Neutron electric dipole moment
with
two flavors of domain wall fermions
Lattice 2005 (Dublin)
Tom Blum
(University of Connecticut and RIKEN BNL Research Center)
F. Berruto, K. Orginos(MIT), A. Soni(BNL)
(RBC Collaboration)
July 26, 2005
2
Outline
1. Introduction: physics and methodology∗
2. Numerical results
3. Summary/Outlook
∗ Our calculation [Lattice 2004] is very similar to a recent quenched
calculation [Aoki, Kikukawa, Kuramashi, and Shintani (2005)]. See
also the talk by E. Shintani at this meeting.
3
Introduction
Consider adding a T- and P-odd (CP-odd) term to the QCD
action
SQCD,θ = −θ
Z
dt
Z
d3x
g2
µν (x)Gρσ (x)
tr
G
[
]
µνρσ
32π 2
where Gµν (x) is the gluon field strength and
~ ·B
~
Tr G(x)G̃(x) ∼ E
(c.f., E 2 − B 2)
1
G̃(x)µν =
µνρσ Gρσ (x)
2
This “θ-term” gives rise to permanent electric dipole moments
to the quarks as well as bound states like the neutron
4
The current experimental bound on dN is
|d~N | < 6.3 × 10−26 e-cm [Harris, et al. (1999)]
New searches: 225Ra (running at ANL) and deuteron (BNL
proposal) to improve sensitivity by 2-3 orders of magnitude.
< 10−10, which is unnaturally small.
+ model calculations implies θ ∼
This is often called the Strong CP problem.
Lattice regularization provides first-principles technique for calculation of dN /θ.
5
The QCD Lagrangian for massless fermions
LQCD,f = ψ̄ (i/
D) ψ
is invariant under chiral transformations of the quark fields
ψ → (1 + iαγ5/2)ψ
ψ̄ → ψ̄(1 + iαγ5/2)
But the measure of the path intergral is not,
DψDψ̄ → DψDψ̄ exp i α
Z
d4x
g 2 µνρσ
TrGµν Gρσ
2
32π
which gives rise to the Adler-Bell-Jackiw anomaly
(a.k.a the axial anomaly, c.f. massive η 0 and π 0 → 2γ).
6
Choosing α = −θ, the θ term can be rotated away, or canceled
exactly in the action [But not for Nf = 1, see Creutz (2004)]
If all the quarks are massive, the chiral rotation generates another
term in the action that can not be canceled by further field redefinitions
mψ̄ψ → mψ̄ψ + iθmψ̄γ5ψ
which is also P- and T- odd.
Thus the CP-violating term in the QCD Lagrangian can be transformed between the gauge and fermion sectors, but it can not
be eliminated
7
The θ term can be written as a total divergence.
Still,
Z
g2
4
µνρσ G G
d x
tr
µν ρσ
32π 2
= Q
where Q is the integral topological charge, Q = 0, ±1, ±2, . . .
Thus the θ term can produce physical effects (like an electric
dipole moment of the neutron)
8
Taking the chiral limit, m → 0
Again, consider the QCD partition function for massive quarks
which can be written
Z=
Z
DAµ det[/
D(m) + iθmγ5]Nf e−SG .
and if θ is small,
−1 ) ]+O(θ 2 ),
det [/
D(m) + iθmγ5] = det[/
D(m)] [1+iθm tr(γ5D(m)
/
−1 and the index theorem
The spectral decomposition of D(m)
/
lead to
−1 =
D(m)
/
X |λihλ|
λ
Nf
X
f =1
tr γ5D
/ −1(m)
h
i
iλ + m
n+ − n−
Q
=
=
m
m
9
What happened? It appears that in the m → 0 limit, the θ term
does not vanish
Correct quark mass dependence is recovered from the usual
CP-even part of the fermionic action,
Nf
det D(m)
/
= Πi(iλi + m)Nf
As m → 0, Q 6= 0 configurations are suppressed since they
support exact zero modes of D
/ with λi = 0.
In other words, the distribution of Q → δ(Q), or hQ2i/V → 0, so
the θ term effectively vanishes.
