Domain wall fermion calculation of
the neutron dipole moments
“From Actions to Experiment”
The 2nd International Lattice Field Theory Network Workshop
Edinburgh
F. Berruto, T. Blum∗, K. Orginos, A. Soni (RBC Collaboration)
March 7, 2005
∗ Presenter
1
Introduction
T- and P-odd term allowed in QCD Lagrangian:
SQCD,θ = iθ
Z
d4x
h
i
g2
tr G(x)G̃(x) = iθQ,
32π 2
(G(x)G̃(x) ∼ E · B)
where Q is the topological charge of the QCD vacuum.
θ-term is CP-odd → neutron electric dipole moment, dN .
Weak interactions: Also violate CP: CKM mechanism: dN ≤
10−30 e-cm (vanishes at one-loop), many orders of magnitude
below the experimental bound [1], |d~N | < 6.3 × 10−26 e-cm.
2
Experimental bound + model calculations imply θ ≤ 10−10,
which is unnaturally small. However, no known symmetry to
say it vanishes. This is often called the Strong CP problem.
To translate the above experimental bound to a constraint on the
fundamental θ parameter requires evaluation of nucleon matrix
elements.
Lattice method is first-principles technique for calculation.
3
Model calculations of dN
dN ' (1 − 10) × 10−16 e θ-cm
Faccioli, Guadagnoli, and Simula recently found
dN = (6 − 14) × 10−16 e θ-cm in the instanton liquid model
4
Using the axial anomaly, one can replace the CP violating gauge
action above with the fermionic action,
Sθ0 = −iθ m
Z
d4xP (x)
¯
P (x) = ū(x)γ5u(x) + d(x)γ
5 d(x) + s̄(x)γ5 s(x)
m =
=
−1
−1
−1
−1
mu + md + ms
mu md ms
mu + md + ms
Note, that the θ term does vanish if one of the quark masses
vanishes, provided P (x) does not go like ∼ 1/m.
5
Remarks on the quenched case
The QCD partition function in the presence of explicit CP violation is
Z=
Z
DAµ det[D(m) + iθmγ5] e−SG .
Setting det[D(m) + iθmγ5] = 1, we lose CP violating physics.
However, if θ is small,
det [D(m) + iθmγ5] = det[D(m)] [1+iθm tr(γ5D(m)−1) ]+O(θ2),
and we quench as usual by setting det [D(m)] = 1.
6
The chiral limit
The spectral decomposition of D−1(m) leads to
D(m)|λii = (λi + m)|λii
Nf
X
f =1
Tr γ5D−1(mf )
h
i
n+ − n−
Q
=
=
m
m
for Nf flavors and n+ and n− the number of right- and lefthanded zero modes of D(m).
If we trade Q for −mP (using the anomaly), m̄ dependence cancels. Correct mass dependence of dN requires (det D(m))Nf . dN
does not vanish in the quenched chiral limit (c.f. topological
susceptibility). Recall that det D(m) ∼ m for Q 6= 0, and contributions to dN vanish for Q = 0.
7
Computational Methodology
Compute the matrix elements of the electromagnetic current
between nucleon states,
hp0, s|J µ|p, siθ = ū(p0, s)Γµ(q 2)u(p, s)
Γµ(q 2) = γµ F1(q 2)
F2(q 2)
ν
+i σµν q
2m
2
+ γµ γ5 q − 2mγ5 qµ FA(q 2)
F3(q 2)
ν
,
+σµν q γ5
2m
q 2 = −2E(~
p)mN + 2m2
N <0
Four terms consistent with Lorentz, gauge, CPT symmetry
8
The physical neutron state in the CP-broken vacuum is a mixture
of the θ = 0 vacuum (opposite parity) eigenstates |N i and |N ∗i.
|N θ i = |N i + iα0(θ)|N ∗i
This gives rise to unphysical mixing in correlation functions.
[Pospelov and Ritz (1999), Aoki, Kuramashi, and Shintani (2004)]
9
The current acting on the states gives
+ |N i + iα0 h0|J + |N ∗ i
h0|J +|N θ i = h0|J
q
q
0
=
ZN u + iα ZN ∗ v
=
q
→
q
ZN u + iα0
q
ZN ∗ γ5u
!
