b. k. sahoo - 国立大学法人...
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B. K. Sahoo Theoretical Physics Division PRL, Ahmedabad, India
A brief discussion on atomic EDMs: NSM and T-PT interactions
Motivation and undergoing measurements
Current experimental status
Probing fundamental particle physics
Atomic many-body methods and their relations
Theoretical results for Xe atom
Summary and Outlook
Outline
T / CP violation EDM
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D =
DJ
J
P : J →
J
D →−
D
T : J →−
J
D →
D
EDM in neutron: N. F. Ramsey (1950s)
Atoms & molecules: P. G. H. Sandars (1960s)
Motivation
Atom: a non-degenerate system
leptonic, semi-leptonic, hadronic CP violations
Not enough CP-violation in the Standard Model (SM) to generate enough matter-antimatter asymmetry of Universe!
+
-
Measurement of EDM in Xe atom
Prof. K. Asahi group, TIT, Tokyo, Japan (T. Inoue et al, Hyperfine Interactions (Springer Netherlands 220, 59 (2013)).
Prof. P. Fierlinger, Cluster of Excellence for Fundamental Physics, Technische Universitaet Muenchen, Germany
Prof. U. Schmidt, Collaboration of the Helium Xenon EDM Experiment, Physikalisches Institut, University of Heidelberg
Experimental Detection
H = – µ ⋅ B – d ⋅ E
Single atom with coherence time τ:
N uncorrelated atoms measured for time T >> τ:
• Statistical Sensitivity:
B E
d µ ω1
B E
d µ ω2
• Larmour spin-precession frequencies
δω = 1 τ
δ d = h 2 E
1 2 τ TN
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ω1 =2 µ • B + 2
d T−PT •
E
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ω2 =2 µ • B − 2
d T−PT •
E
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ω1 −ω2 =4 d T−PT •
E
With 1 year observational data
• d(199Hg) = (0.49±1.29stat±0.76syst)×10−29 e cm
⇒ |d(199Hg)| < 3.1×10−29 e cm (95% C.L.)
199Hg EDM Result
W. C. Griffith, M. D. Swallows, T. H. Loftus, M. V. Romalis, B. R. Heckel, E. N. Fortson
Phys. Rev. Lett. 102, 101601 (2009).
EDM Searches
Quark EDM
Electron EDM
QCD
Nuclear Theory
Atomic Theory
Neutron n
Diamagnetic Atoms Hg, Xe, Rn
Paramagnetic Atoms Tl,Cs, Fr
Quark Chromo-EDM
Molecules PbO, YbF, TlF
Atomic Theory
Atomic Theory
QCD
Fundamental Theory - Supersymmetry, Strings
Nuclear
High Energy
Nuclear
Theory Atom
ic
Atomic Molecular
Experiments
Best limits Fundamental CP-violating phases
neutron EDM
EDMs of diamagnetic systems (Hg,Ra)
EDMs of paramagnetic systems (Tl)
Schiff moment, CT
nucleon level
quark/lepton level
nuclear level
atomic level
Leading mechanisms for EDM generation
|d(199Hg)| < 3 x 10-29 e cm (95% c.l., Seattle, 2009)
|d(205Tl)| < 9.6 x 10-25 e cm (90% c.l., Berkeley, 2002)
|d(n)| < 2.9 x 10-26 e cm (90% c.l., Grenoble, 2006)
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e | ˜ d d − ˜ d u | < 6 ×10-27e cm|d(199Hg)| < 3 x 10-29 e cm
|d(n)| < 2.9 x 10-26 e cm
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| e( ˜ d d + 0.5 ˜ d u) +1.3dd − 0.32du | < 3×10-26e cm
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dq , ˜ d q, θ, cq−eT−PT
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de, cq−eS−PS
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dN , cN−eT−PT
Present status
Interpretation of Schiff moment
gπNN π
n p
-30-20-10102030-20-101020
Exponentially small outside nucleus, zero at two poles
E
nuclear spin
ϕ(R)
R
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ϕ(R)= eρN (r)| R - r |
∫ d3r+ 1Z( d ⋅ ∇ R )
ρN (r)| R - r |∫ d3r
Schiff theorem: No EDM from the point nuleus and only with recoil effect.
SM appears when finite size of the nucleus and screening of the external electric field by the atomic electrons are taken into account.