10
The quenched approximation of lattice QCD
Quenched approximation: set
det[/
D(m)] = 1
(amounts to omitting quark vacuum polarization to all orders)
For small θ,
−1 ) ]+O(θ 2 ),
det [/
D(m) + iθmγ5] = det[/
D(m)] [1+iθm tr(γ5D(m)
/
So, even in quenched case there is CP-violating physics [Aoki,
Gocksch, Manohar, and Sharpe (1990)]. But mass dependence
is completely wrong. Many observables, possibly dN , have
pathological chiral limit (c.f., dN in ILM [Faccioli, Guadagnoli,
Simula (2004)])
11
Computational Methodology
Compute the matrix elements of the electromagnetic current
between nucleon states in the θ 6= 0 vacuum
hp 0, s0|J µ|p, siθ = ūs0 (p 0)Γµ(q 2)us(~
p)
F2(q 2)
2
µν
µ
2
µ
Γ (q ) = γ F1(q ) + i σ qν
2m
F3(q 2)
µ
5
2
5
µ
2
µν
5
+ γ γ q − 2mγ q FA(q ) + σ qν γ
2m
1 µ
1
2 µ
1
¯ d = Jµ + Jµ
ūγ u − dγ
3
3
2 V
6 S
µ
¯ µd
JV = ūγ µu − dγ
µ
¯ µd.
JS = ūγ µu + dγ
Jµ =
q 2 < 0, momentum transfered by the external photon (space-like)
12
The most general matrix element consistent with Lorentz, gauge,
CPT symmetries of QCD. The insertion of J µ probes the electromagnetic structure of the nucleon. For q 2 → 0
• F1(0): electric charge in units of e (+1 proton, 0 neutron)
• F2(0)/2m: magnetic dipole moment in units of e
• FA: anapole moment
• F3(0)/2m: electric dipole moment in units of e
The last two vanish if θ = 0
13
Calculating dipole moments on the lattice
Lattice calculations are done with correlation functions in Euclidean space-time,
and rely on the LSZ reduction formula to project out the desired
matrix element by using exponential dominance of ground state
†
G(t, t0) = hχN (t0, p
~0) J µ(t, q) χN (0, p
~)i
=
X
s,s0
0 −t)E 0 −tE
1
†
0
0
0
0
µ
−(t
e
e
h0|χN |p , s ihp , s |J |p, sihp, s|χN |0i
0
2E 2E
+...
= Gµ(q) ∗ f (t, t0, E, E 0) + . . .
14
With suitable choices of projectors, form factors can be determined from the correlation functions, e.g. for θ = 0,
P xy
=
P4 =
=
trP xy Gx,y (q 2) =
trP 4 G4(q 2) =
i 1 + γ4 x y
−
γ γ
4 2
1 1 + γ4 4
γ
4 2
1 1 + γ4
4 2
±py,x m(F1(q 2) + F2(q 2))
!
2
q
2
2)
m (E + m) F1(q ) +
F
(q
2
(2m)2
Linear combinations of F1 and F2 in ()’s are magnetic and
electric form factors, GM (q 2) and GE (q 2), respectively.
15
To get the desired moment take ratios of three-point functions
(Z, renormalization, kinematical factors all drop out), e.g.,
xy x
0
xy x
2
~)
1 trP GP,N (t, t , E, p
1 trP GP,N (q )
lim
=
4
4
0
2)
0
py trP 4G4
~)
(q
t t0 py TrP GP (t, t , E, p
P
1 F1(q 2) + F2(q 2)
=
(P ) 2
E+m
G
(q )
E
1
lim =
q→0
2m
(
1 + aµ,P
aµ,N
yields the magnetic dipole moments
F1(0) = 1, 0 for the proton, neutron
16
CP violating vacuum, θ 6= 0
The physical neutron 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|N ∗i
α0 ∝ θ
This gives rise to mixing of electric and magnetic dipole moment
terms in projected correlation functions.
[ Pospelov and Ritz (1999), Aoki, Kikukawa, Kuramashi, and Shintani
(2004)]
17
The electric dipole moment is obtained from, e.g.
p2
p2
z
xy
z
2
trP G (q ) = αm(E − m)F1 + α(m(E − m) + )F2 + z F3 + O(θ2)
2
2
E
+
3m
E
+
m
trP xy Gt(q 2) = ipz αmF1(q 2) + α
F2(q 2) +
F3(q 2)
2
2
+O(θ2).