√
q
Z
=
ZN 1 + iα0 √ N ∗ γ5 u
ZN
ZN ei αγ5 u
where
h
i
+
T
J
= abc ua Cγ5db uc
10
The correlation function is
0
e−EN (p )(τ −t) e−EN (p)t
+
+
hθ|J (τ )q̄γµ q J¯ (0)|θi =
hθ|Jp+0 |N θ ihN θ |q̄γµ q|N θ ihN θ |J¯p+ |θi
0
4EN (p )EN (p)
−EN (p0 )(τ −t) e−EN (p)t
p
e
=
ZN (p0 )ZN (p)
uθ (p0 )ūθ (p0 )Γµ (q)uθ (p)ūθ (p)
0
4EN (p )EN (p)
Using the spinor relations
uθ = eiαγ5 u = u + iαv + O(α2)
ūθ = ūeiαγ5 = ū + iαv̄ + O(α2)
X
u ū = i p
/+m
s
11
and expanding to lowest order in α (small θ) yields,
0
hθ|J + (τ )q̄γµ q J¯+ (0)|θi =
Using the projector P =
eiEN (p )(τ −t) eiEN (p)t
0
×
ZN (p )ZN (p)
0
4EN (p )EN (p)
p
/ 0 + mN + 2iγ5 α Γµ (q) p
/ + mN + 2iγ5 α + O(α2 )
p
i1
(1
42
+ γ4 )γx γy we obtain
TrP p
/ 0 + mN + 2iγ5 α Γµ (q) p
/ + mN + 2iγ5 α =
(E+3m)
i
2
2
2
F2 (−q )
− 2 pz (E + m)F3 (−q ) + 2α 2mF1 (−q ) +
2
In the limit q 2 → 0,
= −impz [F3(0) + 2αF2(0)]
12
Mixing angle α is calculated from the ratio of two-point functions
[Aoki, Kuramashi, and Shintani (2004)]
hθ|J +(t)J¯+(0)|θi =
1
e−EN (p)thθ|J +|N θ ihN θ |J¯+|θi
2EN (p)
1
= |ZN |
e−EN (p)tuθ ūθ
2EN (p)
Expand to lowest order in θ,
hθ|J +(t)J¯+(0)|θi ≈ |ZN |
1
eiEN (p)t(uū + iα{γ5, uū})
2EN (p)
1
= |ZN |
eiEN (p)t(p
/ + mN + 2iαmN γ5)
2EN (p)
13
Project:
1 + γ0 θ
Tr
G (t)
2
1 + γ0
Tr −i
γ5Gθ (t)
2
=
=
e−mN t
1 + γ0
ZN
mN Tr
(γ0 + 1 + 2iαγ5)
2mN
2
2 ZN e−mN t
e−mN t
1 + γ0
ZN
mN Tr −i
γ5(γ0 + 1 + 2iαγ5)
2mN
2
2 ZN α e−mN t
=
=
14
Computing with θ 6= 0
Complex action. But since θ 1 in Nature, expand hp0, s0|J µ|p, siθ
to lowest order in θ.
2
1
−S+iθ d4 x g 2 tr[G(x)G̃(x)]
32π
hOi =
Oe
Z(θ) f ields
Z
iθ
=
Q O e−S + O(θ2)
Z(0) f ields
Rewrite (i.e., for the neutron EDM)
R
Z
hp0, s0|J µ|p, si
θ
=
X
P (Qν )Qν hp0, s0|J µ|p, siν
ν
and in large N ,
hp0, s0|J µ|p, siθ = hQ2ihp0, s0|J µ|p, siQ=1
[Diakonov, et al.; Faccioli, et al.]
15
Relation between dN and topology of the vacuum
• Clearly, dn is sensitive to hQ2i
• In the full theory, χPT gives dN , hQ2i ∼ m2
π
• hQ2i = const implies that dN doesn’t vanish in quenched
theory (pathological?)
