)
ˆ BK = C(µ) K0|O∆S=2(µ)|K0
8 3f 2 Km2 K
VDD X! X!
!
0.5 1 1.5 2
"
0.5 1 1.5
2
#
1
#
3
#
!
0.5 1 1.5 2
"
0.5 1 1.5
3
#
3
#
2
#
2
#
d
m $
K
%
K
%
d
m $ &
s
m $
ub
V
1
# sin2
< 0
1#
(excl. at CL > 0.95)
excluded area has CL > 0.95 excluded at CL > 0.95
Summer 2007
CKM
f i t t e r
constraints from |ε| and other experiments using (ρ,η)-plane.
.
Sizable error is one of the dominant uncertainties in light green band.
use BK = 0.79 ± 0.04 ± 0.09 proportional to (
4) [i.e.,
(
the four-quark op receives a multiplicative renormalization only.
lightest pion ⇒ mπ ≈290 MeV, mπ L ≈ 2.8 δ ¯ ψ(x) = ¯ ψ(x)iγ5θbT b, δψ(x) = iθbT bγ5 (1 − aDov) ψ(x),
OMS
LµLµ(µ) = Z(µ, 1/a)
LµLµ + ZP P Olat P P + · · ·
0|OP P |K0
f 2
Km2 K
∼ 1 mq ⇐ (source of uncertainty)
[Golterman and Leung, PRD57(1998)5703]
(4 free parameters)
BP = Bχ
P
P
(4πf)2 ln
P
µ2 + (b1 − b3) m2
P + b2 m2 ss,
m2
ss
∼ B0(msea + msea).
two shortest fit ranges give acceptable χ2/dof.
Our three or four lightest quarks are inside the NLO ChPT regime.
0.05 0.1 0.15 0.2 0.25
(amP)
2
0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
BP
MSbar(2GeV)
0.015 0.025 0.035 0.050 0.070 0.100
test with NLO ChPT with f fixed
[0.015,0.100]: chi/dof=14 [0.015,0.070]: chi/dof=2.6 [0.015,0.050]: chi/dof=0.5 [0.015,0.035]: chi/dof=0.2
To describe heavy region, “O(p4) terms” are added.
[Golterman and Leung, PRD57(1998)5703] (except for O(p4) part)
m2
ij
∼ B0 (mvi + mvj)
All data including non-degenerate quarks (but restricing data with mval ≧ msea) are fit to
B12 = Bχ
12
2 (4πf)2
ss + m2 11 − 3 m4 12 + m4 11
2 m2
12
+ m2
12
m2
12
µ2
m2
22
µ2
2 m2
ss(m2 12 + m2 11)
2 m2
12
+ m2
11(m2 ss − m2 11)
m2
12 − m2 11
m2
22
m2
11
+b1 m2
12 + b3 m2 11
11
m2
12
ss + d1 (m2 12)2,
0.05 0.1 0.15 0.2
(am12)
2
0.2 0.3 0.4 0.5 0.6
B12
MSbar(2 GeV)
msea=0.015 0.025 0.035 0.050 0.070 0.100 using 3 msea 4 msea 5 msea 6 msea
(NLO ChPT + quadratic) fit
Error: stat. and sys.
BMS
K (2 GeV) = 0.537(4)(40)
Andreas Jüttner@Lattice ‘07
ˆ BK = 0.758(6)(60)
Our result:
use BK = 0.79 ± 0.04 ± 0.09 proportional to (
4) [i.e.,
(
Using Unitarity and
are commonly used.
σπN= 40 ~ 70 MeV, y=0.21±0.20
Why sigma term is important?
dark matter interaction with nucleon by higgs exchange in the t-channel
Phys,Rev,D38, 2375(1988) Baltz, Battaglia, Peskin, Wizanksy
Note: strange quark contribution is dominant. Nucleon sigma term for strange quark is most important.
The disconnected diagram is most important!
quenched’ estimate for y because no strange sea quark is present.
The contributions from connected and disconnected diagrams can be separately calculated by partial derivatives of mN with respect to mval and msea!
where N|¯ qq|N ≡ N|¯ qq|N − V 0|¯ qq|0 for simplicity
dmN dmq = N|¯ qq|N
σπN =
mq dmN dmq
.
N = 53(2)stat(+21 −6 )extrap(+5 −0)FSE(+2 −1)mu,d [MeV],
(preliminary)
0.1 0.2 0.3 0.4 0.5 0.6 m!