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ϕ(R)= − 3 S ⋅ R
BρN (R)
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B= ρN (R)R4∫ dR
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S = e
10r2 r − 5
3Zr2 r
g
q q
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S = RN g πNN with g πNN = RQCD ( ˜ d u − ˜ d d )
An electron sees the effective nuclear potential:
Tensor-pseudotensor (T-PT) interaction between quark- electron interaction lead to the nucleon-electron T-PT interaction giving rise to EDM in atom:
Interpretation of T-PT interaction
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HEDMe−q =
iGFCTe−q
2Ψ qσ µνΨq[ ] Ψ eγ
5σ µνΨe[ ]
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HEDMe−N =
iGF
2CT
e−NΨ Nσ µνΨN[ ] Ψ eγ5σ µνΨe[ ]∑
=iGF
2CT
e−NΨN+α iα jΨN[ ]i≠ j
N ,e∑ Ψe
+γ 5βα iα jΨe[ ]i≠ j
=iGF
2CT
e−NΨN+iβεijkσN
kΨN[ ]non−relativistic
N ,e∑ Ψe
+(−i)ε ijlγ lΨe[ ]
≡ 2iGFCT ρN(re ) I N
e∑ . γ e
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(α iα j )i≠ j = (σ iσ j )i≠ j = iεijkσ
k
= iεijkγ5α k
One-body + Two-body operators
Approach: (i) Average over the other two-body interactions as a single particle potential (central field model or mean field model, eg. Hartree-Fock method, )
(ii) For accurate calculations, include the residual interactions perturbatively.
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H = h(ri) + V (rij )i> j∑
i∑
Self-energy correction Vacuum polarization
Atomic Structure Calculations
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H = c α i ⋅ p i + VNuc (ri) + VQED (ri)[ ] + [VC (rij ) + VB (rij )]
i≥ j∑
i∑
Relativistic Hamiltonian:
All Order Many-body Methods
Configuration Interaction (CI) method:
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ΨCI =C0 Φ0 + CI Φ0→I + CII Φ0→II + CIII Φ0→III +…
= CL Φ0→LL∑
Coupled-Cluster (CC) method:
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ΨCC = CL Φ0→LL∑
= Φ0 + tI Φ0→I + tII Φ0→II + 12 tI tI Φ0→II + tIII Φ0→III +…
= Φ0 + TI Φ0 + TII Φ0 + TIII Φ0 +…
= eT Φ0
Many-body Perturbation Theory (MBPT):
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ΨMBPT = Φ0 + Φ0(1) + Φ0
(2) + Φ0(3) +…
= Φ0 + (CI(1) ΦI + CII
(1) ΦII +…) + (CI(2) ΦI + CII
(2) ΦII +…)
= CL Φ0→LL∑
P
Q
Fock space
Bloch’s Description (A well defined prescription)
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P +Q =1
The wave functions of the atoms are known in the model (P) space. For exact ones, need to include contributions from the orthogonal (Q) space.
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Ψ =ΩΦ0
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P = Φ0 Φ0
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Q = Ψ Ψ − Φ0 Φ0
Perturbation approach:
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Ω =Ω(0) +Ω(1) +Ω(2) +Ω(3) +… = Ω(n )
n∑
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[Ω(k ),H0]P =QVΩ(k−1)P − Ω(k−m )
m=1
k−1
∑ PVΩ(k−1)P
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H = H0 +V with H0 Φn = En Φn
Amplitudes of Wave operators:
Effective Hamiltonian:
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Heff = PHΩP
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Ω(0) =1
Double Perturbation
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P +Q =1
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Ψ = Ψ(0) + Ψ(' ) =ΩΦ0
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P = Φ0 Φ0
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Q = Ψ Ψ − Φ0 Φ0
In a perturbation approach:
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Ω =Ω(0,0) +Ω(0,1) +Ω(1,0) +Ω(0,2) +Ω(1,1) +… = Ω(n,m )
n.m∑€
H = H0 +V1 +V2 with H0 Φn = En Φn
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Ψ(k= β +δ ) =Ω(k, 0) Φ0 + Ω(k−δ , δ )
δ =1
k−1
∑ Φ0
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with Ω(0, 0) =1, Ω(1, 0) =V1 and Ω(0, 1) =V2