• mixing angle α must vanish as m2
π → 0; plausible that
α ∝ hQ2i
16
Numerical Results (PRELIMINARY)
• 233 Nf = 2, msea = mval = 0.04, DWF configurations (separated by 20
trajectories) [hep-lat/0411006 (RBC Collaboration)].
Lattice: 163 × 32, Ls = 12, and the inverse lattice spacing in the msea = 0
limit is a−1 ≈ 1.7 GeV.
• 300 quenched DBW2 configurations (separated by 1000 trajectories),
mval = 0.04 valence DWF. [Phys.Rev.D69:074504,2004 (RBC Collaboration)]
• Computed for one source/sink combination t = 0 and 10, Gaussian
smeared source and sink, operator (J µ ) is inserted between source and
sink.
• Averaged over time slices 4-7 and (equivalent) permutations of the momenta p
~ = (1, 0, 0), (1,1,0), and (1,1,1).
17
msea = 0.02
10
-5
msea = 0.03
5
0
-5
-10
10
msea =0.04
Q was computed by integrating the topological charge density after
APE smearing the gauge
fields (20 sweeps with
ape weight 0.45).
0
-10
10
5
0
-5
-10
quenched
Topological charge Q:
5
0
1000
2000
3000
4000
5000
6000
10
0
-10
0
100
200
Trajectory
300
400
18
Topological charge susceptibility χ:
0.06
4
0.04
χ (r0)
hQ2i
χ=
V
should
vanish
like m2
π in dynamical
case.
Errors probably
under-estimated
(blocks of 50
trajecs)
0.02
-1
quenched, a = 1.3 GeV
-1
nf = 2, a =1.7 GeV
0
0
1
2
3
4
5
2
(r0mπ)
Interesting physics for 2+1 flavor DWF lattices (need long evolutions).
19
Form Factor
Ratios
2
(p = 1)
2
Ratio
• Dipole moment ratios
• Non-zero
mixing
implies dN 6= 0
1
0
-1
neutron magnetic ratio
proton magnetic ratio
neutron electric ratio
-2
0
1
2
3
4
5 6 7
time slice
8
9 10 11 12
20
The mixing coefficient α:
nf = 2, msea=mval=0.04
1.1
0.9
0.8
θ=0
θ !=0
0.7
0.6
0
-0.05
α
Systematic difference
in nucleon mass determination (they must
be equal at this order)
makes extraction of α
difficult: fake plateau
MN
1
-0.1
-0.15
-0.2
0
2
4
6
8
tmin
10
12
14
21
mixing coefficient α, continued:
-1
quenched β=0.87 (a = 1.3 GeV)
Things look more sensible on the quenched lattice: better Q distribution (?)
MN
1.1
θ=0
θ !=0
1
0.9
0.8
0
α ∼ −0.2
α
-0.1
-0.2
-0.3
-0.4
0
2
4
6
8
tmin
10
12
14
22
Approximating the q 2 → 0 limit with smallest value of p
~2 (=
(2π/16)2), taking mf = 0.04 as physical yields
aP
µ ≈ 1.75(7)
aN
µ ≈ −1.75(6)
roughly consistent with the experimental values aP
µ = 1.79 and
aN
µ = −1.91.
Error estimates are statistical uncertainties only.
23
Summary/Outlook
Inadequate topological charge distribution limits the accuracy of
the nf = 2 flavor calculation (algorithmic problem)
Quenched calculation avoids this problem, but χ is unphysical,
so quenched dn may also have large systematic error.
Future: 2+1 flavor DWF calculation appears promising if long
HMC evolutions are available.
24
Acknowledgements
The computations described here were done on the RIKEN BNL
Research Center QCDSP supercomputer.
We thank our colleagues in the RBC collaboration, and in particular N. Christ, M. Creutz, and S. Aoki for useful discussions.
The work of FB and AS was supported in part by US DOE grant
# DE-AC02-98CH10886. KO was supported in part by DOE
grant #DFFC02-94ER40818. TB is supported by the RIKEN
BNL Research center.
25