b. k. sahoo - 国立大学法人...

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

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.

b. k. sahoo - 国立大学法人...

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