2 [GeV 2]
0.8 1 1.2 mN [GeV]
This is reasonably consistent with the previous results!
y = 2N|¯ ss|N N|¯ uu + ¯ dd|N = 0.030(16)stat(+6
0 )extrap(+1 −2)ms,
This is much smaller than the previous lattice results! (preliminary)
0.1 0.2 0.3 0.4 0.5 0.6 (m!
ss) 2 [GeV 2]0.8 1 1.2 1.4 mN[GeV]
Connected Disconnected
0.00 0.02 0.04 0.06 0.08 amval = amsea 0.0 1.0 dmN 0.00 0.02 0.04 0.06 0.08 amval = amsea 2.0 3.0 4.0 5.0 dm
2N|¯ ss|N = ∂mN ∂msea
Fitting data to Partially Quenched ChPT formula
receive the additive mass shift through gauge interactions.
⇒ source of large discrepancy
mval(µ) = Zm(µ, a−1)a−1 ˆ mval = Zm(µ, a−1)a−1 ˆ mbare
val + δm( ˆ
msea, a−1)
dmsea = ∂ ˆ msea ∂msea ∂mN ∂ ˆ msea + ∂ ˆ mval ∂msea ∂mN ∂ ˆ mval + · · ·
. (KEK)
spontaneous and explicit chiral symmetry breaking.
[Peskin, Takeuchi (1990,1992)]
(80),Preskill(81)]
S = −16π
10(µ) −
1 192π2
µ2 m2
H
6
m2
PNG = G
∞ dq2q2 Π(1)
T (q2) − Π(1) X (q2)
VV and AA in the massless limit of quarks.
“physics” from〈VV−AA〉. Overlap fermion formalism
in our definition).
i
ν(0)
=
J (q2) − qµqνΠ(0) J (q2),
Jµ(x) =
q1(x)γµq2(x), Aµ(x) = ¯ q1(x)γµγ5 q2(x),
In continuum,
By considering μ=ν and μ≠ν for〈VμVν−AμAν〉, we can extract Since artifacts are expected to cancel in〈VV−AA〉,
µ (x)V ji ν (0) − Aij µ (x)Aji ν (0)
=
V −A − qµqνΠ(0) V −A.
Π(0)
V −A(q2), Π(1) V −A(q2)
0.5 1 1.5 2 0.0 2.010
4.010
6.010
8.010
1.010
mq=0.015 0.025 0.035 0.050 0.5 1 1.5 2
q
2
0.002 0.004 0.006 0.008 0.01
q
2 V-A (0)(q 2)
mq=0.015 0.025 0.035 0.050
Results at mq=0.015 is shown.
compared to the spectral rep.
values are used. In the spectral representation,
q2Π(0)
V −A(q2) =
f 2
πm2 π
q2 + m2
π
+ (excited states ∼ O(m2
q))
0.01 0.02 0.03 0.04 0.05 0.06
mq
q
2 (1) V-A(q 2 min)
finite volume infinite volume
the measured fπ and mπ. L10(mρ)= −5.2(2)(5)×10−3 (χ2/dof=0.5, 2.3) L10(Exp)= −5.09(47)×10−3 (x=4mπ2/q2, H(x) is known function.)
Π(1)
V −A(q2) = −f 2 π
q2 − 8 Lr
10(µχ) −
ln
π
µ2
χ
3 − H(x)
24π2 ,
0.5 1 1.5 2
q
2
q
2 !V-A (1)(q 2)
0.015 0.025 0.035 0.050 1 1.5 2
∆m2
π = − 3α
4πf 2
π
∞ dq2 q2 Π(1)
V −A(q2),
−f 2
π
q2 + f 2
ρ
q2 + m2
ρ
− f 2
a1
q2 + m2
a1
.
Π(1)
V −A(q2) =
− 1 242
π
ln
π
µ2
χ
3 − H(q2) + x6Q2 ρ
1 + x5 (Q2
ρ)4
, (9) − 1 242
π
Q2
ρ(x6 + x7 ln(Q2 ρ))
1 + x5(Q2
ρ)4
, (10) − 1 24π2 x6 Q2
ρ ln(Q2 ρ)
1 + x5 (Q2
ρ)4 ,
(11) − 1 242
π
ln (1 + x6Q2
ρ)
1 + x5 (Q2
ρ)3 ,
(12)
Functional form
Exp: Δmπ2= 1261.2 MeV2
Errors are (statistical)(chiral extrapolation)(large q2)
0.5 1 1.5 2
q
2
q
2 V-A (1)(q 2)
0.015 0.025 0.035 0.050 1 1.5 2
∆m2
π = − 3α
4πf 2
π
∞ dq2 q2 Π(1)
V −A(q2),
is ∆m2
π=
error again. Summing 975(30)(67)(166) MeV2
q2
max
(9) (10) (11) (12) 1.0 687(64) 772(147) 821(79) 829(83) 2.0 676(50) 786(13) 811(12) 806(49)
lue and neglecting the log term e ∆m2
π|q2≥2.0= 232(166) MeV2,
n ∆m2
π|q2≤2.0= 743(30)(67) MeV2.
Π(1)
V −A(q2)
s ∼ (a + b ln(q2))/(q2)3
6
f [−0.001, −0.006] GeV6 lecting the log term contr
a=