The terms proportional to α must be subtracted
(
~)
1 trP xy GtN (t, t0 , E, p
ipz trP tGtP (t, t0 , E, p
~)
−
αmF1
+ α E+3m
F2 (q 2 )
2
(P ) 2
+ m)GE
(q )
(q 2 )
m(E
)
=
lim {· · ·} =
q 2 →0
F3 (q 2 )
) 2
2mG(P
E (q )
F3 (0)
= dN
2m
18
Mixing angle α is calculated from the
ratio of two-point functions
[Aoki, Kikukawa, Kuramashi, and Shintani (2004)]
†
hχN θ (t)χ θ (0)iθ
N
ZN θ (1 + γ 4 + exp i2αγ 5)
=
exp −mN θ t + . . .
2
To lowest order in α (θ) we have
1 + γ4
†
tr
γ5 hχN θ (t)χ θ (0)iθ ≈ i ZN α e−mN t
N
2
1 + γ4
†
tr
hχN θ (t)χ θ (0)iθ ≈ ZN e−mN t
N
2
mN θ = mN + O(θ2)
ZN θ = ZN + O(θ2)
19
Computing with θ 6= 0
1
hOiθ =
Z(θ)
Assuming θ 1
Z
−S(Aµ )−iθ
DAµDψ̄DψO e
R
d4 x
g2
tr[G(x)G̃(x)]
32π 2
1
≈
DAµDψ̄Dψ(1 − iθQ)O e−S(Aµ)
Z(0)
= hOi − iθhQOi
Z
CP odd piece: simple weighted average in CP even vacuum
hQOi =
X
P (Qν ) Qν hOiν ,
ν
Can also weight with the pseudo-scalar density
[Guadagnoli, Lubicz, Martinelli, and Simula (2002)]
For chirally symmetric lattice fermions that have an index, this
is equivalent to weighting with Q. If chiral symmetry is broken,
then the two methods will agree in the limit a → 0.
20
Relation between dN and topology of the vacuum
• dN → 0 as m2
π → 0 since Πi(λi + m) → 0, λi = 0 for Q 6= 0
2
2
2
• χPT gives dN ∼ m2
π log mπ , hQ i/V ∼ mπ
• hQ2i/V = constant implies that dN does not vanish in the
quenched theory (Nf = 0) (pathological?)
• mixing angle α must vanish as m2
π → 0;
α → 0 as hQ2i/V → 0
∂hOi • In large N , hOQi = hQ2i ∂ Q al.
(1996); Faccioli, et al.
ν=0
= hQ2ihOiν=1 [Diakonov, et
(2004)]
21
Numerical Results
• DWF+DBW2, Nf = 2, msea = mval = 0.04
[RBC Collaboration (2004)].
• msea = 0.02 and 0.03 in progress
• a−1 ≈ 1.7 GeV.
• quenched DWF+DBW2, mval = 0.05 valence DWF
[RBC Collaboration (2002)]
• Non-zero momenta for one of the nucleons, p
~ = (±1, 0, 0),
(±1, ±1, 0), and (±1, ±1, ±1) (and permutations) since form
factors multiplied by q ν
(∂/∂(q 2)|0 not accessible on a finite lattice [Wilcox (2002)])
22
Topological charge “history” (distribution):
O(a2) improved Q was
computed by integrating
the topological charge
density after APE smearing the gauge fields.
5
0
-5
msea = 0.03
-10
10
5
0
-5
msea =0.04
-10
10
5
0
-5
-10
quenched
Nf = 2 flavor: small step
hybrid monte-carlo, Q
sampled slowly, longtime correlations
Quenched: big change
heat bath monte-carlo,
Q sampled efficiently,
∼ no correlations
msea = 0.02
10
0
1000
2000
3000
4000
5000
6000
10
0
-10
0
100
200
Trajectory
300
400
23
Topological charge susceptibility χ:
hQ2i
χ =
V
= f 2m2
π /8
0.06
χ (r0)
Statistical
errors may be
under-estimated
(blocks of 50
trajecs)
4
0.04
0.02
-1
quenched, a = 1.3 GeV
-1
nf = 2, a =1.7 GeV
Lowest order Chiral PT
0
0
1
2
3
4
5
2
(r0mπ)
24
The mixing coefficient α
To lowest order in θ, CP even and odd parts of two point function
have the same mass, mN , and amplitude.