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[Ω(β , δ ),H0]P =QV1Ω(β −1, δ )P +QV2Ω
(β , δ −1)P
−m=1
β −1
∑ Ω(β −m, δ − l )PV1Ω(m−1, l )P(
l=1
k−1
∑ −Ω(β −m, δ − l )PV2Ω(m, l−1)P)
EDM of Closed-shell atoms
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Ψ(0) =Ω(n,0) Φ0
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X ≡ ς =DA
S= 2
Ψ(0) DΨNSM(1)
Ψ(0) Ψ(0)
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X ≡η =DA
σN CT
= 2Ψ(0) DΨT−PT
(1)
Ψ(0) Ψ(0)
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Hint (r) = HNSM(r) =3ρN (r)BI
I N • r
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Hint (r) = HT -PT (r) = 2iGFρN (r) I N • γ
rank 1
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Hint (r) =Dodd parity
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X ≡α = 2Ψ(0) DΨD
(1)
Ψ(0) Ψ(0)
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Ψ(1) =Ω(n−1, 1) Φ0
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X ≡ 2Φ0
k= 0
m= k+1,2
∑ Ω(m−k−1, 0)+DΩ(k, 1) Φ0
Φ0k= 0
m= k+1,2
∑ Ω(m−k−1, 0)+Ω(k, 0) Φ0
MBPT(3):
Random Phase Approximation (RPA)
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H = H0 +Ves
= h(ri) + uDHF (ri) + V (rij )i≥ j∑
i∑
i∑ - uDHF (ri)
i∑
= f (0)(ri)i∑ + ves(rij )
i≥ j∑
Hartree-Fock Method:
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Φ0 ⇔ f (0) ϕ0 = ε(0) ϕ0
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Φ0(1) ⇔ ( f (0) −ε(0))ϕ0
(1) = (ε(1) − hint )ϕ0
RPA Method:
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ϕ0(1) = CI
(0, 1)
I∑ ϕ I
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Ψ(1) ⇔ CI(∞,1)
I∑ ( f (0) −ε(0))ϕ I = (ε(∞, 1) − hint )ϕ0
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ΨCP(1) =ΩI
(∞ , 0)
Φ0
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X ≡ 2Φ0 DΨCP
(1)
Φ0 Φ0
= Φ0 DΨCP(1)yielding
Coupled-cluster method (CCM)
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Ψ =ΩΦ0 = eT Φ0
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Ψ(0) = Ω(k, 0) Φ0k
∞
∑ = eT( 0)
Φ0€
Ψ ≅ Ψ(0) + Ψ(1) = Ω(k, 0) +Ω(k−1, 1)[ ]k∑ Φ0
CCM Method:
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T = T (0) + T (' )
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Ψ(1) = Ω(k−1, 1) Φ0k
∞
∑ = eT( 0)
T (1) Φ0
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X ≡ 2Ψ(0) DΨ(1)
Ψ(0) Ψ(0) = Φ0 eT( 0)+
DNeT ( 0)T (1)[ ]
connΦ0
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(H0 − E(0)) Ψ(1) = (E (1) −H int ) Ψ
(0)Equivalent to:
CCSD approximation:
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T = T1 + T2
Results for the Xe atom
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α XT−PT (×1020) XNSM (×1017) α XT−PT (×1020) XNSM (×1017)MBPT(1) 26.918 0.447 0.288 0.45 0.29 [2]MBPT(2) 23.388 0.405 0.266MBPT(3) 18.987 0.515 0.339 0.52 [3]
RPA 26.987 0.562 0.375 27.7 0.57, 0.564 0.38 [2,4]LCCSD 27.484 0.608 0.417CCSD 27.744 0.501 0.336
CCSDpT 27.782 0.501 0.337 [5]Experiment 27.815(27)
Method This work [1] Others Refs.
[1] Y. Singh, B. K. Sahoo and B. P. Das, Phys. Rev. A Rapid Comm (in process). [2] V. A. Dzuba, V. V. Flambaum and S. G. Porsev, Phys. Rev. A 80, 032120 (2009). [3] A. M. Maartensson-Pendrill, Phys. Rev. Lett. 54, 1153 (1985). [4] K. V. P. Latha and P. R. Amjith, Phys. Rev. A 87, 022509 (2013). [5] U. Hohm and K. Kerl, Mol. Phys. 69, 819 (1990).
Summary and Outlook
. Theory of EDMs for the closed-shell atoms are brie!y discussed.
. Relation between various many-body methods are given.
. Relativistic many-body methods to calculate atomic wave functions incorporating electromagnetic Coulomb interactions to all orders and P&T violating weak interaction to "rst order are developed.
. Polarizability and EDMs due to the nuclear Schiff momemt and the nucleus-electron T-PT interactions in Xe are reported.
. The methods are yet to be employed for other atomic systems and a bi-orthogonal RCC method is under development to study atomic EDMs.