1.1
1
MN
G(t) = A e−mN t + · · ·
Gθ (t) = A α e−mN t +· · ·
θ=0
θ !=0
0.8
0.7
0.6
0.2
0.15
α
Systematic difference
in nucleon mass determination makes extraction of α difficult:
fake plateau
0.9
0.1
0.05
0
2
4
tmin
6
8
10
25
mixing coefficient α, quenched
1.1
MN
Things look more sensible on the quenched θlat=0
θ !=0
tice: better sampling of
topological charge.
1
0.9
0.8
0.4
0.3
α
Plateau is (almost) trustworthy.
α = 0.21(3) is rather
large,
c.f. topological
charge susceptibility
0.2
0.1
0
0
2
4
tmin
6
8
10
26
Eigo Shintani’s talk: reweighting with Q is non-trivial, but works!
27
mixing coefficient α, Nf = 2 again
Extract α from fit
Gθ (t) = A α e−mN t.
0.1
α = 0.035(13)
0.08
0.06
α
Can also fix mN to
its correct CP even
value
α = 0.072(15)
Imprecise,
but
much smaller than
quenched
value
(c.f., hQ2/V i)
0.04
0.02
0
1
2
3
4
5
tmin
6
7
8
9
28
10
Three-point correlation function ratios
-0.5
• Neutron
magnetic form
factor ratios
• Nucleon sources
at t = 0 and 10
• Excited
state
contamination
appears small
• lowest q 2 on the
bottom
-1
-1.5
-0.5
-1
-1.5
-0.5
-1
-1.5
2
4
t
6
8
10
29
(P )
Ratio of form factors GM (q 2)/GE (q 2)
q 2 dependence mild
4
3.5
Anomalous µ’s are
equal and opposite
well
within
errors,
implies iso-scalar
contribution ∼ 0;
disconnected
diagrams not
computed, so puzzling
3
2.5
2
2
GM(q )/GE,P(q )
Reasonable agreement
with experiment (diamonds) [Jones, et al.
(2000), JLab]
2
1.5
1
0
1
0.5
2
1.5
2
q (GeV )
(Lattice error estimates are statistical uncertainties only)
30
Subtracted F3(q 2) ratios
• Neutron
electric dipole
form factor ratios
• F1 and F2 terms
subtracted
• Subtraction
term (squares)
is relatively well
resolved
• error dominated
by 3-point function, ∼ 0
0.5
0.25
0
-0.25
0.5
0.25
0
-0.25
0.5
0.25
0
-0.25
2
4
t
6
8
10
31
(P )
Ratio of form factors F3(q 2)/(2mGE (q 2))
F3(0)
2m
F3(0.401 GeV2)
≈
2m
= +0.087(95).
dlat
N =
dN ≈ 0.010(11) θ e fm.
0.3
subtracted ratio
Our conventions have
lead to a positive central value of dN /θ.
Could change as calculation improves.
In physical units,
0.4
0.2
0.1
0
-0.1
0
1
0.5
2
2
q (GeV )
(Error estimates are statistical uncertainties only.)
32
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 unphysically
large and has the wrong quark mass dependence, so quenched
dN will have large systematic error.
Our central value is ∼ 2× leading χPT result [Crewther, Di
Veccio, Veneziano, Witten (1979)], ∼ 5 − 10× sum rules value
[Pospelov and Ritz (1999)], ∼ −0.4× quenched value [Aoki,
Kikukawa, Kuramashi, and Shintani (2005)]
Future: In progress 2+1 flavor DWF calculation (RBC+UKQCD)
may be promising if long HMC evolutions obtained [Talk by Yamaguchi,
this meeting]. Need to carefully study the mass dependence, extrapolate to the physical pion (light quark) mass
33
Acknowledgements
The computations described here were done on the RIKEN BNL
Research Center QCDSP supercomputer.
We thank our colleagues in the RBC collaboration, 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 was supported by the
RIKEN BNL Research center.
34